Network Working Group K. Moriarty, Ed.
InternetDraft EMC Corporation
Obsoletes: 2437 (if approved) B. Kaliski
Intended status: Informational Verisign
Expires: August 5, 2016 J. Jonsson
Subset AB
A. Rusch
RSA
February 2, 2016
PKCS #1 Version 2.2: RSA Cryptography Specifications
draftmoriartypkcs100
Abstract
This memo represents a republication of PKCS #1 v2.2 from RSA
Laboratories' PublicKey Cryptography Standards (PKCS) series. By
publishing this RFC, change control is transferred to the IETF.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Requirements Language . . . . . . . . . . . . . . . . . . 4
2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3. Key types . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1. RSA public key . . . . . . . . . . . . . . . . . . . . . 7
3.2. RSA private key . . . . . . . . . . . . . . . . . . . . . 8
4. Data conversion primitives . . . . . . . . . . . . . . . . . 9
4.1. I2OSP . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2. OS2IP . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5. Cryptographic primitives . . . . . . . . . . . . . . . . . . 11
5.1. Encryption and decryption primitives . . . . . . . . . . 11
5.1.1. RSAEP . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1.2. RSADP . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2. Signature and verification primitives . . . . . . . . . . 13
5.2.1. RSASP1 . . . . . . . . . . . . . . . . . . . . . . . 14
5.2.2. RSAVP1 . . . . . . . . . . . . . . . . . . . . . . . 15
6. Overview of schemes . . . . . . . . . . . . . . . . . . . . . 16
7. Encryption schemes . . . . . . . . . . . . . . . . . . . . . 17
7.1. RSAESOAEP . . . . . . . . . . . . . . . . . . . . . . . 17
7.1.1. Encryption operation . . . . . . . . . . . . . . . . 20
7.1.2. Decryption operation . . . . . . . . . . . . . . . . 23
7.2. RSAESPKCS1v1_5 . . . . . . . . . . . . . . . . . . . . 25
7.2.1. Encryption operation . . . . . . . . . . . . . . . . 26
7.2.2. Decryption Operation . . . . . . . . . . . . . . . . 28
8. Signature scheme with appendix . . . . . . . . . . . . . . . 29
8.1. RSASSAPSS . . . . . . . . . . . . . . . . . . . . . . . 30
8.1.1. Signature generation operation . . . . . . . . . . . 31
8.1.2. Signature verification operation . . . . . . . . . . 32
8.2. RSASSAPKCS1v1_5 . . . . . . . . . . . . . . . . . . . . 33
8.2.1. Signature generation operation . . . . . . . . . . . 35
8.2.2. Signature verification operation . . . . . . . . . . 36
9. Encoding methods for signatures with appendix . . . . . . . . 37
9.1. EMSAPSS . . . . . . . . . . . . . . . . . . . . . . . . 38
9.1.1. Encoding operation . . . . . . . . . . . . . . . . . 40
9.1.2. Verification operation . . . . . . . . . . . . . . . 41
9.2. EMSAPKCS1v1_5 . . . . . . . . . . . . . . . . . . . . . 43
10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 45
11. Security Considerations . . . . . . . . . . . . . . . . . . . 45
12. References . . . . . . . . . . . . . . . . . . . . . . . . . 45
12.1. Normative References . . . . . . . . . . . . . . . . . . 45
12.2. Informative References . . . . . . . . . . . . . . . . . 45
Appendix A. ASN.1 syntax . . . . . . . . . . . . . . . . . . . . 50
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A.1. RSA key representation . . . . . . . . . . . . . . . . . 50
A.1.1. RSA public key syntax . . . . . . . . . . . . . . . . 50
A.1.2. RSA private key syntax . . . . . . . . . . . . . . . 51
A.2. Scheme identification . . . . . . . . . . . . . . . . . . 52
A.2.1. RSAESOAEP . . . . . . . . . . . . . . . . . . . . . 53
A.2.2. RSAESPKCSv1_5 . . . . . . . . . . . . . . . . . . . 55
A.2.3. RSASSAPSS . . . . . . . . . . . . . . . . . . . . . 55
A.2.4. RSASSAPKCSv1_5 . . . . . . . . . . . . . . . . . . 57
Appendix B. Supporting techniques . . . . . . . . . . . . . . . 58
B.1. Hash functions . . . . . . . . . . . . . . . . . . . . . 58
B.2. Mask generation functions . . . . . . . . . . . . . . . . 62
B.2.1. MGF1 . . . . . . . . . . . . . . . . . . . . . . . . 62
Appendix C. ASN.1 module . . . . . . . . . . . . . . . . . . . . 63
Appendix D. Intellectual property considerations . . . . . . . . 71
Appendix E. Revision history . . . . . . . . . . . . . . . . . . 72
Appendix F. About PKCS . . . . . . . . . . . . . . . . . . . . . 73
Appendix G. Acknowledgements . . . . . . . . . . . . . . . . . . 73
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 73
1. Introduction
This document provides recommendations for the implementation of
publickey cryptography based on the RSA algorithm [RSA], covering
the following aspects:
o Cryptographic primitives
o Encryption schemes
o Signature schemes with appendix
o ASN.1 syntax for representing keys and for identifying the schemes
The recommendations are intended for general application within
computer and communications systems, and as such include a fair
amount of flexibility. It is expected that application standards
based on these specifications may include additional constraints.
The recommendations are intended to be compatible with the standards
IEEE 13632000 [IEEE1363], IEEE 1363a2004 [IEEE1363A], and ANSI
X9.442007 [ANSIX944].
This document supersedes PKCS #1 version 2.1 [PKCS1_21] but includes
compatible techniques.
The organization of this document is as follows:
o Section 1 is an introduction.
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o Section 2 defines some notation used in this document.
o Section 3 defines the RSA public and private key types.
o Sections 4 and 5 define several primitives, or basic mathematical
operations. Data conversion primitives are in Section 4, and
cryptographic primitives (encryptiondecryption, signature
verification) are in Section 5.
o Sections 6, 7, and 8 deal with the encryption and signature
schemes in this document. Section 6 gives an overview. Along
with the methods found in PKCS #1 v1.5, Section 7 defines an OAEP
based [OAEP] encryption scheme and Section 8 defines a PSSbased
[RSARABIN][PSS] signature scheme with appendix.
o Section 9 defines the encoding methods for the signature schemes
in Section 8.
o Appendix A defines the ASN.1 syntax for the keys defined in
Section 3 and the schemes in Sections 7 and 8.
o Appendix B defines the hash functions and the mask generation
function used in this document, including ASN.1 syntax for the
techniques.
o Appendix C gives an ASN.1 module.
o Appendices D, E, and F cover intellectual property issues, outline
the revision history of PKCS #1, and provide general information
about the PublicKey Cryptography Standards.
1.1. Requirements Language
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
2. Notation
The notation in this document includes:
c ciphertext representative, an integer between 0 and
n1
C ciphertext, an octet string
d RSA private exponent
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d_i additional factor r_i's CRT exponent, a positive
integer such that
e * d_i == 1 (mod (r_i1)), i = 3, ..., u
dP p's CRT exponent, a positive integer such that
e * dP == 1 (mod (p1))
dQ q's CRT exponent, a positive integer such that
e * dQ == 1 (mod (q1))
e RSA public exponent
EM encoded message, an octet string
emBits (intended) length in bits of an encoded message EM
emLen (intended) length in octets of an encoded message
EM
GCD(. , .) greatest common divisor of two nonnegative integers
Hash hash function
hLen output length in octets of hash function Hash
k length in octets of the RSA modulus n
K RSA private key
L optional RSAESOAEP label, an octet string
LCM(., ..., .) least common multiple of a list of nonnegative
integers
m message representative, an integer between 0 and
n1
M message, an octet string
mask MGF output, an octet string
maskLen (intended) length of the octet string mask
MGF mask generation function
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mgfSeed seed from which mask is generated, an octet string
mLen length in octets of a message M
n RSA modulus, n = r_1 * r_2 * ... * r_u , u >= 2
(n, e) RSA public key
p, q first two prime factors of the RSA modulus n
qInv CRT coefficient, a positive integer less than
p such that q * qInv == 1 (mod p)
r_i prime factors of the RSA modulus n, including
r_1 = p, r_2 = q, and additional factors if any
s signature representative, an integer between 0 and
n1
S signature, an octet string
sLen length in octets of the EMSAPSS salt
t_i additional prime factor r_i's CRT coefficient, a
positive integer less than r_i such that
r_1 * r_2 * ... * r_(i1) * t_i == 1 (mod r_i) ,
i = 3, ... , u
u number of prime factors of the RSA modulus, u >= 2
x a nonnegative integer
X an octet string corresponding to x
xLen (intended) length of the octet string X
0x indicator of hexadecimal representation of an octet
or an octet string; "0x48" denotes the octet with
hexadecimal value 48; "(0x)48 09 0e" denotes the
string of three consecutive octets with hexadecimal
value 48, 09, and 0e, respectively
\lambda(n) LCM(r_11, r_21, ... , r_u1)
\xor bitwise exclusiveor of two octet strings
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\ceil(.) ceiling function; \ceil(x) is the smallest integer
larger than or equal to the real number x
 concatenation operator
== congruence symbol; a == b (mod n) means that the
integer n divides the integer a  b
Note. The CRT can be applied in a nonrecursive as well as a
recursive way. In this document a recursive approach following
Garner's algorithm [GARNER] is used. See also Note 1 in Section 3.2.
3. Key types
Two key types are employed in the primitives and schemes defined in
this document: RSA public key and RSA private key. Together, an RSA
public key and an RSA private key form an RSA key pair.
This specification supports socalled "multiprime" RSA where the
modulus may have more than two prime factors. The benefit of multi
prime RSA is lower computational cost for the decryption and
signature primitives, provided that the CRT (Chinese Remainder
Theorem) is used. Better performance can be achieved on single
processor platforms, but to a greater extent on multiprocessor
platforms, where the modular exponentiations involved can be done in
parallel.
For a discussion on how multiprime affects the security of the RSA
cryptosystem, the reader is referred to [SILVERMAN].
3.1. RSA public key
For the purposes of this document, an RSA public key consists of two
components:
n the RSA modulus, a positive integer
e the RSA public exponent, a positive integer
In a valid RSA public key, the RSA modulus n is a product of u
distinct odd primes r_i, i = 1, 2, ..., u, where u >= 2, and the RSA
public exponent e is an integer between 3 and n  1 satisfying GCD(e,
\lambda(n)) = 1, where \lambda(n) = LCM(r_1  1, ..., r_u  1). By
convention, the first two primes r_1 and r_2 may also be denoted p
and q respectively.
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A recommended syntax for interchanging RSA public keys between
implementations is given in Appendix A.1.1; an implementation's
internal representation may differ.
3.2. RSA private key
For the purposes of this document, an RSA private key may have either
of two representations.
1. The first representation consists of the pair (n, d), where
the components have the following meanings:
n the RSA modulus, a positive integer
d the RSA private exponent, a positive integer
2. The second representation consists of a quintuple (p, q, dP,
dQ, qInv) and a (possibly empty) sequence of triplets (r_i, d_i,
t_i), i = 3, ..., u, one for each prime not in the quintuple,
where the components have the following meanings:
p the first factor, a positive integer
q the second factor, a positive integer
dP the first factor's CRT exponent, a positive integer
dQ the second factor's CRT exponent, a positive integer
qInv the (first) CRT coefficient, a positive integer
r_i the ith factor, a positive integer
d_i the ith factor's CRT exponent, a positive integer
t_i the ith factor's CRT coefficient, a positive integer
In a valid RSA private key with the first representation, the RSA
modulus n is the same as in the corresponding RSA public key and is
the product of u distinct odd primes r_i, i = 1, 2, ..., u, where u
>= 2. The RSA private exponent d is a positive integer less than n
satisfying
e * d == 1 (mod \lambda(n)),
where e is the corresponding RSA public exponent and \lambda(n) is
defined as in Section 3.1.
In a valid RSA private key with the second representation, the two
factors p and q are the first two prime factors of the RSA modulus n
(i.e., r_1 and r_2), the CRT exponents dP and dQ are positive
integers less than p and q respectively satisfying
e * dP == 1 (mod (p1))
e * dQ == 1 (mod (q1)) ,
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and the CRT coefficient qInv is a positive integer less than p
satisfying
q * qInv == 1 (mod p).
If u > 2, the representation will include one or more triplets (r_i,
d_i, t_i), i = 3, ..., u. The factors r_i are the additional prime
factors of the RSA modulus n. Each CRT exponent d_i (i = 3, ..., u)
satisfies
e * d_i == 1 (mod (r_i  1)).
Each CRT coefficient t_i (i = 3, ..., u) is a positive integer less
than r_i satisfying
R_i * t_i == 1 (mod r_i) ,
where R_i = r_1 * r_2 * ... * r_(i1).
A recommended syntax for interchanging RSA private keys between
implementations, which includes components from both representations,
is given in Appendix A.1.2; an implementation's internal
representation may differ.
Notes.
1. The definition of the CRT coefficients here and the formulas that
use them in the primitives in Section 5 generally follow Garner's
algorithm [GARNER] (see also Algorithm 14.71 in [HANDBOOK]).
However, for compatibility with the representations of RSA
private keys in PKCS #1 v2.0 and previous versions, the roles of
p and q are reversed compared to the rest of the primes. Thus,
the first CRT coefficient, qInv, is defined as the inverse of q
mod p, rather than as the inverse of R_1 mod r_2, i.e., of p mod
q.
2. Quisquater and Couvreur [FASTDEC] observed the benefit of
applying the Chinese Remainder Theorem to RSA operations.
4. Data conversion primitives
Two data conversion primitives are employed in the schemes defined in
this document:
o I2OSP  IntegertoOctetString primitive
o OS2IP  OctetStringtoInteger primitive
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For the purposes of this document, and consistent with ASN.1 syntax,
an octet string is an ordered sequence of octets (eightbit bytes).
The sequence is indexed from first (conventionally, leftmost) to last
(rightmost). For purposes of conversion to and from integers, the
first octet is considered the most significant in the following
conversion primitives.
4.1. I2OSP
I2OSP converts a nonnegative integer to an octet string of a
specified length.
I2OSP (x, xLen)
Input:
x nonnegative integer to be converted
xLen intended length of the resulting octet string
Output:
X corresponding octet string of length xLen
Error: "integer too large"
Steps:
1. If x >= 256^xLen, output "integer too large" and stop.
2. Write the integer x in its unique xLendigit representation in
base 256:
x = x_(xLen1) 256^(xLen1) + x_(xLen2) 256^(xLen2) + ...
+ x_1 256 + x_0,
where 0 <= x_i < 256 (note that one or more leading digits
will be zero if x is less than 256^(xLen1)).
3. Let the octet X_i have the integer value x_(xLeni) for 1 <= i
<= xLen. Output the octet string
X = X_1 X_2 ... X_xLen.
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4.2. OS2IP
OS2IP converts an octet string to a nonnegative integer.
OS2IP (X)
Input: X octet string to be converted
Output: x corresponding nonnegative integer
Steps:
1. Let X_1 X_2 ... X_xLen be the octets of X from first to last,
and let x_(xLeni) be the integer value of the octet X_i for 1
<= i <= xLen.
2. Let x = x_(xLen1) 256^(xLen1) + x_(xLen2) 256^(xLen2) +
... + x_1 256 + x_0.
3. Output x.
5. Cryptographic primitives
Cryptographic primitives are basic mathematical operations on which
cryptographic schemes can be built. They are intended for
implementation in hardware or as software modules, and are not
intended to provide security apart from a scheme.
Four types of primitive are specified in this document, organized in
pairs: encryption and decryption; and signature and verification.
The specifications of the primitives assume that certain conditions
are met by the inputs, in particular that RSA public and private keys
are valid.
5.1. Encryption and decryption primitives
An encryption primitive produces a ciphertext representative from a
message representative under the control of a public key, and a
decryption primitive recovers the message representative from the
ciphertext representative under the control of the corresponding
private key.
One pair of encryption and decryption primitives is employed in the
encryption schemes defined in this document and is specified here:
RSAEP/RSADP. RSAEP and RSADP involve the same mathematical
operation, with different keys as input. The primitives defined here
are the same as IFEPRSA/IFDPRSA in IEEE 13632000 [IEEE1363]
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(except that support for multiprime RSA has been added) and are
compatible with PKCS #1 v1.5.
The main mathematical operation in each primitive is exponentiation.
5.1.1. RSAEP
RSAEP ((n, e), m)
Input:
(n, e) RSA public key
m message representative, an integer between 0 and n  1
Output: c ciphertext representative, an integer between 0 and n  1
Error: "message representative out of range"
Assumption: RSA public key (n, e) is valid
Steps:
1. If the message representative m is not between 0 and n  1,
output "message representative out of range" and stop.
2. Let c = m^e mod n.
3. Output c.
5.1.2. RSADP
RSADP (K, c)
Input:
K RSA private key, where K has one of the following forms:
+ a pair (n, d)
+ a quintuple (p, q, dP, dQ, qInv) and a possibly empty
sequence of triplets (r_i, d_i, t_i), i = 3, ..., u
c ciphertext representative, an integer between 0 and n  1
Output: m message representative, an integer between 0 and n  1
Error: "ciphertext representative out of range"
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Assumption: RSA private key K is valid
Steps:
1. If the ciphertext representative c is not between 0 and n  1,
output "ciphertext representative out of range" and stop.
2. The message representative m is computed as follows.
a. If the first form (n, d) of K is used, let m = c^d mod n.
b. If the second form (p, q, dP, dQ, qInv) and (r_i, d_i,
t_i) of K is used, proceed as follows:
i. Let m_1 = c^dP mod p and m_2 = c^dQ mod q.
ii. If u > 2, let m_i = c^(d_i) mod r_i, i = 3, ...,
u.
iii. Let h = (m_1  m_2) * qInv mod p.
iv. Let m = m_2 + q * h.
v. If u > 2, let R = r_1 and for i = 3 to u do
1. Let R = R * r_(i1).
2. Let h = (m_i  m) * t_i mod r_i.
3. Let m = m + R * h.
3. Output m.
Note. Step 2.b can be rewritten as a single loop, provided that one
reverses the order of p and q. For consistency with PKCS #1 v2.0,
however, the first two primes p and q are treated separately from the
additional primes.
5.2. Signature and verification primitives
A signature primitive produces a signature representative from a
message representative under the control of a private key, and a
verification primitive recovers the message representative from the
signature representative under the control of the corresponding
public key. One pair of signature and verification primitives is
employed in the signature schemes defined in this document and is
specified here: RSASP1/RSAVP1.
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The primitives defined here are the same as IFSPRSA1/IFVPRSA1 in
IEEE 13632000 [IEEE1363] (except that support for multiprime RSA
has been added) and are compatible with PKCS #1 v1.5.
The main mathematical operation in each primitive is exponentiation,
as in the encryption and decryption primitives of Section 5.1.
RSASP1 and RSAVP1 are the same as RSADP and RSAEP except for the
names of their input and output arguments; they are distinguished as
they are intended for different purposes.
5.2.1. RSASP1
RSASP1 (K, m)
Input:
K RSA private key, where K has one of the following forms:
 a pair (n, d)
 a quintuple (p, q, dP, dQ, qInv) and a (possibly empty)
sequence of triplets (r_i, d_i, t_i), i = 3, ..., u
m message representative, an integer between 0 and n  1
Output:
s signature representative, an integer between 0 and n  1
Error: "message representative out of range"
Assumption: RSA private key K is valid
Steps:
1. If the message representative m is not between 0 and n  1,
output "message representative out of range" and stop.
2. The signature representative s is computed as follows.
a. If the first form (n, d) of K is used, let s = m^d mod n.
b. If the second form (p, q, dP, dQ, qInv) and (r_i, d_i,
t_i) of K is used, proceed as follows:
1. Let s_1 = m^dP mod p and s_2 = m^dQ mod q.
2. If u > 2, let s_i = m^(d_i) mod r_i, i = 3, ..., u.
3. Let h = (s_1  s_2) * qInv mod p.
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4. Let s = s_2 + q * h.
5. If u > 2, let R = r_1 and for i = 3 to u do
a. Let R = R * r_(i1).
b. Let h = (s_i  s) * t_i mod r_i.
c. Let s = s + R * h.
3. Output s.
Note. Step 2.b can be rewritten as a single loop, provided that one
reverses the order of p and q. For consistency with PKCS #1 v2.0,
however, the first two primes p and q are treated separately from the
additional primes.
5.2.2. RSAVP1
RSAVP1 ((n, e), s)
Input:
(n, e) RSA public key
s signature representative, an integer between 0 and n  1
Output:
m message representative, an integer between 0 and n  1
Error: "signature representative out of range"
Assumption: RSA public key (n, e) is valid
Steps:
1. If the signature representative s is not between 0 and n  1,
output "signature representative out of range" and stop.
2. Let m = s^e mod n.
3. Output m.
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6. Overview of schemes
A scheme combines cryptographic primitives and other techniques to
achieve a particular security goal. Two types of scheme are
specified in this document: encryption schemes and signature schemes
with appendix.
The schemes specified in this document are limited in scope in that
their operations consist only of steps to process data with an RSA
public or private key, and do not include steps for obtaining or
validating the key. Thus, in addition to the scheme operations, an
application will typically include key management operations by which
parties may select RSA public and private keys for a scheme
operation. The specific additional operations and other details are
outside the scope of this document.
As was the case for the cryptographic primitives (Section 5), the
specifications of scheme operations assume that certain conditions
are met by the inputs, in particular that RSA public and private keys
are valid. The behavior of an implementation is thus unspecified
when a key is invalid. The impact of such unspecified behavior
depends on the application. Possible means of addressing key
validation include explicit key validation by the application; key
validation within the publickey infrastructure; and assignment of
liability for operations performed with an invalid key to the party
who generated the key.
A generally good cryptographic practice is to employ a given RSA key
pair in only one scheme. This avoids the risk that vulnerability in
one scheme may compromise the security of the other, and may be
essential to maintain provable security. While RSAESPKCS1v1_5
(Section 7.2) and RSASSAPKCS1v1_5 (Section 8.2) have traditionally
been employed together without any known bad interactions (indeed,
this is the model introduced by PKCS #1 v1.5), such a combined use of
an RSA key pair is NOT RECOMMENDED for new applications.
To illustrate the risks related to the employment of an RSA key pair
in more than one scheme, suppose an RSA key pair is employed in both
RSAESOAEP (Section 7.1) and RSAESPKCS1v1_5. Although RSAESOAEP
by itself would resist attack, an opponent might be able to exploit a
weakness in the implementation of RSAESPKCS1v1_5 to recover
messages encrypted with either scheme. As another example, suppose
an RSA key pair is employed in both RSASSAPSS (Section 8.1) and
RSASSAPKCS1v1_5. Then the security proof for RSASSAPSS would no
longer be sufficient since the proof does not account for the
possibility that signatures might be generated with a second scheme.
Similar considerations may apply if an RSA key pair is employed in
one of the schemes defined here and in a variant defined elsewhere.
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7. Encryption schemes
For the purposes of this document, an encryption scheme consists of
an encryption operation and a decryption operation, where the
encryption operation produces a ciphertext from a message with a
recipient's RSA public key, and the decryption operation recovers the
message from the ciphertext with the recipient's corresponding RSA
private key.
An encryption scheme can be employed in a variety of applications. A
typical application is a key establishment protocol, where the
message contains key material to be delivered confidentially from one
party to another. For instance, PKCS #7 [RFC2315] employs such a
protocol to deliver a contentencryption key from a sender to a
recipient; the encryption schemes defined here would be suitable key
encryption algorithms in that context.
Two encryption schemes are specified in this document: RSAESOAEP and
RSAESPKCS1v1_5. RSAESOAEP is required to be supported for new
applications; RSAESPKCS1v1_5 is included only for compatibility
with existing applications.
The encryption schemes given here follow a general model similar to
that employed in IEEE 13632000 [IEEE1363], combining encryption and
decryption primitives with an encoding method for encryption. The
encryption operations apply a message encoding operation to a message
to produce an encoded message, which is then converted to an integer
message representative. An encryption primitive is applied to the
message representative to produce the ciphertext. Reversing this,
the decryption operations apply a decryption primitive to the
ciphertext to recover a message representative, which is then
converted to an octet string encoded message. A message decoding
operation is applied to the encoded message to recover the message
and verify the correctness of the decryption.
To avoid implementation weaknesses related to the way errors are
handled within the decoding operation (see [BLEICHENBACHER] and
[MANGER]), the encoding and decoding operations for RSAESOAEP and
RSAESPKCS1v1_5 are embedded in the specifications of the respective
encryption schemes rather than defined in separate specifications.
Both encryption schemes are compatible with the corresponding schemes
in PKCS #1 v2.1.
7.1. RSAESOAEP
RSAESOAEP combines the RSAEP and RSADP primitives (Sections 5.1.1
and 5.1.2) with the EMEOAEP encoding method (step 2 in Section 7.1.1
and step 3 in Section 7.1.2). EMEOAEP is based on Bellare and
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Rogaway's Optimal Asymmetric Encryption scheme [OAEP]. (OAEP stands
for "Optimal Asymmetric Encryption Padding"). It is compatible with
the IFES scheme defined in IEEE 13632000 [IEEE1363], where the
encryption and decryption primitives are IFEPRSA and IFDPRSA and
the message encoding method is EMEOAEP. RSAESOAEP can operate on
messages of length up to k  2hLen 2 octets, where hLen is the
length of the output from the underlying hash function and k is the
length in octets of the recipient'sRSA modulus.
Assuming that computing eth roots modulo n is infeasible and the
mask generation function in RSAESOAEP has appropriate properties,
RSAESOAEP is semantically secure against adaptive chosenciphertext
attacks. This assurance is provable in the sense that the difficulty
of breaking RSAESOAEP can be directly related to the difficulty of
inverting the RSA function, provided that the mask generation
function is viewed as a black box or random oracle; see [FOPS] and
the note below for further discussion.
Both the encryption and the decryption operations of RSAESOAEP take
the value of a label L as input. In this version of PKCS #1, L is
the empty string; other uses of the label are outside the scope of
this document. See Appendix A.2.1 for the relevant ASN.1 syntax.
RSAESOAEP is parameterized by the choice of hash function and mask
generation function. This choice should be fixed for a given RSA
key. Suggested hash and mask generation functions are given in
Appendix B.
Note. Past results have helpfully clarified the security properties
of the OAEP encoding method [OAEP] (roughly the procedure described
in step 2 in Section 7.1.1). The background is as follows. In 1994,
Bellare and Rogaway [OAEP] introduced a security concept that they
denoted plaintext awareness (PA94). They proved that if a
deterministic publickey encryption primitive (e.g., RSAEP) is hard
to invert without the private key, then the corresponding OAEPbased
encryption scheme is plaintextaware (in the random oracle model),
meaning roughly that an adversary cannot produce a valid ciphertext
without actually "knowing"the underlying plaintext. Plaintext
awareness of an encryption scheme is closely related to the
resistance of the scheme against chosenciphertext attacks. In such
attacks, an adversary is given the opportunity to send queries to an
oracle simulating the decryption primitive. Using the results of
these queries, the adversary attempts to decrypt a challenge
ciphertext.
However, there are two flavors of chosenciphertext attacks, and PA94
implies security against only one of them. The difference relies on
what the adversary is allowed to do after she is given the challenge
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ciphertext. The indifferent attack scenario (denoted CCA1) does not
admit any queries to the decryption oracle after the adversary is
given the challenge ciphertext, whereas the adaptive scenario
(denoted CCA2) does (except that the decryption oracle refuses to
decrypt the challenge ciphertext once it is published). In 1998,
Bellare and Rogaway, together with Desai and Pointcheval [PA98], came
up with a new, stronger notion of plaintext awareness (PA98) that
does imply security against CCA2.
To summarize, there have been two potential sources for
misconception: that PA94 and PA98 are equivalent concepts; or that
CCA1 and CCA2 are equivalent concepts. Either assumption leads to
the conclusion that the BellareRogaway paper implies security of
OAEP against CCA2, which it does not.
(Footnote: It might be fair to mention that PKCS #1 v2.0 cites [OAEP]
and claims that "a chosen ciphertext attack is ineffective against a
plaintextaware encryption scheme such as RSAESOAEP" without
specifying the kind of plaintext awareness or chosen ciphertext
attack considered.)
OAEP has never been proven secure against CCA2; in fact, Victor Shoup
[SHOUP] has demonstrated that such a proof does not exist in the
general case. Put briefly, Shoup showed that an adversary in the
CCA2 scenario who knows how to partially invert the encryption
primitive but does not know how to invert it completely may well be
able to break the scheme. For example, one may imagine an attacker
who is able to break RSAESOAEP if she knows how to recover all but
the first 20 bytes of a random integer encrypted with RSAEP. Such an
attacker does not need to be able to fully invert RSAEP, because she
does not use the first 20 octets in her attack.
Still, RSAESOAEP is secure against CCA2, which was proved by
Fujisaki, Okamoto, Pointcheval, and Stern [FOPS] shortly after the
announcement of Shoup's result. Using clever lattice reduction
techniques, they managed to show how to invert RSAEP completely given
a sufficiently large part of the preimage. This observation,
combined with a proof that OAEP is secure against CCA2 if the
underlying encryption primitive is hard to partially invert, fills
the gap between what Bellare and Rogaway proved about RSAESOAEP and
what some may have believed that they proved. Somewhat
paradoxically, we are hence saved by an ostensible weakness in RSAEP
(i.e., the whole inverse can be deduced from parts of it).
Unfortunately however, the security reduction is not efficient for
concrete parameters. While the proof successfully relates an
adversary A against the CCA2 security of RSAESOAEP to an algorithm I
inverting RSA, the probability of success for I is only approximately
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\epsilon^2 / 2^18, where \epsilon is the probability of success for
A.
(Footnote: In [FOPS] the probability of success for the inverter was
\epsilon^2 / 4. The additional factor 1 / 2^16 is due to the eight
fixed zero bits at the beginning of the encoded message EM, which are
not present in the variant of OAEP considered in [FOPS] (Inv must
apply Adv twice to invert RSA, and each application corresponds to a
factor 1 / 2^8).)
In addition, the running time for I is approximately t^2, where t is
the running time of the adversary. The consequence is that we cannot
exclude the possibility that attacking RSAESOAEP is considerably
easier than inverting RSA for concrete parameters. Still, the
existence of a security proof provides some assurance that the RSAES
OAEP construction is sounder than ad hoc constructions such as RSAES
PKCS1v1_5.
Hybrid encryption schemes based on the RSAKEM key encapsulation
paradigm offer tight proofs of security directly applicable to
concrete parameters; see [ISO18033] for discussion. Future versions
of PKCS #1 may specify schemes based on this paradigm.
7.1.1. Encryption operation
RSAESOAEPENCRYPT ((n, e), M, L)
Options:
Hash hash function (hLen denotes the length in octets of the hash
function output)
MGF mask generation function
Input:
(n, e) recipient's RSA public key (k denotes the length in octets
of the RSA modulus n)
M message to be encrypted, an octet string of length mLen,
where mLen <= k  2hLen  2
L optional label to be associated with the message; the
default value for L, if L is not provided, is the empty
string
Output:
C ciphertext, an octet string of length k
Errors: "message too long"; "label too long"
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Assumption: RSA public key (n, e) is valid
Steps:
1. Length checking:
a. If the length of L is greater than the input limitation
for the hash function (2^61  1 octets for SHA1), output
"label too long" and stop.
b. If mLen > k  2hLen  2, output "message too long" and
stop.
2. EMEOAEP encoding (see Figure 1 below):
a. If the label L is not provided, let L be the empty string.
Let lHash = Hash(L), an octet string of length hLen (see
the note below).
b. Generate an octet string PS consisting of k  mLen  2hLen
 2 zero octets. The length of PS may be zero.
c. Concatenate lHash, PS, a single octet with hexadecimal
value 0x01, and the message M to form a data block DB of
length k  hLen  1 octets as
DB = lHash  PS  0x01  M.
d. Generate a random octet string seed of length hLen.
e. Let dbMask = MGF(seed, k  hLen  1).
f. Let maskedDB = DB \xor dbMask.
g. Let seedMask = MGF(maskedDB, hLen).
h. Let maskedSeed = seed \xor seedMask.
i. Concatenate a single octet with hexadecimal value 0x00,
maskedSeed, and maskedDB to form an encoded message EM of
length k octets as
EM = 0x00  maskedSeed  maskedDB.
3. RSA encryption:
a. Convert the encoded message EM to an integer message
representative m (see Section 4.2):
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m = OS2IP (EM).
b. Apply the RSAEP encryption primitive (Section 5.1.1) to
the RSA public key (n, e) and the message representative m
to produce an integer ciphertext representative c:
c = RSAEP ((n, e), m).
c. Convert the ciphertext representative c to a ciphertext C
of length k octets (see Section 4.1):
C = I2OSP (c, k).
4. Output the ciphertext C.
_________________________________________________________________
+++++
DB =  lHash  PS 01 M 
+++++

++ 
 seed  
++ 
 
> MGF > xor
 
++ V 
00 xor < MGF <
++  
  
V V V
++++
EM = 00maskedSeed maskedDB 
++++
_________________________________________________________________
Figure 1: EMEOAEP encoding operation. lHash is the hash of the
optional label L. Decoding operation follows reverse steps to
recover M and verify lHash and PS.
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Note. If L is the empty string, the corresponding hash value lHash
has the following hexadecimal representation for different choices of
Hash:
SHA1: (0x)da39a3ee 5e6b4b0d 3255bfef 95601890 afd80709
SHA256: (0x)e3b0c442 98fc1c14 9afbf4c8 996fb924 27ae41e4 649b934c
a495991b 7852b855
SHA384: (0x)38b060a7 51ac9638 4cd9327e b1b1e36a 21fdb711 14be0743
4c0cc7bf 63f6e1da 274edebf e76f65fb d51ad2f1 4898b95b
SHA512: (0x)cf83e135 7eefb8bd f1542850 d66d8007 d620e405 0b5715dc
83f4a921 d36ce9ce 47d0d13c 5d85f2b0 ff8318d2 877eec2f
63b931bd 47417a81 a538327a f927da3e
7.1.2. Decryption operation
RSAESOAEPDECRYPT (K, C, L)
Options:
Hash hash function (hLen denotes the length in octets of the hash
function output)
MGF mask generation function
Input:
K recipient's RSA private key (k denotes the length in octets
of the RSA modulus n)
C ciphertext to be decrypted, an octet string of length k,
where k = 2hLen + 2
L optional label whose association with the message is to be
verified; the default value for L, if L is not provided, is
the empty string
Output:
M message, an octet string of length mLen, where mLen <= k 
2hLen  2
Error: "decryption error"
Steps:
1. Length checking:
a. If the length of L is greater than the input limitation
for the hash function (2^61  1 octets for SHA1), output
"decryption error" and stop.
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b. If the length of the ciphertext C is not k octets, output
"decryption error" and stop.
c. If k < 2hLen + 2, output "decryption error" and stop.
2. RSA decryption:
a. Convert the ciphertext C to an integer ciphertext
representative c (see Section 4.2):
c = OS2IP (C).
b. Apply the RSADP decryption primitive (Section 5.1.2) to
the RSA private key K and the ciphertext representative c
to produce an integer message representative m:
m = RSADP (K, c).
If RSADP outputs "ciphertext representative out of range"
(meaning that c >= n), output "decryption error" and stop.
c. Convert the message representative m to an encoded message
EM of length k octets (see Section 4.1):
EM = I2OSP (m, k).
3. EMEOAEP decoding:
a. If the label L is not provided, let L be the empty string.
Let lHash = Hash(L), an octet string of length hLen (see
the note in Section 7.1.1).
b. Separate the encoded message EM into a single octet Y, an
octet string maskedSeed of length hLen, and an octet
string maskedDB of length k  hLen  1 as
EM = Y  maskedSeed  maskedDB.
c. Let seedMask = MGF(maskedDB, hLen).
d. Let seed = maskedSeed \xor seedMask.
e. Let dbMask = MGF(seed, k  hLen  1).
f. Let DB = maskedDB \xor dbMask.
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g. Separate DB into an octet string lHash' of length hLen, a
(possibly empty) padding string PS consisting of octets
with hexadecimal value 0x00, and a message M as
DB = lHash'  PS  0x01  M.
If there is no octet with hexadecimal value 0x01 to
separate PS from M, if lHash does not equal lHash', or if
Y is nonzero, output "decryption error" and stop. (See
the note below.)
4. Output the message M.
Note. Care must be taken to ensure that an opponent cannot
distinguish the different error conditions in Step 3.g, whether by
error message or timing, or, more generally, learn partial
information about the encoded message EM. Otherwise an opponent
may be able to obtain useful information about the decryption of
the ciphertext C, leading to a chosenciphertext attack such as
the one observed by Manger [MANGER].
7.2. RSAESPKCS1v1_5
RSAESPKCS1v1_5 combines the RSAEP and RSADP primitives (Sections
5.1.1 and 5.1.2) with the EMEPKCS1v1_5 encoding method (step 1 in
Section 7.2.1 and step 3 in Section 7.2.2). It is mathematically
equivalent to the encryption scheme in PKCS #1 v1.5. RSAESPKCS1
v1_5 can operate on messages of length up to k  11 octets (k is the
octet length of the RSA modulus), although care should be taken to
avoid certain attacks on lowexponent RSA due to Coppersmith,
Franklin, Patarin, and Reiter when long messages are encrypted (see
the third bullet in the notes below and [LOWEXP]; [NEWATTACK]
contains an improved attack). As a general rule, the use of this
scheme for encrypting an arbitrary message, as opposed to a randomly
generated key, is NOT RECOMMENDED.
It is possible to generate valid RSAESPKCS1v1_5 ciphertexts without
knowing the corresponding plaintexts, with a reasonable probability
of success. This ability can be exploited in a chosen ciphertext
attack as shown in [BLEICHENBACHER]. Therefore, if RSAESPKCS1v1_5
is to be used, certain easily implemented countermeasures should be
taken to thwart the attack found in [BLEICHENBACHER]. Typical
examples include the addition of structure to the data to be encoded,
rigorous checking of PKCS #1 v1.5 conformance (and other redundancy)
in decrypted messages, and the consolidation of error messages in a
clientserver protocol based on PKCS #1 v1.5. These can all be
effective countermeasures and do not involve changes to a PKCS #1
v1.5based protocol. See [BKS] for a further discussion of these and
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other countermeasures. It has recently been shown that the security
of the SSL/TLS handshake protocol [RFC5246], which uses RSAES
PKCS1v1_5 and certain countermeasures, can be related to a variant
of the RSA problem; see [RSATLS] for discussion.
Note. The following passages describe some security recommendations
pertaining to the use of RSAESPKCS1v1_5. Recommendations from
version 1.5 of this document are included as well as new
recommendations motivated by cryptanalytic advances made in the
intervening years.
o It is RECOMMENDED that the pseudorandom octets in step 2 in
Section 7.2.1 be generated independently for each encryption
process, especially if the same data is input to more than one
encryption process. Haastad's results [HAASTAD] are one
motivation for this recommendation.
o The padding string PS in step 2 in Section 7.2.1 is at least eight
octets long, which is a security condition for publickey
operations that makes it difficult for an attacker to recover data
by trying all possible encryption blocks.
o The pseudorandom octets can also help thwart an attack due to
Coppersmith et al. [LOWEXP] (see [NEWATTACK] for an improvement
of the attack) when the size of the message to be encrypted is
kept small. The attack works on lowexponent RSA when similar
messages are encrypted with the same RSA public key. More
specifically, in one flavor of the attack, when two inputs to
RSAEP agree on a large fraction of bits (8/9) and lowexponent RSA
(e = 3) is used to encrypt both of them, it may be possible to
recover both inputs with the attack. Another flavor of the attack
is successful in decrypting a single ciphertext when a large
fraction (2/3) of the input to RSAEP is already known. For
typical applications, the message to be encrypted is short (e.g.,
a 128bit symmetric key) so not enough information will be known
or common between two messages to enable the attack. However, if
a long message is encrypted, or if part of a message is known,
then the attack may be a concern. In any case, the RSAESOAEP
scheme overcomes the attack.
7.2.1. Encryption operation
RSAESPKCS1V1_5ENCRYPT ((n, e), M)
Input:
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(n, e) recipient's RSA public key (k denotes the length in octets
of the modulus n)
M message to be encrypted, an octet string of length mLen,
where mLen <= k  11
Output:
C ciphertext, an octet string of length k
Error: "message too long"
Steps:
1. Length checking: If mLen > k  11, output "message too long"
and stop.
2. EMEPKCS1v1_5 encoding:
a. Generate an octet string PS of length k  mLen  3
consisting of pseudorandomly generated nonzero octets.
The length of PS will be at least eight octets.
b. Concatenate PS, the message M, and other padding to form
an encoded message EM of length k octets as
EM = 0x00  0x02  PS  0x00  M.
3. RSA encryption:
a. Convert the encoded message EM to an integer message
representative m (see Section 4.2):
m = OS2IP (EM).
b. Apply the RSAEP encryption primitive (Section 5.1.1) to
the RSA public key (n, e) and the message representative m
to produce an integer ciphertext representative c:
c = RSAEP ((n, e), m).
c. Convert the ciphertext representative c to a ciphertext C
of length k octets (see Section 4.1):
C = I2OSP (c, k).
4. Output the ciphertext C.
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7.2.2. Decryption Operation
RSAESPKCS1V1_5DECRYPT (K, C)
Input:
K recipient's RSA private key
C ciphertext to be decrypted, an octet string of length k,
where k is the length in octets of the RSA modulus n
Output:
M message, an octet string of length at most k  11
Error: "decryption error"
Steps:
1. Length checking: If the length of the ciphertext C is not k
octets (or if k < 11), output "decryption error" and stop.
2. RSA decryption:
a. Convert the ciphertext C to an integer ciphertext
representative c (see Section 4.2):
c = OS2IP (C).
b. Apply the RSADP decryption primitive (Section 5.1.2) to
the RSA private key (n, d) and the ciphertext
representative c to produce an integer message
representative m:
m = RSADP ((n, d), c).
If RSADP outputs "ciphertext representative out of range"
(meaning that c >= n), output "decryption error" and stop.
c. Convert the message representative m to an encoded message
EM of length k octets (see Section 4.1):
EM = I2OSP (m, k).
3. EMEPKCS1v1_5 decoding: Separate the encoded message EM into
an octet string PS consisting of nonzero octets and a message
M as
EM = 0x00  0x02  PS  0x00  M.
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If the first octet of EM does not have hexadecimal value 0x00,
if the second octet of EM does not have hexadecimal value
0x02, if there is no octet with hexadecimal value 0x00 to
separate PS from M, or if the length of PS is less than 8
octets, output "decryption error" and stop. (See the note
below.)
4. Output M.
Note. Care shall be taken to ensure that an opponent cannot
distinguish the different error conditions in Step 3, whether by
error message or timing. Otherwise an opponent may be able to
obtain useful information about the decryption of the ciphertext
C, leading to a strengthened version of Bleichenbacher's attack
[BLEICHENBACHER]; compare to Manger's attack [MANGER].
8. Signature scheme with appendix
For the purposes of this document, a signature scheme with appendix
consists of a signature generation operation and a signature
verification operation, where the signature generation operation
produces a signature from a message with a signer's RSA private key,
and the signature verification operation verifies the signature on
the message with the signer's corresponding RSA public key. To
verify a signature constructed with this type of scheme it is
necessary to have the message itself. In this way, signature schemes
with appendix are distinguished from signature schemes with message
recovery, which are not supported in this document.
A signature scheme with appendix can be employed in a variety of
applications. For instance, the signature schemes with appendix
defined here would be suitable signature algorithms for X.509
certificates [ISO9594]. Related signature schemes could be employed
in PKCS #7 [RFC2315], although for technical reasons the current
version of PKCS #7 separates a hash function from a signature scheme,
which is different than what is done here; see the note in
Appendix A.2.3 for more discussion.
Two signature schemes with appendix are specified in this document:
RSASSAPSS and RSASSAPKCS1v1_5. Although no attacks are known
against RSASSAPKCS1v1_5, in the interest of increased robustness,
RSASSAPSS is required in new applications. RSASSAPKCS1v1_5 is
included only for compatibility with existing applications.
The signature schemes with appendix given here follow a general model
similar to that employed in IEEE 13632000 [IEEE1363], combining
signature and verification primitives with an encoding method for
signatures. The signature generation operations apply a message
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encoding operation to a message to produce an encoded message, which
is then converted to an integer message representative. A signature
primitive is applied to the message representative to produce the
signature. Reversing this, the signature verification operations
apply a signature verification primitive to the signature to recover
a message representative, which is then converted to an octet string
encoded message. A verification operation is applied to the message
and the encoded message to determine whether they are consistent.
If the encoding method is deterministic (e.g., EMSAPKCS1v1_5), the
verification operation may apply the message encoding operation to
the message and compare the resulting encoded message to the
previously derived encoded message. If there is a match, the
signature is considered valid. If the method is randomized (e.g.,
EMSAPSS), the verification operation is typically more complicated.
For example, the verification operation in EMSAPSS extracts the
random salt and a hash output from the encoded message and checks
whether the hash output, the salt, and the message are consistent;
the hash output is a deterministic function in terms of the message
and the salt. For both signature schemes with appendix defined in
this document, the signature generation and signature verification
operations are readily implemented as "singlepass" operations if the
signature is placed after the message. See PKCS #7 [RFC2315] for an
example format in the case of RSASSAPKCS1v1_5.
8.1. RSASSAPSS
RSASSAPSS combines the RSASP1 and RSAVP1 primitives with the EMSA
PSS encoding method. It is compatible with the IFSSA scheme as
amended in the IEEE 1363a2004 [IEEE1363A], where the signature and
verification primitives are IFSPRSA1 and IFVPRSA1 as defined in
IEEE 13632000 [IEEE1363] and the message encoding method is EMSA4.
EMSA4 is slightly more general than EMSAPSS as it acts on bit
strings rather than on octet strings. EMSAPSS is equivalent to
EMSA4 restricted to the case that the operands as well as the hash
and salt values are octet strings.
The length of messages on which RSASSAPSS can operate is either
unrestricted or constrained by a very large number, depending on the
hash function underlying the EMSAPSS encoding method.
Assuming that computing eth roots modulo n is infeasible and the
hash and mask generation functions in EMSAPSS have appropriate
properties, RSASSAPSS provides secure signatures. This assurance is
provable in the sense that the difficulty of forging signatures can
be directly related to the difficulty of inverting the RSA function,
provided that the hash and mask generation functions are viewed as
black boxes or random oracles. The bounds in the security proof are
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essentially "tight", meaning that the success probability and running
time for the best forger against RSASSAPSS are very close to the
corresponding parameters for the best RSA inversion algorithm; see
[RSARABIN][PSSPROOF][JONSSON] for further discussion.
In contrast to the RSASSAPKCS1v1_5 signature scheme, a hash
function identifier is not embedded in the EMSAPSS encoded message,
so in theory it is possible for an adversary to substitute a
different (and potentially weaker) hash function than the one
selected by the signer. Therefore, it is RECOMMENDED that the EMSA
PSS mask generation function be based on the same hash function. In
this manner the entire encoded message will be dependent on the hash
function and it will be difficult for an opponent to substitute a
different hash function than the one intended by the signer. This
matching of hash functions is only for the purpose of preventing hash
function substitution, and is not necessary if hash function
substitution is addressed by other means (e.g., the verifier accepts
only a designated hash function). See [HASHID] for further
discussion of these points. The provable security of RSASSAPSS does
not rely on the hash function in the mask generation function being
the same as the hash function applied to the message.
RSASSAPSS is different from other RSAbased signature schemes in
that it is probabilistic rather than deterministic, incorporating a
randomly generated salt value. The salt value enhances the security
of the scheme by affording a "tighter" security proof than
deterministic alternatives such as Full Domain Hashing (FDH); see
[RSARABIN] for discussion. However, the randomness is not critical
to security. In situations where random generation is not possible,
a fixed value or a sequence number could be employed instead, with
the resulting provable security similar to that of FDH [FDH].
8.1.1. Signature generation operation
RSASSAPSSSIGN (K, M)
Input:
K signer's RSA private key
M message to be signed, an octet string
Output:
S signature, an octet string of length k, where k is the
length in octets of the RSA modulus n
Errors: "message too long;" "encoding error"
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Steps:
1. EMSAPSS encoding: Apply the EMSAPSS encoding operation
(Section 9.1.1) to the message M to produce an encoded message
EM of length \ceil ((modBits  1)/8) octets such that the bit
length of the integer OS2IP (EM) (see Section 4.2) is at most
modBits  1, where modBits is the length in bits of the RSA
modulus n:
EM = EMSAPSSENCODE (M, modBits  1).
Note that the octet length of EM will be one less than k if
modBits  1 is divisible by 8 and equal to k otherwise. If
the encoding operation outputs "message too long," output
"message too long" and stop. If the encoding operation
outputs "encoding error," output "encoding error" and stop.
2. RSA signature:
a. Convert the encoded message EM to an integer message
representative m (see Section 4.2):
m = OS2IP (EM).
b. Apply the RSASP1 signature primitive (Section 5.2.1) to
the RSA private key K and the message representative m to
produce an integer signature representative s:
s = RSASP1 (K, m).
c. Convert the signature representative s to a signature S of
length k octets (see Section 4.1):
S = I2OSP (s, k).
3. Output the signature S.
8.1.2. Signature verification operation
RSASSAPSSVERIFY ((n, e), M, S)
Input:
(n, e) signer's RSA public key
M message whose signature is to be verified, an octet string
S signature to be verified, an octet string of length k, where
k is the length in octets of the RSA modulus n
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Output: "valid signature" or "invalid signature"
Steps:
1. Length checking: If the length of the signature S is not k
octets, output "invalid signature" and stop.
2. RSA verification:
a. Convert the signature S to an integer signature
representative s (see Section 4.2):
s = OS2IP (S).
b. Apply the RSAVP1 verification primitive (Section 5.2.2) to
the RSA public key (n, e) and the signature representative
s to produce an integer message representative m:
m = RSAVP1 ((n, e), s).
If RSAVP1 output "signature representative out of range,"
output "invalid signature" and stop.
c. Convert the message representative m to an encoded message
EM of length emLen = \ceil ((modBits  1)/8) octets, where
modBits is the length in bits of the RSA modulus n (see
Section 4.1):
EM = I2OSP (m, emLen).
Note that emLen will be one less than k if modBits  1 is
divisible by 8 and equal to k otherwise. If I2OSP outputs
"integer too large," output "invalid signature" and stop.
3. EMSAPSS verification: Apply the EMSAPSS verification
operation (Section 9.1.2) to the message M and the encoded
message EM to determine whether they are consistent:
Result = EMSAPSSVERIFY (M, EM, modBits  1).
4. If Result = "consistent," output "valid signature."
Otherwise, output "invalid signature."
8.2. RSASSAPKCS1v1_5
RSASSAPKCS1v1_5 combines the RSASP1 and RSAVP1 primitives with the
EMSAPKCS1v1_5 encoding method. It is compatible with the IFSSA
scheme defined in IEEE 13632000 [IEEE1363], where the signature and
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verification primitives are IFSPRSA1 and IFVPRSA1 and the message
encoding method is EMSAPKCS1v1_5 (which is not defined in IEEE
13632000, but is in the IEEE 1363a2004 [IEEE1363A]).
The length of messages on which RSASSAPKCS1v1_5 can operate is
either unrestricted or constrained by a very large number, depending
on the hash function underlying the EMSAPKCS1v1_5 method.
Assuming that computing eth roots modulo n is infeasible and the
hash function in EMSAPKCS1v1_5 has appropriate properties, RSASSA
PKCS1v1_5 is conjectured to provide secure signatures. More
precisely, forging signatures without knowing the RSA private key is
conjectured to be computationally infeasible. Also, in the encoding
method EMSAPKCS1v1_5, a hash function identifier is embedded in the
encoding. Because of this feature, an adversary trying to find a
message with the same signature as a previously signed message must
find collisions of the particular hash function being used; attacking
a different hash function than the one selected by the signer is not
useful to the adversary. See [HASHID] for further discussion.
Note: As noted in PKCS #1 v1.5, the EMSAPKCS1v1_5 encoding method
has the property that the encoded message, converted to an integer
message representative, is guaranteed to be large and at least
somewhat "random". This prevents attacks of the kind proposed by
Desmedt and Odlyzko [CHOSEN] where multiplicative relationships
between message representatives are developed by factoring the
message representatives into a set of small values (e.g., a set of
small primes). Coron, Naccache, and Stern [PADDING] showed that a
stronger form of this type of attack could be quite effective against
some instances of the ISO/IEC 97962 signature scheme. They also
analyzed the complexity of this type of attack against the EMSA
PKCS1v1_5 encoding method and concluded that an attack would be
impractical, requiring more operations than a collision search on the
underlying hash function (i.e., more than 2^80 operations).
Coppersmith, Halevi, and Jutla [FORGERY] subsequently extended Coron
et al.'s attack to break the ISO/IEC 97961 signature scheme with
message recovery. The various attacks illustrate the importance of
carefully constructing the input to the RSA signature primitive,
particularly in a signature scheme with message recovery.
Accordingly, the EMSAPKCSv1_5 encoding method explicitly includes a
hash operation and is not intended for signature schemes with message
recovery. Moreover, while no attack is known against the EMSAPKCS
v1_5 encoding method, a gradual transition to EMSAPSS is recommended
as a precaution against future developments.
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8.2.1. Signature generation operation
RSASSAPKCS1V1_5SIGN (K, M)
Input:
K signer's RSA private key
M message to be signed, an octet string
Output:
S signature, an octet string of length k, where k is the
length in octets of the RSA modulus n
Errors: "message too long"; "RSA modulus too short"
Steps:
1. EMSAPKCS1v1_5 encoding: Apply the EMSAPKCS1v1_5 encoding
operation (Section 9.2) to the message M to produce an encoded
message EM of length k octets:
EM = EMSAPKCS1V1_5ENCODE (M, k).
If the encoding operation outputs "message too long," output
"message too long" and stop. If the encoding operation
outputs "intended encoded message length too short," output
"RSA modulus too short" and stop.
2. RSA signature:
a. Convert the encoded message EM to an integer message
representative m (see Section 4.2):
m = OS2IP (EM).
b. Apply the RSASP1 signature primitive (Section 5.2.1) to
the RSA private key K and the message representative m to
produce an integer signature representative s:
s = RSASP1 (K, m).
c. Convert the signature representative s to a signature S of
length k octets (see Section 4.1):
S = I2OSP (s, k).
3. Output the signature S.
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8.2.2. Signature verification operation
RSASSAPKCS1V1_5VERIFY ((n, e), M, S)
Input:
(n, e) signer's RSA public key
M message whose signature is to be verified, an octet string
S signature to be verified, an octet string of length k, where
k is the length in octets of the RSA modulus n
Output "valid signature" or "invalid signature"
Errors: "message too long"; "RSA modulus too short"
Steps:
1. Length checking: If the length of the signature S is not k
octets, output "invalid signature" and stop.
2. RSA verification:
a. Convert the signature S to an integer signature
representative s (see Section 4.2):
s = OS2IP (S).
b. Apply the RSAVP1 verification primitive (Section 5.2.2) to
the RSA public key (n, e) and the signature representative
s to produce an integer message representative m:
m = RSAVP1 ((n, e), s).
If RSAVP1 outputs "signature representative out of range,"
output "invalid signature" and stop.
c. Convert the message representative m to an encoded message
EM of length k octets (see Section 4.1):
EM' = I2OSP (m, k).
If I2OSP outputs "integer too large," output "invalid
signature" and stop.
3. EMSAPKCS1v1_5 encoding: Apply the EMSAPKCS1v1_5 encoding
operation (Section 9.2) to the message M to produce a second
encoded message EM' of length k octets:
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EM' = EMSAPKCS1V1_5ENCODE (M, k).
If the encoding operation outputs "message too long," output
"message too long" and stop. If the encoding operation
outputs "intended encoded message length too short," output
"RSA modulus too short" and stop.
4. Compare the encoded message EM and the second encoded message
EM'. If they are the same, output "valid signature";
otherwise, output "invalid signature."
Note. Another way to implement the signature verification
operation is to apply a "decoding" operation (not specified in
this document) to the encoded message to recover the underlying
hash value, and then to compare it to a newly computed hash value.
This has the advantage that it requires less intermediate storage
(two hash values rather than two encoded messages), but the
disadvantage that it requires additional code.
9. Encoding methods for signatures with appendix
Encoding methods consist of operations that map between octet string
messages and octet string encoded messages, which are converted to
and from integer message representatives in the schemes. The integer
message representatives are processed via the primitives. The
encoding methods thus provide the connection between the schemes,
which process messages, and the primitives.
An encoding method for signatures with appendix, for the purposes of
this document, consists of an encoding operation and optionally a
verification operation. An encoding operation maps a message M to an
encoded message EM of a specified length. A verification operation
determines whether a message M and an encoded message EM are
consistent, i.e., whether the encoded message EM is a valid encoding
of the message M.
The encoding operation may introduce some randomness, so that
different applications of the encoding operation to the same message
will produce different encoded messages, which has benefits for
provable security. For such an encoding method, both an encoding and
a verification operation are needed unless the verifier can reproduce
the randomness (e.g., by obtaining the salt value from the signer).
For a deterministic encoding method only an encoding operation is
needed.
Two encoding methods for signatures with appendix are employed in the
signature schemes and are specified here: EMSAPSS and EMSA
PKCS1v1_5.
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9.1. EMSAPSS
This encoding method is parameterized by the choice of hash function,
mask generation function, and salt length. These options should be
fixed for a given RSA key, except that the salt length can be
variable (see [JONSSON] for discussion). Suggested hash and mask
generation functions are given in Appendix B. The encoding method is
based on Bellare and Rogaway's Probabilistic Signature Scheme (PSS)
[RSARABIN][PSS]. It is randomized and has an encoding operation and
a verification operation.
Figure 2 illustrates the encoding operation.
__________________________________________________________________
++
 M 
++

V
Hash

V
++++
M' = Padding1 mHash  salt 
++++

+++ V
DB = Padding2 salt  Hash
+++ 
 
V 
xor < MGF <
 
 
V V
++++
EM =  maskedDB  H bc
++++
__________________________________________________________________
Figure 2: EMSAPSS encoding operation. Verification operation
follows reverse steps to recover salt, then forward steps to
recompute and compare H.
Notes.
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1. The encoding method defined here differs from the one in Bellare
and Rogaway's submission to IEEE 1363a2004 [PSS] in three
respects:
* It applies a hash function rather than a mask generation
function to the message. Even though the mask generation
function is based on a hash function, it seems more natural to
apply a hash function directly.
* The value that is hashed together with the salt value is the
string (0x)00 00 00 00 00 00 00 00  mHash rather than the
message M itself. Here, mHash is the hash of M. Note that
the hash function is the same in both steps. See Note 3 below
for further discussion. (Also, the name "salt" is used
instead of "seed", as it is more reflective of the value's
role.)
* The encoded message in EMSAPSS has nine fixed bits; the first
bit is 0 and the last eight bits form a "trailer field", the
octet 0xbc. In the original scheme, only the first bit is
fixed. The rationale for the trailer field is for
compatibility with the RabinWilliams IFSPRW signature
primitive in IEEE 13632000 [IEEE1363] and the corresponding
primitive in ISO/IEC 97962:2010 [ISO9796].
2. Assuming that the mask generation function is based on a hash
function, it is RECOMMENDED that the hash function be the same as
the one that is applied to the message; see Section 8.1 for
further discussion.
3. Without compromising the security proof for RSASSAPSS, one may
perform steps 1 and 2 of EMSAPSSENCODE and EMSAPSSVERIFY (the
application of the hash function to the message) outside the
module that computes the rest of the signature operation, so that
mHash rather than the message M itself is input to the module.
In other words, the security proof for RSASSAPSS still holds
even if an opponent can control the value of mHash. This is
convenient if the module has limited I/O bandwidth, e.g., a smart
card. Note that previous versions of PSS [RSARABIN][PSS] did not
have this property. Of course, it may be desirable for other
security reasons to have the module process the full message.
For instance, the module may need to "see" what it is signing if
it does not trust the component that computes the hash value.
4. Typical salt lengths in octets are hLen (the length of the output
of the hash function Hash) and 0. In both cases the security of
RSASSAPSS can be closely related to the hardness of inverting
RSAVP1. Bellare and Rogaway [RSARABIN] give a tight lower bound
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for the security of the original RSAPSS scheme, which
corresponds roughly to the former case, while Coron [FDH] gives a
lower bound for the related Full Domain Hashing scheme, which
corresponds roughly to the latter case. In [PSSPROOF] Coron
provides a general treatment with various salt lengths ranging
from 0 to hLen; see [IEEE1363A] for discussion. See also
[JONSSON], which adapts the security proofs in
[RSARABIN][PSSPROOF] to address the differences between the
original and the present version of RSAPSS as listed in Note 1
above.
5. As noted in IEEE 1363a2004 [IEEE1363A], the use of randomization
in signature schemes  such as the salt value in EMSAPSS  may
provide a "covert channel" for transmitting information other
than the message being signed. For more on covert channels, see
[SIMMONS].
9.1.1. Encoding operation
EMSAPSSENCODE (M, emBits)
Options:
Hash hash function (hLen denotes the length in octets of the hash
function output)
MGF mask generation function
sLen intended length in octets of the salt
Input:
M message to be encoded, an octet string
emBits maximal bit length of the integer OS2IP (EM) (see Section
4.2), at least 8hLen + 8sLen + 9
Output:
EM encoded message, an octet string of length emLen = \ceil
(emBits/8)
Errors: "Encoding error"; "message too long"
Steps:
1. If the length of M is greater than the input limitation for
the hash function (2^61  1 octets for SHA1), output
"message too long" and stop.
2. Let mHash = Hash(M), an octet string of length hLen.
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3. If emLen < hLen + sLen + 2, output "encoding error" and stop.
4. Generate a random octet string salt of length sLen; if sLen =
0, then salt is the empty string.
5. Let
M' = (0x)00 00 00 00 00 00 00 00  mHash  salt;
M' is an octet string of length 8 + hLen + sLen with eight
initial zero octets.
6. Let H = Hash(M'), an octet string of length hLen.
7. Generate an octet string PS consisting of emLen  sLen  hLen
 2 zero octets. The length of PS may be 0.
8. Let DB = PS  0x01  salt; DB is an octet string of length
emLen  hLen  1.
9. Let dbMask = MGF(H, emLen  hLen  1).
10. Let maskedDB = DB \xor dbMask.
11. Set the leftmost 8emLen  emBits bits of the leftmost octet
in maskedDB to zero.
12. Let EM = maskedDB  H  0xbc.
13. Output EM.
9.1.2. Verification operation
EMSAPSSVERIFY (M, EM, emBits)
Options:
Hash hash function (hLen denotes the length in octets of the hash
function output)
MGF mask generation function
sLen intended length in octets of the salt
Input:
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M message to be verified, an octet string
EM encoded message, an octet string of length emLen = \ceil
(emBits/8)
emBits maximal bit length of the integer OS2IP (EM) (see Section
4.2), at least 8hLen + 8sLen + 9
Output: "consistent" or "inconsistent"
Steps:
1. If the length of M is greater than the input limitation for
the hash function (2^61  1 octets for SHA1), output
"inconsistent" and stop.
2. Let mHash = Hash(M), an octet string of length hLen.
3. If emLen < hLen + sLen + 2, output "inconsistent" and stop.
4. If the rightmost octet of EM does not have hexadecimal value
0xbc, output "inconsistent" and stop.
5. Let maskedDB be the leftmost emLen  hLen  1 octets of EM,
and let H be the next hLen octets.
6. If the leftmost 8emLen  emBits bits of the leftmost octet in
maskedDB are not all equal to zero, output "inconsistent" and
stop.
7. Let dbMask = MGF(H, emLen  hLen  1).
8. Let DB = maskedDB \xor dbMask.
9. Set the leftmost 8emLen  emBits bits of the leftmost octet
in DB to zero.
10. If the emLen  hLen  sLen  2 leftmost octets of DB are not
zero or if the octet at position emLen  hLen  sLen  1 (the
leftmost position is "position 1") does not have hexadecimal
value 0x01, output "inconsistent" and stop.
11. Let salt be the last sLen octets of DB.
12. Let
M' = (0x)00 00 00 00 00 00 00 00  mHash  salt ;
M' is an octet string of length 8 + hLen + sLen with eight
initial zero octets.
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13. Let H' = Hash(M'), an octet string of length hLen.
14. If H = H', output "consistent." Otherwise, output
"inconsistent."
9.2. EMSAPKCS1v1_5
This encoding method is deterministic and only has an encoding
operation.
EMSAPKCS1v1_5ENCODE (M, emLen)
Option:
Hash hash function (hLen denotes the length in octets of the hash
function output)
Input:
M message to be encoded
emLen intended length in octets of the encoded message, at least
tLen + 11, where tLen is the octet length of the DER
encoding T of a certain value computed during the encoding
operation
Output:
EM encoded message, an octet string of length emLen
Errors: "message too long"; "intended encoded message length too
short"
Steps:
1. Apply the hash function to the message M to produce a hash
value H:
H = Hash(M).
If the hash function outputs "message too long," output
"message too long" and stop.
2. Encode the algorithm ID for the hash function and the hash
value into an ASN.1 value of type DigestInfo (see
Appendix A.2.4) with the Distinguished Encoding Rules (DER),
where the type DigestInfo has the syntax
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DigestInfo ::= SEQUENCE {
digestAlgorithm AlgorithmIdentifier,
digest OCTET STRING
}
The first field identifies the hash function and the second
contains the hash value. Let T be the DER encoding of the
DigestInfo value (see the notes below) and let tLen be the
length in octets of T.
3. If emLen < tLen + 11, output "intended encoded message length
too short" and stop.
4. Generate an octet string PS consisting of emLen  tLen  3
octets with hexadecimal value 0xff. The length of PS will be
at least 8 octets.
5. Concatenate PS, the DER encoding T, and other padding to form
the encoded message EM as
EM = 0x00  0x01  PS  0x00  T.
6. Output EM.
Notes.
1. For the six hash functions mentioned in Appendix B.1, the DER
encoding T of the DigestInfo value is equal to the following:
MD2: (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 02 05 00 04
10  H.
MD5: (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 05 05 00 04
10  H.
SHA1: (0x)30 21 30 09 06 05 2b 0e 03 02 1a 05 00 04 14  H.
SHA256: (0x)30 31 30 0d 06 09 60 86 48 01 65 03 04 02 01 05 00
04 20  H.
SHA384: (0x)30 41 30 0d 06 09 60 86 48 01 65 03 04 02 02 05 00
04 30  H.
SHA512: (0x)30 51 30 0d 06 09 60 86 48 01 65 03 04 02 03 05 00
04 40  H.
SHA512/224: (0x)30 2d 30 0d 06 09 60 86 48 01 65 03 04 02 05
05 00 04 1c  H.
SHA512/256: (0x)30 31 30 0d 06 09 60 86 48 01 65 03 04 02 06
05 00 04 20  H.
2. In version 1.5 of this document, T was defined as the BER
encoding, rather than the DER encoding, of the DigestInfo value.
In particular, it is possible  at least in theory  that the
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verification operation defined in this document (as well as in
version 2.0) rejects a signature that is valid with respect to
the specification given in PKCS #1 v1.5. This occurs if other
rules than DER are applied to DigestInfo (e.g., an indefinite
length encoding of the underlying SEQUENCE type). While this is
unlikely to be a concern in practice, a cautious implementer may
choose to employ a verification operation based on a BER decoding
operation as specified in PKCS #1 v1.5. In this manner,
compatibility with any valid implementation based on PKCS #1 v1.5
is obtained. Such a verification operation should indicate
whether the underlying BER encoding is a DER encoding and hence
whether the signature is valid with respect to the specification
given in this document.
10. IANA Considerations
This memo includes no request to IANA.
11. Security Considerations
Security considerations are discussed throughout this memo.
12. References
12.1. Normative References
[GARNER] Garner, H., "The Residue Number System", IRE Transactions
on Electronic Computers EC8 (6), pp. 140147, June 1959.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RSA] Rivest, R., Shamir, A., and L. Adleman, "A Method for
Obtaining Digital Signatures and PublicKey
Cryptosystems", Communications of the ACM 21 (2), pp.
120126, February 1978.
12.2. Informative References
[ANSIX944]
ANSI, "ANSI X9.442007: Key Extablishment Using Integer
Factorization Cryptography", March 2002.
[BKS] Bleichenbacher, D., Kaliski, B., and J. Staddon, "Recent
Results on PKCS #1: RSA Encryption Standard", RSA
Laboratories' Bulletin No. 7, June 1998.
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[BLEICHENBACHER]
Bleichenbacher, D., "Chosen Ciphertext Attacks Against
Protocols Based on the RSA Encryption Standard PKCS #1",
Lecture Notes in Computer Science Volume 1462, pp. 112,
1998.
[CHOSEN] Desmedt, Y. and A. Odlyzko, "A Chosen Text Attack on the
RSA Cryptosystem and Some Discrete Logarithm schemes",
Lecture Notes in Computer Science Volume 218, pp. 516522,
1986.
[COCHRAN] Cochran, M., "Notes on the Wang et al. 2^63 SHA1
Differential Path", August 2008,
.
[FASTDEC] Quisquater, J. and C. Couvreur, "Fast Decipherment
Algorithm for RSA PublicKey Cryptosystem", Electronics
Letters 18 (21), pp 905907, October 1982.
[FDH] Coron, J., "On the Exact Security of Full Domain Hashing",
Lecture Notes in Computer Science Volume 1880, pp.
229235, 2000.
[FOPS] Fujisaki, E., Okamoto, T., Pointcheval, D., and J. Stern,
"RSAOAEP is Secure under the RSA Assumption", Lecture
Notes in Computer Science Volume 2139, pp. 260274, 2001.
[FORGERY] Coppersmith, D., Halevi, S., and C. Jutla, "ISO 97961 and
the New Forgery Strategy", August 1999.
[GRELL] Grell, M., "Re: Encoding Methods PSS/PSSR", June 1999,
.
[HAASTAD] Haastad, J., "Solving Simultaneous Modular Equations of
Low Degree", SIAM Journal of Computing Volume 17, pp.
336341, 1998.
[HANDBOOK]
Menezes, A., van Oorschot, P., and S. Vanstone, "Handbook
of Applied Cryptography", 1996.
[HASHID] Kaliski, B., "On Hash Function Identification in Signature
Schemes", Lecture Notes in Computer Science Volume 2271,
pp. 116, 2002.
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[IEEE1363]
IEEE, "IEEE Std 13632000: Standard Specifications for
Public Key Cryptography", August 2000.
[IEEE1363A]
IEEE, "IEEE Std P1363a2004: Standard Specifications for
Public Key Cryptography: Additional Techniques", September
2004.
[ISO18033]
ISO/IEC, "ISO/IEC 180332:2006: Information Technology 
Security Techniques  Encryption Algorithms  Part 2:
Asymmetric Ciphers", 2006.
[ISO9594] ISO/IEC, "ISO/IEC 95948:2008: Information technology 
Open Systems Interconnection  The Directory:
Authentication Framework", 2008.
[ISO9796] ISO/IEC, "ISO/IEC 97962:2010: Information technology 
Security Techniques  Digital Signature Schemes Giving
Message Recovery  Part 2: Integer Factorization Based
Mechanisms", 2010.
[JONSSON] Jonsson, J., "Security Proofs for the RSAPSS Signature
Scheme and Its Variants", March 2002,
.
[LOWEXP] Coppersmith, D., Franklin, M., Patarin, J., and M. Reiter,
"LowExponent RSA with Related Messages", Lecture Notes in
Computer Science Volume 1070, pp. 19, 1996.
[MANGER] Manger, J., "A Chosen Ciphertext Attack on RSA Optimal
Asymmetric Encryption Padding (OAEP) as Standardized in
PKCS #1 v2.0", Lecture Notes in Computer Science Volume
2139, pp. 260274, 2001.
[MD4] Dobbertin, H., "Cryptanalysis of MD4", Lecture Notes in
Computer Science Volume 1039, pp. 5572, 1996.
[MD4FIRST]
Dobbertin, H., "The First Two Rounds of MD4 are Not One
Way", Lecture Notes in Computer Science Volume 1372, pp.
284292, 1998.
[MD4LAST] den Boer, B. and A. Bosselaers, "An Attack on the Last Two
Rounds of MD4", Lecture Notes in Computer Science Volume
576, pp. 194203, 1992.
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[NEWATTACK]
Coron, J., Joye, M., Naccache, D., and P. Paillier, "New
Attacks on PKCS #1 v1.5 Encryption", Lecture Notes in
Computer Science Volume 1807, pp. 369379, 2000.
[OAEP] Bellare, M. and P. Rogaway, "Optimal Asymmetric Encryption
 How to Encrypt with RSA", Lecture Notes in Computer
Science Volume 950, pp. 92111, 1996.
[PA98] Bellare, M., Desai, A., Pointcheval, D., and P. Rogaway,
"Relations Among Notions of Security for PublicKey
Encryption Schemes", Lecture Notes in Computer
Science Volume 1462, pp. 2645, 1998.
[PADDING] Coron, J., Naccache, D., and J. Stern, "On the Security of
RSA Padding", Lecture Notes in Computer Science Volume
1666, pp. 118, 1999.
[PKCS1_21]
RSA Laboratories, "PKCS #1v2.1: RSA Encryption Standard",
June 2002.
[PREFIX] Stevens, M., Lenstra, A., and B. de Weger, "Chosenprefix
collisions for MD5 and applications", International
Journal of Applied Cryptography Vol. 2, No. 4, pp.
322359, July 2012.
[PSS] Bellare, M. and P. Rogaway, "PSS: Provably Secure Encoding
Method for Digital Signatures", August 1998,
.
[PSSPROOF]
Coron, J., "Optimal Security Proofs for PSS and Other
Signature Schemes", Lecture Notes in Computer
Science Volume 2332, pp. 272287, 2002.
[RFC1319] Kaliski, B., "The MD2 MessageDigest Algorithm", RFC 1319,
DOI 10.17487/RFC1319, April 1992,
.
[RFC1321] Rivest, R., "The MD5 MessageDigest Algorithm", RFC 1321,
DOI 10.17487/RFC1321, April 1992,
.
[RFC2315] Kaliski, B., "PKCS #7: Cryptographic Message Syntax
Version 1.5", RFC 2315, DOI 10.17487/RFC2315, March 1998,
.
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[RFC2437] Kaliski, B. and J. Staddon, "PKCS #1: RSA Cryptography
Specifications Version 2.0", RFC 2437,
DOI 10.17487/RFC2437, October 1998,
.
[RFC5208] Kaliski, B., "PublicKey Cryptography Standards (PKCS) #8:
PrivateKey Information Syntax Specification Version 1.2",
RFC 5208, DOI 10.17487/RFC5208, May 2008,
.
[RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer Security
(TLS) Protocol Version 1.2", RFC 5246,
DOI 10.17487/RFC5246, August 2008,
.
[RFC5652] Housley, R., "Cryptographic Message Syntax (CMS)", STD 70,
RFC 5652, DOI 10.17487/RFC5652, September 2009,
.
[RFC7292] Moriarty, K., Ed., Nystrom, M., Parkinson, S., Rusch, A.,
and M. Scott, "PKCS #12: Personal Information Exchange
Syntax v1.1", RFC 7292, DOI 10.17487/RFC7292, July 2014,
.
[RSARABIN]
Bellare, M. and P. Rogaway, "The Exact Security of Digital
Signatures  How to Sign with RSA and Rabin", Lecture
Notes in Computer Science Volume 1070, pp. 399416, 1996.
[RSATLS] Jonsson, J. and B. Kaliski, "On the Security of RSA
Encryption in TLS", Lecture Notes in Computer
Science Volume 2442, pp. 127142, 2002.
[SHA1CRYPT]
Wang, X., Yao, A., and F. Yao, "Cryptanalysis on SHA1",
Lecture Notes in Computer Science Volume 2442, pp.
127142, February 2005,
.
[SHOUP] Shoup, V., "OAEP Reconsidered", Lecture Notes in Computer
Science Volume 2139, pp. 239259, 2001.
[SHS] National Institute of Standards and Technology (NIST),
"FIPS Publication 1804: Secure hash Standard", March
2012.
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[SILVERMAN]
Silverman, R., "A CostBased Security Analysis of
Symmetric and Asymmetric Key Lengths", RSA Laboratories
Bulletin No. 13, 2000.
[SIMMONS] Simmons, G., "Subliminal Communication is Easy Using the
DSA", Lecture Notes in Computer Science Volume 765, pp.
218232, 1993.
Appendix A. ASN.1 syntax
A.1. RSA key representation
This section defines ASN.1 object identifiers for RSA public and
private keys, and defines the types RSAPublicKey and RSAPrivateKey.
The intended application of these definitions includes X.509
certificates, PKCS #8 [RFC5208], and PKCS #12 [RFC7292].
The object identifier rsaEncryption identifies RSA public and private
keys as defined in Appendices A.1.1 and A.1.2. The parameters field
associated with this OID in a value of type AlgorithmIdentifier SHALL
have a value of type NULL.
rsaEncryption OBJECT IDENTIFIER ::= { pkcs1 1 }
The definitions in this section have been extended to support multi
prime RSA, but are backward compatible with previous versions.
A.1.1. RSA public key syntax
An RSA public key should be represented with the ASN.1 type
RSAPublicKey:
RSAPublicKey ::= SEQUENCE {
modulus INTEGER,  n
publicExponent INTEGER  e
}
The fields of type RSAPublicKey have the following meanings:
o modulus is the RSA modulus n.
o publicExponent is the RSA public exponent e.
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A.1.2. RSA private key syntax
An RSA private key should be represented with the ASN.1 type
RSAPrivateKey:
RSAPrivateKey ::= SEQUENCE {
version Version,
modulus INTEGER,  n
publicExponent INTEGER,  e
privateExponent INTEGER,  d
prime1 INTEGER,  p
prime2 INTEGER,  q
exponent1 INTEGER,  d mod (p1)
exponent2 INTEGER,  d mod (q1)
coefficient INTEGER,  (inverse of q) mod p
otherPrimeInfos OtherPrimeInfos OPTIONAL
}
The fields of type RSAPrivateKey have the following meanings:
o version is the version number, for compatibility with future
revisions of this document. It SHALL be 0 for this version of the
document, unless multiprime is used, in which case it SHALL be 1.
Version ::= INTEGER { twoprime(0), multi(1) }
(CONSTRAINED BY
{ version must be multi if otherPrimeInfos present })
o modulus is the RSA modulus n.
o publicExponent is the RSA public exponent e.
o privateExponent is the RSA private exponent d.
o prime1 is the prime factor p of n.
o prime2 is the prime factor q of n.
o exponent1 is d mod (p  1).
o exponent2 is d mod (q  1).
o coefficient is the CRT coefficient q^(1) mod p.
o otherPrimeInfos contains the information for the additional primes
r_3, ..., r_u, in order. It SHALL be omitted if version is 0 and
SHALL contain at least one instance of OtherPrimeInfo if version
is 1.
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OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo
OtherPrimeInfo ::= SEQUENCE {
prime INTEGER,  ri
exponent INTEGER,  di
coefficient INTEGER  ti
}
The fields of type OtherPrimeInfo have the following meanings:
o prime is a prime factor r_i of n, where i >= 3.
o exponent is d_i = d mod (r_i  1).
o coefficient is the CRT coefficient t_i = (r_1 * r_2 * ... *
r_(i1))^(1) mod r_i.
Note. It is important to protect the RSA private key against both
disclosure and modification. Techniques for such protection are
outside the scope of this document. Methods for storing and
distributing private keys and other cryptographic data are described
in PKCS #12 and #15.
A.2. Scheme identification
This section defines object identifiers for the encryption and
signature schemes. The schemes compatible with PKCS #1 v1.5 have the
same definitions as in PKCS #1 v1.5. The intended application of
these definitions includes X.509 certificates and PKCS #7.
Here are type identifier definitions for the PKCS #1 OIDs:
PKCS1Algorithms ALGORITHMIDENTIFIER ::= {
{ OID rsaEncryption PARAMETERS NULL } 
{ OID md2WithRSAEncryption PARAMETERS NULL } 
{ OID md5WithRSAEncryption PARAMETERS NULL } 
{ OID sha1WithRSAEncryption PARAMETERS NULL } 
{ OID sha224WithRSAEncryption PARAMETERS NULL } 
{ OID sha256WithRSAEncryption PARAMETERS NULL } 
{ OID sha384WithRSAEncryption PARAMETERS NULL } 
{ OID sha512WithRSAEncryption PARAMETERS NULL } 
{ OID sha512224WithRSAEncryption PARAMETERS NULL } 
{ OID sha512256WithRSAEncryption PARAMETERS NULL } 
{ OID idRSAESOAEP PARAMETERS RSAESOAEPparams } 
PKCS1PSourceAlgorithms 
{ OID idRSASSAPSS PARAMETERS RSASSAPSSparams },
...  Allows for future expansion 
}
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A.2.1. RSAESOAEP
The object identifier idRSAESOAEP identifies the RSAESOAEP
encryption scheme.
idRSAESOAEP OBJECT IDENTIFIER ::= { pkcs1 7 }
The parameters field associated with this OID in a value of type
AlgorithmIdentifier SHALL have a value of type RSAESOAEPparams:
RSAESOAEPparams ::= SEQUENCE {
hashAlgorithm [0] HashAlgorithm DEFAULT sha1,
maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1,
pSourceAlgorithm [2] PSourceAlgorithm DEFAULT pSpecifiedEmpty
}
The fields of type RSAESOAEPparams have the following meanings:
o hashAlgorithm identifies the hash function. It SHALL be an
algorithm ID with an OID in the set OAEPPSSDigestAlgorithms. For
a discussion of supported hash functions, see Appendix B.1.
HashAlgorithm ::= AlgorithmIdentifier {
{OAEPPSSDigestAlgorithms}
}
OAEPPSSDigestAlgorithms ALGORITHMIDENTIFIER ::= {
{ OID idsha1 PARAMETERS NULL }
{ OID idsha224 PARAMETERS NULL }
{ OID idsha256 PARAMETERS NULL }
{ OID idsha384 PARAMETERS NULL }
{ OID idsha512 PARAMETERS NULL }
{ OID idsha512224 PARAMETERS NULL }
{ OID idsha512256 PARAMETERS NULL },
...  Allows for future expansion 
}
The default hash function is SHA1:
sha1 HashAlgorithm ::= {
algorithm idsha1,
parameters SHA1Parameters : NULL
}
SHA1Parameters ::= NULL
o maskGenAlgorithm identifies the mask generation function. It
SHALL be an algorithm ID with an OID in the set
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PKCS1MGFAlgorithms, which for this version SHALL consist of id
mgf1, identifying the MGF1 mask generation function (see
Appendix B.2.1). The parameters field associated with idmgf1
SHALL be an algorithm ID with an OID in the set OAEP
PSSDigestAlgorithms, identifying the hash function on which MGF1
is based.
MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }
PKCS1MGFAlgorithms ALGORITHMIDENTIFIER ::= {
{ OID idmgf1 PARAMETERS HashAlgorithm },
...  Allows for future expansion 
}
o The default mask generation function is MGF1 with SHA1:
mgf1SHA1 MaskGenAlgorithm ::= {
algorithm idmgf1,
parameters HashAlgorithm : sha1
}
o pSourceAlgorithm identifies the source (and possibly the value) of
the label L. It SHALL be an algorithm ID with an OID in the set
PKCS1PSourceAlgorithms, which for this version SHALL consist of
idpSpecified, indicating that the label is specified explicitly.
The parameters field associated with idpSpecified SHALL have a
value of type OCTET STRING, containing the label. In previous
versions of this specification, the term "encoding parameters" was
used rather than "label", hence the name of the type below.
PSourceAlgorithm ::= AlgorithmIdentifier {
{PKCS1PSourceAlgorithms}
}
PKCS1PSourceAlgorithms ALGORITHMIDENTIFIER ::= {
{ OID idpSpecified PARAMETERS EncodingParameters },
...  Allows for future expansion 
}
idpSpecified OBJECT IDENTIFIER ::= { pkcs1 9 }
EncodingParameters ::= OCTET STRING(SIZE(0..MAX))
o The default label is an empty string (so that lHash will contain
the hash of the empty string):
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pSpecifiedEmpty PSourceAlgorithm ::= {
algorithm idpSpecified,
parameters EncodingParameters : emptyString
}
emptyString EncodingParameters ::= ''H
If all of the default values of the fields in RSAESOAEPparams are
used, then the algorithm identifier will have the following value:
rSAESOAEPDefaultIdentifier RSAESAlgorithmIdentifier ::= {
algorithm idRSAESOAEP,
parameters RSAESOAEPparams : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
pSourceAlgorithm pSpecifiedEmpty
}
}
RSAESAlgorithmIdentifier ::= AlgorithmIdentifier {
{PKCS1Algorithms}
}
A.2.2. RSAESPKCSv1_5
The object identifier rsaEncryption (see Appendix A.1) identifies the
RSAESPKCS1v1_5 encryption scheme. The parameters field associated
with this OID in a value of type AlgorithmIdentifier SHALL have a
value of type NULL. This is the same as in PKCS #1 v1.5.
rsaEncryption OBJECT IDENTIFIER ::= { pkcs1 1 }
A.2.3. RSASSAPSS
The object identifier idRSASSAPSS identifies the RSASSAPSS
encryption scheme.
idRSASSAPSS OBJECT IDENTIFIER ::= { pkcs1 10 }
The parameters field associated with this OID in a value of type
AlgorithmIdentifier SHALL have a value of type RSASSAPSSparams:
RSASSAPSSparams ::= SEQUENCE {
hashAlgorithm [0] HashAlgorithm DEFAULT sha1,
maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1,
saltLength [2] INTEGER DEFAULT 20,
trailerField [3] TrailerField DEFAULT trailerFieldBC
}
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The fields of type RSASSAPSSparams have the following meanings:
o hashAlgorithm identifies the hash function. It SHALL be an
algorithm ID with an OID in the set OAEPPSSDigestAlgorithms (see
Appendix A.2.1). The default hash function is SHA1.
o maskGenAlgorithm identifies the mask generation function. It
SHALL be an algorithm ID with an OID in the set PKCS1MGFAlgorithms
(see Appendix A.2.1). The default mask generation function is
MGF1 with SHA1. For MGF1 (and more generally, for other mask
generation functions based on a hash function), it is RECOMMENDED
that the underlying hash function be the same as the one
identified by hashAlgorithm; see Note 2 in Section 9.1 for further
comments.
o saltLength is the octet length of the salt. It SHALL be an
integer. For a given hashAlgorithm, the default value of
saltLength is the octet length of the hash value. Unlike the
other fields of type RSASSAPSSparams, saltLength does not need
to be fixed for a given RSA key pair.
o trailerField is the trailer field number, for compatibility with
the draft IEEE 1363a2004 [IEEE1363A]. It SHALLL be 1 for this
version of the document, which represents the trailer field with
hexadecimal value 0xbc. Other trailer fields (including the
trailer field HashID  0xcc in IEEE 1363a2004) are not supported
in this document.
TrailerField ::= INTEGER { trailerFieldBC(1) }
If the default values of the hashAlgorithm, maskGenAlgorithm, and
trailerField fields of RSASSAPSSparams are used, then the algorithm
identifier will have the following value:
rSASSAPSSDefaultIdentifier RSASSAAlgorithmIdentifier ::= {
algorithm idRSASSAPSS,
parameters RSASSAPSSparams : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
saltLength 20,
trailerField trailerFieldBC
}
}
RSASSAAlgorithmIdentifier ::= AlgorithmIdentifier {
{PKCS1Algorithms}
}
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Note. In some applications, the hash function underlying a signature
scheme is identified separately from the rest of the operations in
the signature scheme. For instance, in PKCS #7 [RFC2315], a hash
function identifier is placed before the message and a "digest
encryption" algorithm identifier (indicating the rest of the
operations) is carried with the signature. In order for PKCS #7 to
support the RSASSAPSS signature scheme, an object identifier would
need to be defined for the operations in RSASSAPSS after the hash
function (analogous to the RSAEncryption OID for the RSASSA
PKCS1v1_5 scheme). S/MIME CMS [RFC5652] takes a different approach.
Although a hash function identifier is placed before the message, an
algorithm identifier for the full signature scheme may be carried
with a CMS signature (this is done for DSA signatures). Following
this convention, the idRSASSAPSS OID can be used to identify
RSASSAPSS signatures in CMS. Since CMS is considered the successor
to PKCS #7 and new developments such as the addition of support for
RSASSAPSS will be pursued with respect to CMS rather than PKCS #7,
an OID for the "rest of" RSASSAPSS is not defined in this version of
PKCS #1.
A.2.4. RSASSAPKCSv1_5
The object identifier for RSASSAPKCS1v1_5 SHALL be one of the
following. The choice of OID depends on the choice of hash
algorithm: MD2, MD5, SHA1, SHA224, SHA256, SHA384, SHA512, SHA
512/224, or SHA512/256. Note that if either MD2 or MD5 is used,
then the OID is just as in PKCS #1 v1.5. For each OID, the
parameters field associated with this OID in a value of type
AlgorithmIdentifier SHALL have a value of type NULL. The OID should
be chosen in accordance with the following table:
Hash algorithm OID

MD2 md2WithRSAEncryption ::= {pkcs1 2}
MD5 md5WithRSAEncryption ::= {pkcs1 4}
SHA1 sha1WithRSAEncryption ::= {pkcs1 5}
SHA256 sha256WithRSAEncryption ::= {pkcs1 11}
SHA384 sha384WithRSAEncryption ::= {pkcs1 12}
SHA512 sha512WithRSAEncryption ::= {pkcs1 13}
SHA512/224 sha512224WithRSAEncryption ::= {pkcs1 15}
SHA512/256 sha512256WithRSAEncryption ::= {pkcs1 16}
The EMSAPKCS1v1_5 encoding method includes an ASN.1 value of type
DigestInfo, where the type DigestInfo has the syntax
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DigestInfo ::= SEQUENCE {
digestAlgorithm DigestAlgorithm,
digest OCTET STRING
}
digestAlgorithm identifies the hash function and SHALL be an
algorithm ID with an OID in the set PKCS1v15DigestAlgorithms. For
a discussion of supported hash functions, see Appendix B.1.
DigestAlgorithm ::= AlgorithmIdentifier {
{PKCS1v15DigestAlgorithms}
}
PKCS1v15DigestAlgorithms ALGORITHMIDENTIFIER ::= {
{ OID idmd2 PARAMETERS NULL }
{ OID idmd5 PARAMETERS NULL }
{ OID idsha1 PARAMETERS NULL }
{ OID idsha224 PARAMETERS NULL }
{ OID idsha256 PARAMETERS NULL }
{ OID idsha384 PARAMETERS NULL }
{ OID idsha512 PARAMETERS NULL }
{ OID idsha512224 PARAMETERS NULL }
{ OID idsha512256 PARAMETERS NULL }
}
Appendix B. Supporting techniques
This section gives several examples of underlying functions
supporting the encryption schemes in Section 7 and the encoding
methods in Section 9. A range of techniques is given here to allow
compatibility with existing applications as well as migration to new
techniques. While these supporting techniques are appropriate for
applications to implement, none of them is required to be
implemented. It is expected that profiles for PKCS #1 v2.2 will be
developed that specify particular supporting techniques.
This section also gives object identifiers for the supporting
techniques.
B.1. Hash functions
Hash functions are used in the operations contained in Sections 7 and
9. Hash functions are deterministic, meaning that the output is
completely determined by the input. Hash functions take octet
strings of variable length, and generate fixed length octet strings.
The hash functions used in the operations contained in Sections 7 and
9 should generally be collisionresistant. This means that it is
infeasible to find two distinct inputs to the hash function that
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produce the same output. A collisionresistant hash function also
has the desirable property of being oneway; this means that given an
output, it is infeasible to find an input whose hash is the specified
output. In addition to the requirements, the hash function should
yield a mask generation function (Appendix B.2) with pseudorandom
output.
Nine hash functions are given as examples for the encoding methods in
this document: MD2 [RFC1319], MD5 [RFC1321], SHA1, SHA224, SHA256,
SHA384, SHA512, SHA512/224, and SHA512/256 [SHS]. For the RSAES
OAEP encryption scheme and EMSAPSS encoding method, only SHA1, SHA
224, SHA256, SHA384, SHA512, SHA512/224, and SHA512/256 are
RECOMMENDED. For the EMSAPKCS1v1_5 encoding method, SHA224, SHA
256, SHA384, SHA512, SHA512/224, and SHA512/256 are RECOMMENDED
for new applications. MD2, MD5 and SHA1 are recommended only for
compatibility with existing applications based on PKCS #1 v1.5.
The object identifiers idmd2, idmd5, idsha1, idsha224, idsha256,
idsha384, idsha512, idsha512/224, and idsha512/256, identify the
respective hash functions:
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idmd2 OBJECT IDENTIFIER ::= {
iso (1) memberbody (2) us (840) rsadsi (113549)
digestAlgorithm (2) 2
}
idmd5 OBJECT IDENTIFIER ::= {
iso (1) memberbody (2) us (840) rsadsi (113549)
digestAlgorithm (2) 5
}
idsha1 OBJECT IDENTIFIER ::= {
iso(1) identifiedorganization(3) oiw(14) secsig(3)
algorithms(2) 26
}
idsha224 OBJECT IDENTIFIER ::= {
jointisoitut (2) country (16) us (840) organization (1)
gov (101) csor (3) nistalgorithm (4) hashalgs (2) 4
}
idsha256 OBJECT IDENTIFIER ::= {
jointisoitut (2) country (16) us (840) organization (1)
gov (101) csor (3) nistalgorithm (4) hashalgs (2) 1
}
idsha384 OBJECT IDENTIFIER ::= {
jointisoitut (2) country (16) us (840) organization (1)
gov (101) csor (3) nistalgorithm (4) hashalgs (2) 2
}
idsha512 OBJECT IDENTIFIER ::= {
jointisoitut (2) country (16) us (840) organization (1)
gov (101) csor (3) nistalgorithm (4) hashalgs (2) 3
}
idsha512224 OBJECT IDENTIFIER ::= {
jointisoitut (2) country (16) us (840) organization (1)
gov (101) csor (3) nistalgorithm (4) hashalgs (2) 5
}
idsha512256 OBJECT IDENTIFIER ::= {
jointisoitut (2) country (16) us (840) organization (1)
gov (101) csor (3) nistalgorithm (4) hashalgs (2) 6
}
The parameters field associated with these OIDs in a value of type
AlgorithmIdentifier SHALL have a value of type NULL.
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The parameters field associated with idmd2 and idmd5 in a value of
type AlgorithmIdentifier shall have a value of type NULL.
The parameters field associated with idsha1, idsha224, idsha256,
idsha384, idsha512, idsha512/224, and idsha512/256 should
generally be omitted, but if present, shall have a value of type
NULL.
This is to align with the definitions originally promulgated by NIST.
For the SHA algorithms, implementations MUST accept
AlgorithmIdentifier values both without parameters and with NULL
parameters.
Exception: When formatting the DigestInfoValue in EMSAPKCS1v1_5
(see 9.2), the parameters field associated with idsha1, idsha224,
idsha256, idsha384, idsha512, idsha512/224, and idsha512/256
shall have a value of type NULL. This is to maintain compatibility
with existing implementations and with the numeric information values
already published for EMSAPKCS1v1_5 which are also reflected in
IEEE 1363a2004 [IEEE1363A].
Note. Version 1.5 of PKCS #1 also allowed for the use of MD4 in
signature schemes. The cryptanalysis of MD4 has progressed
significantly in the intervening years. For example, Dobbertin [MD4]
demonstrated how to find collisions for MD4 and that the first two
rounds of MD4 are not oneway [MD4FIRST]. Because of these results
and others (e.g. [MD4LAST]), MD4 is NOT RECOMMENDED.
Further advances have been made in the cryptanalysis of MD2 and MD5,
and especially after Stevens et al. findings [PREFIX] on chosen
prefix collisions on MD5. MD2 and MD5 should be considered
cryptographically broken and removed from existing applications.
This version of the standard supports MD2 and MD5 just for backwards
compatibility reasons.
There have also been advances in the cryptanalysis of SHA1.
Particularly the results of Wang et al. [SHA1CRYPT] (which have been
independently verified by M. Cochran in his analysis [COCHRAN]) on
using a differential path to find collisions in SHA1, which conclude
that the security strength of the SHA1 hashing algorithm is
significantly reduced. However, this reduction is not significant
enough to warrant the removal of SHA1 from existing applications,
but its usage is only recommended for backwards compatibility
reasons.
To address these concerns, only SHA224, SHA256, SHA384, SHA512,
SHA512/224, and SHA512/256 are RECOMMENDED for new applications.
As of today, the best (known) collision attacks against these hash
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functions are generic attacks with complexity 2L/2, where L is the
bit length of the hash output. For the signature schemes in this
document, a collision attack is easily translated into a signature
forgery. Therefore, the value L / 2 should be at least equal to the
desired security level in bits of the signature scheme (a security
level of B bits means that the best attack has complexity 2B). The
same rule of thumb can be applied to RSAESOAEP; it is RECOMMENDED
that the bit length of the seed (which is equal to the bit length of
the hash output) be twice the desired security level in bits.
B.2. Mask generation functions
A mask generation function takes an octet string of variable length
and a desired output length as input, and outputs an octet string of
the desired length. There may be restrictions on the length of the
input and output octet strings, but such bounds are generally very
large. Mask generation functions are deterministic; the octet string
output is completely determined by the input octet string. The
output of a mask generation function should be pseudorandom: Given
one part of the output but not the input, it should be infeasible to
predict another part of the output. The provable security of RSAES
OAEP and RSASSAPSS relies on the random nature of the output of the
mask generation function, which in turn relies on the random nature
of the underlying hash.
One mask generation function is given here: MGF1, which is based on a
hash function. MGF1 coincides with the mask generation functions
defined in IEEE 13632000 [IEEE1363] and ANSI X9.442007 [ANSIX944].
Future versions of this document may define other mask generation
functions.
B.2.1. MGF1
MGF1 is a Mask Generation Function based on a hash function.
MGF1 (mgfSeed, maskLen)
Options:
Hash hash function (hLen denotes the length in octets of the hash
function output)
Input:
mgfSeed seed from which mask is generated, an octet string
maskLen intended length in octets of the mask, at most 2^32 hLen
Output:
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mask mask, an octet string of length maskLen
Error:
"mask too long"
Steps:
1. If maskLen > 2^32 hLen, output "mask too long" and stop.
2. Let T be the empty octet string.
3. For counter from 0 to \ceil (maskLen / hLen)  1, do the
following:
A. Convert counter to an octet string C of length 4 octets (see
Section 4.1):
C = I2OSP (counter, 4) .
B. Concatenate the hash of the seed mgfSeed and C to the octet
string T:
T = T  Hash(mgfSeed  C) .
4. Output the leading maskLen octets of T as the octet string mask.
The object identifier idmgf1 identifies the MGF1 mask generation
function:
idmgf1 OBJECT IDENTIFIER ::= { pkcs1 8 }
The parameters field associated with this OID in a value of type
AlgorithmIdentifier shall have a value of type hashAlgorithm,
identifying the hash function on which MGF1 is based.
Appendix C. ASN.1 module
 PKCS #1 v2.2 ASN.1 Module
 Revised October 27, 2012
 This module has been checked for conformance with the
 ASN.1 standard by the OSS ASN.1 Tools
PKCS1 {
iso(1) memberbody(2) us(840) rsadsi(113549) pkcs(1) pkcs1(1)
modules(0) pkcs1(1)
}
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DEFINITIONS EXPLICIT TAGS ::=
BEGIN
 EXPORTS ALL
 All types and values defined in this module are exported for use
 in other ASN.1 modules.
IMPORTS
idsha224, idsha256, idsha384, idsha512, idsha512224,
idsha512256
FROM NISTSHA2 {
jointisoitut(2) country(16) us(840) organization(1)
gov(101) csor(3) nistalgorithm(4) hashAlgs(2)
};
 ============================
 Basic object identifiers
 ============================
 The DER encoding of this in hexadecimal is:
 (0x)06 08
 2A 86 48 86 F7 0D 01 01

pkcs1 OBJECT IDENTIFIER ::= {
iso(1) memberbody(2) us(840) rsadsi(113549) pkcs(1) 1
}

 When rsaEncryption is used in an AlgorithmIdentifier the
 parameters MUST be present and MUST be NULL.

rsaEncryption OBJECT IDENTIFIER ::= { pkcs1 1 }

 When idRSAESOAEP is used in an AlgorithmIdentifier the
 parameters MUST be present and MUST be RSAESOAEPparams.

idRSAESOAEP OBJECT IDENTIFIER ::= { pkcs1 7 }

 When idpSpecified is used in an AlgorithmIdentifier the
 parameters MUST be an OCTET STRING.

idpSpecified OBJECT IDENTIFIER ::= { pkcs1 9 }

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 When idRSASSAPSS is used in an AlgorithmIdentifier the
 parameters MUST be present and MUST be RSASSAPSSparams.

idRSASSAPSS OBJECT IDENTIFIER ::= { pkcs1 10 }

 When the following OIDs are used in an AlgorithmIdentifier the
 parameters MUST be present and MUST be NULL.

md2WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs1 2 }
md5WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs1 4 }
sha1WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs1 5 }
sha224WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs1 14 }
sha256WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs1 11 }
sha384WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs1 12 }
sha512WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs1 13 }
sha512224WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs1 15 }
sha512256WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs1 16 }

 This OID really belongs in a module with the secsig OIDs.

idsha1 OBJECT IDENTIFIER ::= {
iso(1) identifiedorganization(3) oiw(14) secsig(3) algorithms(2)
26
}

 OIDs for MD2 and MD5, allowed only in EMSAPKCS1v1_5.

idmd2 OBJECT IDENTIFIER ::= {
iso(1) memberbody(2) us(840) rsadsi(113549) digestAlgorithm(2) 2
}
idmd5 OBJECT IDENTIFIER ::= {
iso(1) memberbody(2) us(840) rsadsi(113549) digestAlgorithm(2) 5
}

 When idmgf1 is used in an AlgorithmIdentifier the parameters MUST
 be present and MUST be a HashAlgorithm, for example sha1.

idmgf1 OBJECT IDENTIFIER ::= { pkcs1 8 }
 ================
 Useful types
 ================
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ALGORITHMIDENTIFIER ::= CLASS {
&id OBJECT IDENTIFIER UNIQUE,
&Type OPTIONAL
}
WITH SYNTAX { OID &id [PARAMETERS &Type] }
 Note: the parameter InfoObjectSet in the following definitions
 allows a distinct information object set to be specified for sets
 of algorithms such as:
 DigestAlgorithms ALGORITHMIDENTIFIER ::= {
 { OID idmd2 PARAMETERS NULL }
 { OID idmd5 PARAMETERS NULL }
 { OID idsha1 PARAMETERS NULL }
 }

AlgorithmIdentifier { ALGORITHMIDENTIFIER:InfoObjectSet } ::=
SEQUENCE {
algorithm
ALGORITHMIDENTIFIER.&id({InfoObjectSet}),
parameters
ALGORITHMIDENTIFIER.&Type({InfoObjectSet}{@.algorithm})
OPTIONAL
}
 ==============
 Algorithms
 ==============

 Allowed EMEOAEP and EMSAPSS digest algorithms.

OAEPPSSDigestAlgorithms ALGORITHMIDENTIFIER ::= {
{ OID idsha1 PARAMETERS NULL }
{ OID idsha224 PARAMETERS NULL }
{ OID idsha256 PARAMETERS NULL }
{ OID idsha384 PARAMETERS NULL }
{ OID idsha512 PARAMETERS NULL }
{ OID idsha512224 PARAMETERS NULL }
{ OID idsha512256 PARAMETERS NULL },
...  Allows for future expansion 
}

 Allowed EMSAPKCS1v1_5 digest algorithms.

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PKCS1v15DigestAlgorithms ALGORITHMIDENTIFIER ::= {
{ OID idmd2 PARAMETERS NULL }
{ OID idmd5 PARAMETERS NULL }
{ OID idsha1 PARAMETERS NULL }
{ OID idsha224 PARAMETERS NULL }
{ OID idsha256 PARAMETERS NULL }
{ OID idsha384 PARAMETERS NULL }
{ OID idsha512 PARAMETERS NULL }
{ OID idsha512224 PARAMETERS NULL }
{ OID idsha512256 PARAMETERS NULL }
}
 When idmd2 and idmd5 are used in an AlgorithmIdentifier the
 parameters field shall have a value of type NULL.
 When idsha1, idsha224, idsha256, idsha384, idsha512,
 idsha512224, and idsha512256 are used in an
 AlgorithmIdentifier the parameters (which are optional) SHOULD be
 omitted, but if present, SHALL have a value of type NULL. However,
 implementations MUST accept AlgorithmIdentifier values both
 without parameters and with NULL parameters.
 Exception: When formatting the DigestInfoValue in EMSAPKCS1v1_5
 (see 9.2), the parameters field associated with idsha1,
 idsha224, idsha256, idsha384 , idsha512 , idsha512224, and
 idsha512256 SHALL have a value of type NULL. This is to maintain
 compatibility with existing implementations and with the numeric
 information values already published for EMSAPKCS1v1_5, which
 are also reflected in IEEE 1363a2004.
sha1 HashAlgorithm ::= {
algorithm idsha1,
parameters SHA1Parameters : NULL
}
HashAlgorithm ::= AlgorithmIdentifier { {OAEPPSSDigestAlgorithms} }
SHA1Parameters ::= NULL

 Allowed mask generation function algorithms.
 If the identifier is idmgf1, the parameters are a HashAlgorithm.

PKCS1MGFAlgorithms ALGORITHMIDENTIFIER ::= {
{ OID idmgf1 PARAMETERS HashAlgorithm },
...  Allows for future expansion 
}
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 Default AlgorithmIdentifier for idRSAESOAEP.maskGenAlgorithm and
 idRSASSAPSS.maskGenAlgorithm.

mgf1SHA1 MaskGenAlgorithm ::= {
algorithm idmgf1,
parameters HashAlgorithm : sha1
}
MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} }

 Allowed algorithms for pSourceAlgorithm.

PKCS1PSourceAlgorithms ALGORITHMIDENTIFIER ::= {
{ OID idpSpecified PARAMETERS EncodingParameters },
...  Allows for future expansion 
}
EncodingParameters ::= OCTET STRING(SIZE(0..MAX))

 This identifier means that the label L is an empty string, so the
 digest of the empty string appears in the RSA block before
 masking.

pSpecifiedEmpty PSourceAlgorithm ::= {
algorithm idpSpecified,
parameters EncodingParameters : emptyString
}
PSourceAlgorithm ::= AlgorithmIdentifier { {PKCS1PSourceAlgorithms} }
emptyString EncodingParameters ::= ''H

 Type identifier definitions for the PKCS #1 OIDs.

PKCS1Algorithms ALGORITHMIDENTIFIER ::= {
{ OID rsaEncryption PARAMETERS NULL } 
{ OID md2WithRSAEncryption PARAMETERS NULL } 
{ OID md5WithRSAEncryption PARAMETERS NULL } 
{ OID sha1WithRSAEncryption PARAMETERS NULL } 
{ OID sha224WithRSAEncryption PARAMETERS NULL } 
{ OID sha256WithRSAEncryption PARAMETERS NULL } 
{ OID sha384WithRSAEncryption PARAMETERS NULL } 
{ OID sha512WithRSAEncryption PARAMETERS NULL } 
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{ OID sha512224WithRSAEncryption PARAMETERS NULL } 
{ OID sha512256WithRSAEncryption PARAMETERS NULL } 
{ OID idRSAESOAEP PARAMETERS RSAESOAEPparams } 
PKCS1PSourceAlgorithms 
{ OID idRSASSAPSS PARAMETERS RSASSAPSSparams },
...  Allows for future expansion 
}
 ===================
 Main structures
 ===================
RSAPublicKey ::= SEQUENCE {
modulus INTEGER,  n
publicExponent INTEGER  e
}

 Representation of RSA private key with information for the CRT
 algorithm.

RSAPrivateKey ::= SEQUENCE {
version Version,
modulus INTEGER,  n
publicExponent INTEGER,  e
privateExponent INTEGER,  d
prime1 INTEGER,  p
prime2 INTEGER,  q
exponent1 INTEGER,  d mod (p1)
exponent2 INTEGER,  d mod (q1)
coefficient INTEGER,  (inverse of q) mod p
otherPrimeInfos OtherPrimeInfos OPTIONAL
}
Version ::= INTEGER { twoprime(0), multi(1) }
(CONSTRAINED BY
{ version MUST
be multi if otherPrimeInfos present })
OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo
OtherPrimeInfo ::= SEQUENCE {
prime INTEGER,  ri
exponent INTEGER,  di
coefficient INTEGER  ti
}
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 AlgorithmIdentifier.parameters for idRSAESOAEP.
 Note that the tags in this Sequence are explicit.

RSAESOAEPparams ::= SEQUENCE {
hashAlgorithm [0] HashAlgorithm DEFAULT sha1,
maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1,
pSourceAlgorithm [2] PSourceAlgorithm DEFAULT pSpecifiedEmpty
}

 Identifier for default RSAESOAEP algorithm identifier.
 The DER Encoding of this is in hexadecimal:
 (0x)30 0D
 06 09
 2A 86 48 86 F7 0D 01 01 07
 30 00
 Notice that the DER encoding of default values is "empty".

rSAESOAEPDefaultIdentifier RSAESAlgorithmIdentifier ::= {
algorithm idRSAESOAEP,
parameters RSAESOAEPparams : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
pSourceAlgorithm pSpecifiedEmpty
}
}
RSAESAlgorithmIdentifier ::= AlgorithmIdentifier {
{PKCS1Algorithms}
}

 AlgorithmIdentifier.parameters for idRSASSAPSS.
 Note that the tags in this Sequence are explicit.

RSASSAPSSparams ::= SEQUENCE {
hashAlgorithm [0] HashAlgorithm DEFAULT sha1,
maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1,
saltLength [2] INTEGER DEFAULT 20,
trailerField [3] TrailerField DEFAULT trailerFieldBC
}
TrailerField ::= INTEGER { trailerFieldBC(1) }

 Identifier for default RSASSAPSS algorithm identifier
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 The DER Encoding of this is in hexadecimal:
 (0x)30 0D
 06 09
 2A 86 48 86 F7 0D 01 01 0A
 30 00
 Notice that the DER encoding of default values is "empty".

rSASSAPSSDefaultIdentifier RSASSAAlgorithmIdentifier ::= {
algorithm idRSASSAPSS,
parameters RSASSAPSSparams : {
hashAlgorithm sha1,
maskGenAlgorithm mgf1SHA1,
saltLength 20,
trailerField trailerFieldBC
}
}
RSASSAAlgorithmIdentifier ::= AlgorithmIdentifier {
{PKCS1Algorithms}
}

 Syntax for the EMSAPKCS1v1_5 hash identifier.

DigestInfo ::= SEQUENCE {
digestAlgorithm DigestAlgorithm,
digest OCTET STRING
}
DigestAlgorithm ::= AlgorithmIdentifier {
{PKCS1v15DigestAlgorithms}
}
END
Appendix D. Intellectual property considerations
The RSA publickey cryptosystem is described in U.S. Patent
4,405,829, which expired on September 20, 2000. EMC Corporation
makes no other patent claims on the constructions described in this
document, although specific underlying techniques may be covered.
Multiprime RSA is described in U.S. Patent 5,848,159.
The PSS signature scheme [PSS] is described in U.S. Patent
6,266,771, which expired on July 24, 2009, and U.S. Patent
7,036,014, which expired on April 25, 2010, and was held by the
University of California. The PSS signature scheme is specified in
IEEE P1363a2004 [IEEE1363A]. Prior to the finalization of this
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standard, the University of California provided a letter to the IEEE
P1363 working group stating that if the PSS signature scheme is
included in an IEEE standard, "the University of California will,
when that standard is adopted, FREELY license any conforming
implementation of PSS as a technique for achieving a digital
signature with appendix" [GRELL].
EMC Corporation makes no other representations regarding intellectual
property claims by other parties. Such determination is the
responsibility of the user.
Appendix E. Revision history
Versions 1.0  1.3
Versions 1.0  1.3 were distributed to participants in RSA Data
Security, Inc.'s PublicKey Cryptography Standards meetings in
February and March 1991. Version 1.4
Version 1.4 was part of the June 3, 1991 initial public release of
PKCS. Version 1.4 was published as NIST/OSI Implementors' Workshop
document SECSIG9118. Version 1.5
Version 1.5 incorporated several editorial changes, including updates
to the references and the addition of a revision history. The
following substantive changes were made:
o Section 10: "MD4 with RSA" signature and verification processes
were added.
o Section 11: md4WithRSAEncryption object identifier was added.
Version 1.5 was republished as IETF RFC 2313.
Version 2.0
Version 2.0 incorporated major editorial changes in terms of the
document structure and introduced the RSAESOAEP encryption scheme.
This version continued to support the encryption and signature
processes in version 1.5, although the hash algorithm MD4 was no
longer allowed due to cryptanalytic advances in the intervening
years. Version 2.0 was republished as RFC 2437 [RFC2437].
Version 2.1
Version 2.1 introduces multiprime RSA and the RSASSAPSS signature
scheme with appendix along with several editorial improvements. This
version continues to support the schemes in version 2.0.
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Version 2.2
Version 2.2 updates the list of allowed hashing algorithms to align
them with FIPS 1804 [SHS], therefore adding SHA224, SHA512/224 and
SHA512/256. The following substantive changes were made:
o Object identifiers for sha224WithRSAEncryption,
sha512224WithRSAEncryption and sha512256WithRSAEncryption were
added.
o This version continues to support the schemes in version 2.1.
Appendix F. About PKCS
The PublicKey Cryptography Standards are specifications produced by
RSA Laboratories in cooperation with secure systems developers
worldwide for the purpose of accelerating the deployment of public
key cryptography. First published in 1991 as a result of meetings
with a small group of early adopters of publickey technology, the
PKCS documents have become widely referenced and implemented.
Contributions from the PKCS series have become part of many formal
and de facto standards, including ANSI X9 and IEEE P1363 documents,
PKIX, SET, S/MIME, SSL/TLS, and WAP/WTLS.
Further development of PKCS occurs through the IETF. Suggestions for
improvement are welcome.
Appendix G. Acknowledgements
This document is based on a contribution of RSA Laboratories, the
research center of RSA Security Inc.
Authors' Addresses
Kathleen M. Moriarty (editor)
EMC Corporation
176 South Street
Hopkinton, MA 01748
US
Email: kathleen.moriarty@emc.com
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Burt Kaliski
Verisign
12061 Bluemont Way
Reston, VA 20190
US
Email: bkaliski@verisign.com
URI: http://verisignlabs.com
Jakob Jonsson
Subset AB
Munkbrogtan 4
Stockholm SE11127
SE
Phone: +46 8 428 687 43
Email: jakob.jonsson@subset.se
Andreas Rusch
RSA
345 Queen Street
Brisbane, QLD 4000
AU
Email: Andreas.Rusch@rsa.com
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