< draft-hoffman-pkcs-rsa-encrypt-01.txt   draft-hoffman-pkcs-rsa-encrypt-02.txt >
Internet Draft RSA Laboratories Internet Draft Burt Kaliski
Expires 11/5/97 Expires March 16, 1998
<draft-hoffman-pkcs-rsa-encrypt-01.txt> <draft-hoffman-pkcs-rsa-encrypt-02.txt>
PKCS #1: RSA Encryption PKCS #1: RSA Encryption
Version 1.5 Version 1.5
Status of this Memo Status of this Memo
This document is an Internet-Draft. Internet-Drafts are working This document is an Internet-Draft. Internet-Drafts are working
documents of the Internet Engineering Task Force (IETF), its areas, documents of the Internet Engineering Task Force (IETF), its areas,
and its working groups. Note that other groups may also distribute and its working groups. Note that other groups may also distribute
working documents as Internet-Drafts. working documents as Internet-Drafts.
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Directories on ftp.is.co.za (Africa), nic.nordu.net (Europe), Directories on ftp.is.co.za (Africa), nic.nordu.net (Europe),
munnari.oz.au (Pacific Rim), ds.internic.net (US East Coast), or munnari.oz.au (Pacific Rim), ds.internic.net (US East Coast), or
ftp.isi.edu (US West Coast). ftp.isi.edu (US West Coast).
This memo provides information for the Internet community. This memo This memo provides information for the Internet community. This memo
does not specify an Internet standard of any kind. Distribution of does not specify an Internet standard of any kind. Distribution of
this memo is unlimited. this memo is unlimited.
Overview Overview
This standard describes a method for encrypting data using the RSA This document describes a method for encrypting data using the RSA
public-key cryptosystem. public-key cryptosystem.
Please note: The information in this document is historical material
being published for the public record. It is not an IETF standard.
The use of the word "standard" in this document indicates a standard
for RSA Laboratories and its customers, not an IETF standard.
1. Scope 1. Scope
This standard describes a method for encrypting data using the RSA This document describes a method for encrypting data using the RSA
public-key cryptosystem. Its intended use is in the construction of public-key cryptosystem. Its intended use is in the construction of
digital signatures and digital envelopes, as described in PKCS #7: digital signatures and digital envelopes, as described in PKCS #7:
o For digital signatures, the content to be signed o For digital signatures, the content to be signed
PKCS #1: RSA Encryption
is first reduced to a message digest with a is first reduced to a message digest with a
message-digest algorithm (such as MD5), and then message-digest algorithm (such as MD5), and then
an octet string containing the message digest is an octet string containing the message digest is
encrypted with the RSA private key of the signer encrypted with the RSA private key of the signer
of the content. The content and the encrypted of the content. The content and the encrypted
message digest are represented together according message digest are represented together according
to the syntax in PKCS #7 to yield a digital to the syntax in PKCS #7 to yield a digital
RFC nnn PKCS #1: RSA Encryption November 1993
signature. This application is compatible with signature. This application is compatible with
Privacy-Enhanced Mail (PEM) methods. Privacy-Enhanced Mail (PEM) methods.
o For digital envelopes, the content to be enveloped o For digital envelopes, the content to be enveloped
is first encrypted under a content-encryption key is first encrypted under a content-encryption key
with a content-encryption algorithm (such as DES), with a content-encryption algorithm (such as DES),
and then the content-encryption key is encrypted and then the content-encryption key is encrypted
with the RSA public keys of the recipients of the with the RSA public keys of the recipients of the
content. The encrypted content and the encrypted content. The encrypted content and the encrypted
content-encryption key are represented together content-encryption key are represented together
according to the syntax in PKCS #7 to yield a according to the syntax in PKCS #7 to yield a
digital envelope. This application is also digital envelope. This application is also
compatible with PEM methods. compatible with PEM methods.
The standard also describes a syntax for RSA public keys and private The document also describes a syntax for RSA public keys and private
keys. The public-key syntax would be used in certificates; the keys. The public-key syntax would be used in certificates; the
private-key syntax would be used typically in PKCS #8 private-key private-key syntax would be used typically in PKCS #8 private-key
information. The public-key syntax is identical to that in both X.509 information. The public-key syntax is identical to that in both X.509
and Privacy-Enhanced Mail. Thus X.509/PEM RSA keys can be used in and Privacy-Enhanced Mail. Thus X.509/PEM RSA keys can be used in
this standard. this document.
The standard also defines three signature algorithms for use in The document also defines three signature algorithms for use in
signing X.509/PEM certificates and certificate-revocation lists, PKCS signing X.509/PEM certificates and certificate-revocation lists, PKCS
#6 extended certificates, and other objects employing digital #6 extended certificates, and other objects employing digital
signatures such as X.401 message tokens. signatures such as X.401 message tokens.
Details on message-digest and content-encryption algorithms are Details on message-digest and content-encryption algorithms are
outside the scope of this standard, as are details on sources of the outside the scope of this document, as are details on sources of the
pseudorandom bits required by certain methods in this standard. pseudorandom bits required by certain methods in this document.
2. References 2. References
FIPS PUB 46-1 National Bureau of Standards. FIPS PUB 46-1: FIPS PUB 46-1 National Bureau of Standards. FIPS PUB 46-1:
Data Encryption Standard. January 1988. Data Encryption Standard. January 1988.
PKCS #6 RSA Laboratories. PKCS #6: Extended-Certificate PKCS #6 RSA Laboratories. PKCS #6: Extended-Certificate
Syntax Standard. Version 1.5, November 1993. Syntax. Version 1.5, November 1993.
PKCS #7 RSA Laboratories. PKCS #7: Cryptographic Message PKCS #7 RSA Laboratories. PKCS #7: Cryptographic Message
Syntax Standard. Version 1.5, November 1993. Syntax. Version 1.5, November 1993.
PKCS #1: RSA Encryption
PKCS #8 RSA Laboratories. PKCS #8: Private-Key Information PKCS #8 RSA Laboratories. PKCS #8: Private-Key Information
Syntax Standard. Version 1.2, November 1993. Syntax. Version 1.2, November 1993.
RFC 1319 B. Kaliski. RFC 1319: The MD2 Message-Digest RFC 1319 B. Kaliski. RFC 1319: The MD2 Message-Digest
Algorithm. April 1992. Algorithm. April 1992.
RFC 1320 R. Rivest. RFC 1320: The MD4 Message-Digest RFC 1320 R. Rivest. RFC 1320: The MD4 Message-Digest
RFC nnn PKCS #1: RSA Encryption November 1993
Algorithm. April 1992. Algorithm. April 1992.
RFC 1321 R. Rivest. RFC 1321: The MD5 Message-Digest RFC 1321 R. Rivest. RFC 1321: The MD5 Message-Digest
Algorithm. April 1992. Algorithm. April 1992.
RFC 1423 D. Balenson. RFC 1423: Privacy Enhancement for RFC 1423 D. Balenson. RFC 1423: Privacy Enhancement for
Internet Electronic Mail: Part III: Algorithms, Internet Electronic Mail: Part III: Algorithms,
Modes, and Identifiers. February 1993. Modes, and Identifiers. February 1993.
X.208 CCITT. Recommendation X.208: Specification of X.208 CCITT. Recommendation X.208: Specification of
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EUROCRYPT '93 (Lofthus, Norway, May 24-27, 1993). EUROCRYPT '93 (Lofthus, Norway, May 24-27, 1993).
[DO86] Y. Desmedt and A.M. Odlyzko. A chosen text attack [DO86] Y. Desmedt and A.M. Odlyzko. A chosen text attack
on the RSA cryptosystem and some discrete on the RSA cryptosystem and some discrete
logarithm schemes. In H.C. Williams, editor, logarithm schemes. In H.C. Williams, editor,
Advances in Cryptology---CRYPTO '85 Proceedings, Advances in Cryptology---CRYPTO '85 Proceedings,
volume 218 of Lecture Notes in Computer Science, volume 218 of Lecture Notes in Computer Science,
pages 516-521. Springer-Verlag, New York, 1986. pages 516-521. Springer-Verlag, New York, 1986.
[Has88] Johan Hastad. Solving simultaneous modular [Has88] Johan Hastad. Solving simultaneous modular
PKCS #1: RSA Encryption
equations. SIAM Journal on Computing, equations. SIAM Journal on Computing,
17(2):336-341, April 1988. 17(2):336-341, April 1988.
[IM90] Colin I'Anson and Chris Mitchell. Security defects [IM90] Colin I'Anson and Chris Mitchell. Security defects
in CCITT Recommendation X.509--The directory in CCITT Recommendation X.509--The directory
authentication framework. Computer Communications authentication framework. Computer Communications
Review, :30-34, April 1990. Review, :30-34, April 1990.
RFC nnn PKCS #1: RSA Encryption November 1993
[Mer90] R.C. Merkle. Note on MD4. Unpublished manuscript, [Mer90] R.C. Merkle. Note on MD4. Unpublished manuscript,
1990. 1990.
[Mil76] G.L. Miller. Riemann's hypothesis and tests for [Mil76] G.L. Miller. Riemann's hypothesis and tests for
primality. Journal of Computer and Systems primality. Journal of Computer and Systems
Sciences, 13(3):300-307, 1976. Sciences, 13(3):300-307, 1976.
[QC82] J.-J. Quisquater and C. Couvreur. Fast [QC82] J.-J. Quisquater and C. Couvreur. Fast
decipherment algorithm for RSA public-key decipherment algorithm for RSA public-key
cryptosystem. Electronics Letters, 18(21):905-907, cryptosystem. Electronics Letters, 18(21):905-907,
October 1982. October 1982.
[RSA78] R.L. Rivest, A. Shamir, and L. Adleman. A method [RSA78] R.L. Rivest, A. Shamir, and L. Adleman. A method
for obtaining digital signatures and public-key for obtaining digital signatures and public-key
cryptosystems. Communications of the ACM, cryptosystems. Communications of the ACM,
21(2):120-126, February 1978. 21(2):120-126, February 1978.
3. Definitions 3. Definitions
For the purposes of this standard, the following definitions apply. For the purposes of this document, the following definitions apply.
AlgorithmIdentifier: A type that identifies an algorithm (by object AlgorithmIdentifier: A type that identifies an algorithm (by object
identifier) and associated parameters. This type is defined in X.509. identifier) and associated parameters. This type is defined in X.509.
ASN.1: Abstract Syntax Notation One, as defined in X.208. ASN.1: Abstract Syntax Notation One, as defined in X.208.
BER: Basic Encoding Rules, as defined in X.209. BER: Basic Encoding Rules, as defined in X.209.
DES: Data Encryption Standard, as defined in FIPS PUB 46-1. DES: Data Encryption Standard, as defined in FIPS PUB 46-1.
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defined in RFC 1319. defined in RFC 1319.
MD4: RSA Data Security, Inc.'s MD4 message-digest algorithm, as MD4: RSA Data Security, Inc.'s MD4 message-digest algorithm, as
defined in RFC 1320. defined in RFC 1320.
MD5: RSA Data Security, Inc.'s MD5 message-digest algorithm, as MD5: RSA Data Security, Inc.'s MD5 message-digest algorithm, as
defined in RFC 1321. defined in RFC 1321.
modulus: Integer constructed as the product of two primes. modulus: Integer constructed as the product of two primes.
PKCS #1: RSA Encryption
PEM: Internet Privacy-Enhanced Mail, as defined in RFC 1423 and PEM: Internet Privacy-Enhanced Mail, as defined in RFC 1423 and
related documents. related documents.
RSA: The RSA public-key cryptosystem, as defined in [RSA78]. RSA: The RSA public-key cryptosystem, as defined in [RSA78].
private key: Modulus and private exponent. private key: Modulus and private exponent.
RFC nnn PKCS #1: RSA Encryption November 1993
public key: Modulus and public exponent. public key: Modulus and public exponent.
4. Symbols and abbreviations 4. Symbols and abbreviations
Upper-case italic symbols (e.g., BT) denote octet strings and bit Upper-case symbols (e.g., BT) denote octet strings and bit strings
strings (in the case of the signature S); lower-case italic symbols (in the case of the signature S); lower-case symbols (e.g., c) denote
(e.g., c) denote integers. integers.
ab hexadecimal octet value c exponent ab hexadecimal octet value c exponent
BT block type d private exponent BT block type d private exponent
D data e public exponent D data e public exponent
EB encryption block k length of modulus in EB encryption block k length of modulus in
octets octets
ED encrypted data n modulus ED encrypted data n modulus
M message p, q prime factors of modulus M message p, q prime factors of modulus
MD message digest x integer encryption block MD message digest x integer encryption block
MD' comparative message y integer encrypted data MD' comparative message y integer encrypted data
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Thus the encryption process can be either a public-key operation or a Thus the encryption process can be either a public-key operation or a
private-key operation, and so can the decryption process. Both private-key operation, and so can the decryption process. Both
processes transform an octet string to another octet string. The processes transform an octet string to another octet string. The
processes are inverses of each other if one process uses an entity's processes are inverses of each other if one process uses an entity's
public key and the other process uses the same entity's private key. public key and the other process uses the same entity's private key.
The encryption and decryption processes can implement either the The encryption and decryption processes can implement either the
classic RSA transformations, or variations with padding. classic RSA transformations, or variations with padding.
6. Key generation 6. Key generation
PKCS #1: RSA Encryption
This section describes RSA key generation. This section describes RSA key generation.
Each entity shall select a positive integer e as its public exponent. Each entity shall select a positive integer e as its public exponent.
Each entity shall privately and randomly select two distinct odd Each entity shall privately and randomly select two distinct odd
primes p and q such that (p-1) and e have no common divisors, and primes p and q such that (p-1) and e have no common divisors, and
RFC nnn PKCS #1: RSA Encryption November 1993
(q-1) and e have no common divisors. (q-1) and e have no common divisors.
The public modulus n shall be the product of the private prime The public modulus n shall be the product of the private prime
factors p and q: factors p and q:
n = pq . n = pq .
The private exponent shall be a positive integer d such that de-1 is The private exponent shall be a positive integer d such that de-1 is
divisible by both p-1 and q-1. divisible by both p-1 and q-1.
The length of the modulus n in octets is the integer k satisfying The length of the modulus n in octets is the integer k satisfying
2^(8(k-1)) <= n < 2^(8k) . 2^(8(k-1)) <= n < 2^(8k) .
The length k of the modulus must be at least 12 octets to accommodate The length k of the modulus must be at least 12 octets to accommodate
the block formats in this standard (see Section 8). the block formats in this document (see Section 8).
Notes. Notes.
1. The public exponent may be standardized in 1. The public exponent may be standardized in
specific applications. The values 3 and F4 (65537) specific applications. The values 3 and F4 (65537)
may have some practical advantages, as noted in may have some practical advantages, as noted in
X.509 Annex C. X.509 Annex C.
2. Some additional conditions on the choice of primes 2. Some additional conditions on the choice of primes
may well be taken into account in order to deter may well be taken into account in order to deter
factorization of the modulus. These security factorization of the modulus. These security
conditions fall outside the scope of this conditions fall outside the scope of this
standard. The lower bound on the length k is to document. The lower bound on the length k is to
accommodate the block formats, not for security. accommodate the block formats, not for security.
7. Key syntax 7. Key syntax
This section gives the syntax for RSA public and private keys. This section gives the syntax for RSA public and private keys.
7.1 Public-key syntax 7.1 Public-key syntax
An RSA public key shall have ASN.1 type RSAPublicKey: An RSA public key shall have ASN.1 type RSAPublicKey:
RSAPublicKey ::= SEQUENCE { RSAPublicKey ::= SEQUENCE {
modulus INTEGER, -- n modulus INTEGER, -- n
publicExponent INTEGER -- e } publicExponent INTEGER -- e }
PKCS #1: RSA Encryption
(This type is specified in X.509 and is retained here for (This type is specified in X.509 and is retained here for
compatibility.) compatibility.)
The fields of type RSAPublicKey have the following meanings: The fields of type RSAPublicKey have the following meanings:
RFC nnn PKCS #1: RSA Encryption November 1993
o modulus is the modulus n. o modulus is the modulus n.
o publicExponent is the public exponent e. o publicExponent is the public exponent e.
7.2 Private-key syntax 7.2 Private-key syntax
An RSA private key shall have ASN.1 type RSAPrivateKey: An RSA private key shall have ASN.1 type RSAPrivateKey:
RSAPrivateKey ::= SEQUENCE { RSAPrivateKey ::= SEQUENCE {
version Version, version Version,
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prime2 INTEGER, -- q prime2 INTEGER, -- q
exponent1 INTEGER, -- d mod (p-1) exponent1 INTEGER, -- d mod (p-1)
exponent2 INTEGER, -- d mod (q-1) exponent2 INTEGER, -- d mod (q-1)
coefficient INTEGER -- (inverse of q) mod p } coefficient INTEGER -- (inverse of q) mod p }
Version ::= INTEGER Version ::= INTEGER
The fields of type RSAPrivateKey have the following meanings: The fields of type RSAPrivateKey have the following meanings:
o version is the version number, for compatibility o version is the version number, for compatibility
with future revisions of this standard. It shall with future revisions of this document. It shall
be 0 for this version of the standard. be 0 for this version of the document.
o modulus is the modulus n. o modulus is the modulus n.
o publicExponent is the public exponent e. o publicExponent is the public exponent e.
o privateExponent is the private exponent d. o privateExponent is the private exponent d.
o prime1 is the prime factor p of n. o prime1 is the prime factor p of n.
o prime2 is the prime factor q of n. o prime2 is the prime factor q of n.
o exponent1 is d mod (p-1). o exponent1 is d mod (p-1).
o exponent2 is d mod (q-1). o exponent2 is d mod (q-1).
o coefficient is the Chinese Remainder Theorem o coefficient is the Chinese Remainder Theorem
coefficient q-1 mod p. coefficient q-1 mod p.
PKCS #1: RSA Encryption
Notes. Notes.
1. An RSA private key logically consists of only the 1. An RSA private key logically consists of only the
RFC nnn PKCS #1: RSA Encryption November 1993
modulus n and the private exponent d. The presence modulus n and the private exponent d. The presence
of the values p, q, d mod (p-1), d mod (p-1), and of the values p, q, d mod (p-1), d mod (p-1), and
q-1 mod p is intended for efficiency, as q-1 mod p is intended for efficiency, as
Quisquater and Couvreur have shown [QC82]. A Quisquater and Couvreur have shown [QC82]. A
private-key syntax that does not include all the private-key syntax that does not include all the
extra values can be converted readily to the extra values can be converted readily to the
syntax defined here, provided the public key is syntax defined here, provided the public key is
known, according to a result by Miller [Mil76]. known, according to a result by Miller [Mil76].
2. The presence of the public exponent e is intended 2. The presence of the public exponent e is intended
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from the encryption process shall be an octet string ED, the from the encryption process shall be an octet string ED, the
encrypted data. encrypted data.
The length of the data D shall not be more than k-11 octets, which is The length of the data D shall not be more than k-11 octets, which is
positive since the length k of the modulus is at least 12 octets. positive since the length k of the modulus is at least 12 octets.
This limitation guarantees that the length of the padding string PS This limitation guarantees that the length of the padding string PS
is at least eight octets, which is a security condition. is at least eight octets, which is a security condition.
Notes. Notes.
1. In typical applications of this standard to 1. In typical applications of this document to
encrypt content-encryption keys and message encrypt content-encryption keys and message
digests, one would have ||D|| <= 30. Thus the digests, one would have ||D|| <= 30. Thus the
length of the RSA modulus will need to be at least length of the RSA modulus will need to be at least
328 bits (41 octets), which is reasonable and 328 bits (41 octets), which is reasonable and
consistent with security recommendations. consistent with security recommendations.
2. The encryption process does not provide an 2. The encryption process does not provide an
explicit integrity check to facilitate error explicit integrity check to facilitate error
detection should the encrypted data be corrupted detection should the encrypted data be corrupted
PKCS #1: RSA Encryption
in transmission. However, the structure of the in transmission. However, the structure of the
encryption block guarantees that the probability encryption block guarantees that the probability
that corruption is undetected is less than 2-16, that corruption is undetected is less than 2-16,
RFC nnn PKCS #1: RSA Encryption November 1993
which is an upper bound on the probability that a which is an upper bound on the probability that a
random encryption block looks like block type 02. random encryption block looks like block type 02.
3. Application of private-key operations as defined 3. Application of private-key operations as defined
here to data other than an octet string containing here to data other than an octet string containing
a message digest is not recommended and is subject a message digest is not recommended and is subject
to further study. to further study.
4. This standard may be extended to handle data of 4. This document may be extended to handle data of
length more than k-11 octets. length more than k-11 octets.
8.1 Encryption-block formatting 8.1 Encryption-block formatting
A block type BT, a padding string PS, and the data D shall be A block type BT, a padding string PS, and the data D shall be
formatted into an octet string EB, the encryption block. formatted into an octet string EB, the encryption block.
EB = 00 || BT || PS || 00 || D . (1) EB = 00 || BT || PS || 00 || D . (1)
The block type BT shall be a single octet indicating the structure of The block type BT shall be a single octet indicating the structure of
the encryption block. For this version of the standard it shall have the encryption block. For this version of the document it shall have
value 00, 01, or 02. For a private- key operation, the block type value 00, 01, or 02. For a private- key operation, the block type
shall be 00 or 01. For a public-key operation, it shall be 02. shall be 00 or 01. For a public-key operation, it shall be 02.
The padding string PS shall consist of k-3-||D|| octets. For block The padding string PS shall consist of k-3-||D|| octets. For block
type 00, the octets shall have value 00; for block type 01, they type 00, the octets shall have value 00; for block type 01, they
shall have value FF; and for block type 02, they shall be shall have value FF; and for block type 02, they shall be
pseudorandomly generated and nonzero. This makes the length of the pseudorandomly generated and nonzero. This makes the length of the
encryption block EB equal to k. encryption block EB equal to k.
Notes. Notes.
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2. For block type 00, the data D must begin with a 2. For block type 00, the data D must begin with a
nonzero octet or have known length so that the nonzero octet or have known length so that the
encryption block can be parsed unambiguously. For encryption block can be parsed unambiguously. For
block types 01 and 02, the encryption block can be block types 01 and 02, the encryption block can be
parsed unambiguously since the padding string PS parsed unambiguously since the padding string PS
contains no octets with value 00 and the padding contains no octets with value 00 and the padding
string is separated from the data D by an octet string is separated from the data D by an octet
with value 00. with value 00.
PKCS #1: RSA Encryption
3. Block type 01 is recommended for private-key 3. Block type 01 is recommended for private-key
operations. Block type 01 has the property that operations. Block type 01 has the property that
the encryption block, converted to an integer, is the encryption block, converted to an integer, is
RFC nnn PKCS #1: RSA Encryption November 1993
guaranteed to be large, which prevents certain guaranteed to be large, which prevents certain
attacks of the kind proposed by Desmedt and attacks of the kind proposed by Desmedt and
Odlyzko [DO86]. Odlyzko [DO86].
4. Block types 01 and 02 are compatible with PEM RSA 4. Block types 01 and 02 are compatible with PEM RSA
encryption of content-encryption keys and message encryption of content-encryption keys and message
digests as described in RFC 1423. digests as described in RFC 1423.
5. For block type 02, it is recommended that the 5. For block type 02, it is recommended that the
pseudorandom octets be generated independently for pseudorandom octets be generated independently for
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Hastad's results [Has88] motivate this Hastad's results [Has88] motivate this
recommendation. recommendation.
6. For block type 02, the padding string is at least 6. For block type 02, the padding string is at least
eight octets long, which is a security condition eight octets long, which is a security condition
for public-key operations that prevents an for public-key operations that prevents an
attacker from recoving data by trying all possible attacker from recoving data by trying all possible
encryption blocks. For simplicity, the minimum encryption blocks. For simplicity, the minimum
length is the same for block type 01. length is the same for block type 01.
7. This standard may be extended in the future to 7. This document may be extended in the future to
include other block types. include other block types.
8.2 Octet-string-to-integer conversion 8.2 Octet-string-to-integer conversion
The encryption block EB shall be converted to an integer x, the The encryption block EB shall be converted to an integer x, the
integer encryption block. Let EB1, ..., EBk be the octets of EB from integer encryption block. Let EB1, ..., EBk be the octets of EB from
first to last. Then the integer x shall satisfy first to last. Then the integer x shall satisfy
k k
x = SUM 2^(8(k-i)) EBi . (2) x = SUM 2^(8(k-i)) EBi . (2)
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the integer and the last octet of EB has the least significance. the integer and the last octet of EB has the least significance.
Note. The integer encryption block x satisfies 0 <= x < n since EB1 Note. The integer encryption block x satisfies 0 <= x < n since EB1
= 00 and 2^(8(k-1)) <= n. = 00 and 2^(8(k-1)) <= n.
8.3 RSA computation 8.3 RSA computation
The integer encryption block x shall be raised to the power c modulo The integer encryption block x shall be raised to the power c modulo
n to give an integer y, the integer encrypted data. n to give an integer y, the integer encrypted data.
PKCS #1: RSA Encryption
y = x^c mod n, 0 <= y < n . y = x^c mod n, 0 <= y < n .
RFC nnn PKCS #1: RSA Encryption November 1993
This is the classic RSA computation. This is the classic RSA computation.
8.4 Integer-to-octet-string conversion 8.4 Integer-to-octet-string conversion
The integer encrypted data y shall be converted to an octet string ED The integer encrypted data y shall be converted to an octet string ED
of length k, the encrypted data. The encrypted data ED shall satisfy of length k, the encrypted data. The encrypted data ED shall satisfy
k k
y = SUM 2^(8(k-i)) EDi . (3) y = SUM 2^(8(k-i)) EDi . (3)
i = 1 i = 1
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The encrypted data ED shall be converted to an integer y, the integer The encrypted data ED shall be converted to an integer y, the integer
encrypted data, according to Equation (3). encrypted data, according to Equation (3).
It is an error if the integer encrypted data y does not satisfy 0 <= It is an error if the integer encrypted data y does not satisfy 0 <=
y < n. y < n.
9.2 RSA computation 9.2 RSA computation
The integer encrypted data y shall be raised to the power c modulo n The integer encrypted data y shall be raised to the power c modulo n
PKCS #1: RSA Encryption
to give an integer x, the integer encryption block. to give an integer x, the integer encryption block.
RFC nnn PKCS #1: RSA Encryption November 1993
x = y^c mod n, 0 <= x < n . x = y^c mod n, 0 <= x < n .
This is the classic RSA computation. This is the classic RSA computation.
9.3 Integer-to-octet-string conversion 9.3 Integer-to-octet-string conversion
The integer encryption block x shall be converted to an octet string The integer encryption block x shall be converted to an octet string
EB of length k, the encryption block, according to Equation (2). EB of length k, the encryption block, according to Equation (2).
9.4 Encryption-block parsing 9.4 Encryption-block parsing
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(informally, "MD2 with RSA") combines the MD2 message-digest (informally, "MD2 with RSA") combines the MD2 message-digest
algorithm with RSA, the second (informally, "MD4 with RSA") combines algorithm with RSA, the second (informally, "MD4 with RSA") combines
the MD4 message-digest algorithm with RSA, and the third (informally, the MD4 message-digest algorithm with RSA, and the third (informally,
"MD5 with RSA") combines the MD5 message- digest algorithm with RSA. "MD5 with RSA") combines the MD5 message- digest algorithm with RSA.
This section describes the signature process and the verification This section describes the signature process and the verification
process for the two algorithms. The "selected" message-digest process for the two algorithms. The "selected" message-digest
algorithm shall be either MD2 or MD5, depending on the signature algorithm shall be either MD2 or MD5, depending on the signature
algorithm. The signature process shall be performed with an entity's algorithm. The signature process shall be performed with an entity's
private key and the verification process shall be performed with an private key and the verification process shall be performed with an
PKCS #1: RSA Encryption
entity's public key. The signature process transforms an octet string entity's public key. The signature process transforms an octet string
(the message) to a bit string (the signature); the verification (the message) to a bit string (the signature); the verification
RFC nnn PKCS #1: RSA Encryption November 1993
process determines whether a bit string (the signature) is the process determines whether a bit string (the signature) is the
signature of an octet string (the message). signature of an octet string (the message).
Note. The only difference between the signature algorithms defined Note. The only difference between the signature algorithms defined
here and one of the the methods by which signatures (encrypted here and one of the the methods by which signatures (encrypted
message digests) are constructed in PKCS #7 is that signatures here message digests) are constructed in PKCS #7 is that signatures here
are represented here as bit strings, for consistency with the X.509 are represented here as bit strings, for consistency with the X.509
SIGNED macro. In PKCS #7 encrypted message digests are octet strings. SIGNED macro. In PKCS #7 encrypted message digests are octet strings.
10.1 Signature process 10.1 Signature process
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The fields of type DigestInfo have the following meanings: The fields of type DigestInfo have the following meanings:
o digestAlgorithm identifies the message-digest o digestAlgorithm identifies the message-digest
algorithm (and any associated parameters). For algorithm (and any associated parameters). For
this application, it should identify the selected this application, it should identify the selected
message-digest algorithm, MD2, MD4 or MD5. For message-digest algorithm, MD2, MD4 or MD5. For
reference, the relevant object identifiers are the reference, the relevant object identifiers are the
following: following:
md2 OBJECT IDENTIFIER ::= md2 OBJECT IDENTIFIER ::=
PKCS #1: RSA Encryption
{ iso(1) member-body(2) US(840) rsadsi(113549) { iso(1) member-body(2) US(840) rsadsi(113549)
digestAlgorithm(2) 2 } md4 OBJECT IDENTIFIER ::= digestAlgorithm(2) 2 } md4 OBJECT IDENTIFIER ::=
RFC nnn PKCS #1: RSA Encryption November 1993
{ iso(1) member-body(2) US(840) rsadsi(113549) { iso(1) member-body(2) US(840) rsadsi(113549)
digestAlgorithm(2) 4 } md5 OBJECT IDENTIFIER ::= digestAlgorithm(2) 4 } md5 OBJECT IDENTIFIER ::=
{ iso(1) member-body(2) US(840) rsadsi(113549) { iso(1) member-body(2) US(840) rsadsi(113549)
digestAlgorithm(2) 5 } digestAlgorithm(2) 5 }
For these object identifiers, the parameters field For these object identifiers, the parameters field
of the digestAlgorithm value should be NULL. of the digestAlgorithm value should be NULL.
o digest is the result of the message-digesting o digest is the result of the message-digesting
process, i.e., the message digest MD. process, i.e., the message digest MD.
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3. No reason is known that MD4 would not be 3. No reason is known that MD4 would not be
sufficient for very high security digital sufficient for very high security digital
signature schemes, but because MD4 was designed to signature schemes, but because MD4 was designed to
be exceptionally fast, it is "at the edge" in be exceptionally fast, it is "at the edge" in
terms of risking successful cryptanalytic attack. terms of risking successful cryptanalytic attack.
A message-digest algorithm can be considered A message-digest algorithm can be considered
"broken" if someone can find a collision: two "broken" if someone can find a collision: two
messages with the same digest. While collisions messages with the same digest. While collisions
have been found in variants of MD4 with only two have been found in variants of MD4 with only two
PKCS #1: RSA Encryption
digesting "rounds" [Mer90][dBB92], none have been digesting "rounds" [Mer90][dBB92], none have been
found in MD4 itself, which has three rounds. After found in MD4 itself, which has three rounds. After
RFC nnn PKCS #1: RSA Encryption November 1993
further critical review, it may be appropriate to further critical review, it may be appropriate to
consider MD4 for very high security applications. consider MD4 for very high security applications.
MD5, which has four rounds and is proportionally MD5, which has four rounds and is proportionally
slower than MD4, is recommended until the slower than MD4, is recommended until the
completion of MD4's review. The reported completion of MD4's review. The reported
"pseudocollisions" in MD5's internal compression "pseudocollisions" in MD5's internal compression
function [dBB93] do not appear to have any function [dBB93] do not appear to have any
practical impact on MD5's security. practical impact on MD5's security.
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signer's public key; and a bit string S, the signature. The output signer's public key; and a bit string S, the signature. The output
from the verification process shall be an indication of success or from the verification process shall be an indication of success or
failure. failure.
10.2.1 Bit-string-to-octet-string conversion 10.2.1 Bit-string-to-octet-string conversion
The signature S shall be converted into an octet string ED, the The signature S shall be converted into an octet string ED, the
encrypted data. Specifically, assuming that the length in bits of the encrypted data. Specifically, assuming that the length in bits of the
signature S is a multiple of eight, the first bit of the signature signature S is a multiple of eight, the first bit of the signature
shall become the most significant bit of the first octet of the shall become the most significant bit of the first octet of the
PKCS #1: RSA Encryption
encrypted data, and so on through the last bit of the signature, encrypted data, and so on through the last bit of the signature,
which shall become the least significant bit of the last octet of the which shall become the least significant bit of the last octet of the
RFC nnn PKCS #1: RSA Encryption November 1993
encrypted data. encrypted data.
It is an error if the length in bits of the signature S is not a It is an error if the length in bits of the signature S is not a
multiple of eight. multiple of eight.
10.2.2 RSA decryption 10.2.2 RSA decryption
The encrypted data ED shall be decrypted with the signer's RSA public The encrypted data ED shall be decrypted with the signer's RSA public
key as described in Section 8 to give an octet string D, the data. key as described in Section 8 to give an octet string D, the data.
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10.2.4 Message digesting and comparison 10.2.4 Message digesting and comparison
The message M shall be digested with the selected message- digest The message M shall be digested with the selected message- digest
algorithm to give an octet string MD', the comparative message algorithm to give an octet string MD', the comparative message
digest. The verification process shall succeed if the comparative digest. The verification process shall succeed if the comparative
message digest MD' is the same as the message digest MD, and the message digest MD' is the same as the message digest MD, and the
verification process shall fail otherwise. verification process shall fail otherwise.
11. Object identifiers 11. Object identifiers
This standard defines five object identifiers: pkcs-1, rsaEncryption, This document defines five object identifiers: pkcs-1, rsaEncryption,
md2WithRSAEncryption, md4WithRSAEncryption, and md5WithRSAEncryption. md2WithRSAEncryption, md4WithRSAEncryption, and md5WithRSAEncryption.
The object identifier pkcs-1 identifies this standard. The object identifier pkcs-1 identifies this document.
pkcs-1 OBJECT IDENTIFIER ::= pkcs-1 OBJECT IDENTIFIER ::=
{ iso(1) member-body(2) US(840) rsadsi(113549) { iso(1) member-body(2) US(840) rsadsi(113549)
pkcs(1) 1 } pkcs(1) 1 }
The object identifier rsaEncryption identifies RSA public and private The object identifier rsaEncryption identifies RSA public and private
keys as defined in Section 7 and the RSA encryption and decryption keys as defined in Section 7 and the RSA encryption and decryption
PKCS #1: RSA Encryption
processes defined in Sections 8 and 9. processes defined in Sections 8 and 9.
RFC nnn PKCS #1: RSA Encryption November 1993
rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 } rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 }
The rsaEncryption object identifier is intended to be used in the The rsaEncryption object identifier is intended to be used in the
algorithm field of a value of type AlgorithmIdentifier. The algorithm field of a value of type AlgorithmIdentifier. The
parameters field of that type, which has the algorithm-specific parameters field of that type, which has the algorithm-specific
syntax ANY DEFINED BY algorithm, would have ASN.1 type NULL for this syntax ANY DEFINED BY algorithm, would have ASN.1 type NULL for this
algorithm. algorithm.
The object identifiers md2WithRSAEncryption, md4WithRSAEncryption, The object identifiers md2WithRSAEncryption, md4WithRSAEncryption,
md5WithRSAEncryption, identify, respectively, the "MD2 with RSA," md5WithRSAEncryption, identify, respectively, the "MD2 with RSA,"
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These object identifiers are intended to be used in the algorithm These object identifiers are intended to be used in the algorithm
field of a value of type AlgorithmIdentifier. The parameters field of field of a value of type AlgorithmIdentifier. The parameters field of
that type, which has the algorithm- specific syntax ANY DEFINED BY that type, which has the algorithm- specific syntax ANY DEFINED BY
algorithm, would have ASN.1 type NULL for these algorithms. algorithm, would have ASN.1 type NULL for these algorithms.
Note. X.509's object identifier rsa also identifies RSA public keys Note. X.509's object identifier rsa also identifies RSA public keys
as defined in Section 7, but does not identify private keys, and as defined in Section 7, but does not identify private keys, and
identifies different encryption and decryption processes. It is identifies different encryption and decryption processes. It is
expected that some applications will identify public keys by rsa. expected that some applications will identify public keys by rsa.
Such public keys are compatible with this standard; an rsaEncryption Such public keys are compatible with this document; an rsaEncryption
process under an rsa public key is the same as the rsaEncryption process under an rsa public key is the same as the rsaEncryption
process under an rsaEncryption public key. process under an rsaEncryption public key.
Revision history Revision history
Versions 1.0-1.3 Versions 1.0-1.3
Versions 1.0-1.3 were distributed to participants in RSA Data Versions 1.0-1.3 were distributed to participants in RSA Data
Security, Inc.'s Public-Key Cryptography Standards meetings in Security, Inc.'s Public-Key Cryptography Standards meetings in
February and March 1991. February and March 1991.
Version 1.4 Version 1.4
Version 1.4 is part of the June 3, 1991 initial public release of Version 1.4 is part of the June 3, 1991 initial public release of
PKCS. Version 1.4 was published as NIST/OSI Implementors' Workshop PKCS. Version 1.4 was published as NIST/OSI Implementors' Workshop
document SEC-SIG-91-18. document SEC-SIG-91-18.
PKCS #1: RSA Encryption
Version 1.5 Version 1.5
RFC nnn PKCS #1: RSA Encryption November 1993
Version 1.5 incorporates several editorial changes, including updates Version 1.5 incorporates several editorial changes, including updates
to the references and the addition of a revision history. The to the references and the addition of a revision history. The
following substantive changes were made: following substantive changes were made:
o Section 10: "MD4 with RSA" signature and o Section 10: "MD4 with RSA" signature and
verification processes are added. verification processes are added.
o Section 11: md4WithRSAEncryption object identifier o Section 11: md4WithRSAEncryption object identifier
is added. is added.
Supersedes June 3, 1991 version, which was also published as NIST/OSI Supersedes June 3, 1991 version, which was also published as NIST/OSI
Implementors' Workshop document SEC-SIG-91-18. Implementors' Workshop document SEC-SIG-91-18.
Copyright Copyright
Copyright (C) 1991-1993 RSA Laboratories, a division of RSA Data Copyright (c) 1991-1993 RSA Laboratories, a division of RSA Data
Security, Inc. License to copy this document is granted provided that Security, Inc. Any substantial use of the text from this document
it is identified as "RSA Data Security, Inc. Public-Key Cryptography must acknowledge RSA Data Security, Inc. RSA Data Security, Inc.
Standards (PKCS)" in all material mentioning or referencing this requests that all material mentioning or referencing this document
document. identify this as "RSA Data Security, Inc. PKCS #1".
Author's Address Author's Address
RSA Laboratories Burt Kaliski
100 Marine Parkway RSA Laboratories East
Redwood City, CA 94065 USA 20 Crosby Drive
Tel: (415) 595-7703 Bedford, MA 01730
Fax: (415) 595-4126 (617) 687-7000
pkcs-editor@rsa.com burt@rsa.com
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