Crypto Forum Research Group D. McGrew
InternetDraft M. Curcio
Intended status: Informational S. Fluhrer
Expires: July 11, 2019 Cisco Systems
January 7, 2019
HashBased Signatures
draftmcgrewhashsigs15
Abstract
This note describes a digital signature system based on cryptographic
hash functions, following the seminal work in this area of Lamport,
Diffie, Winternitz, and Merkle, as adapted by Leighton and Micali in
1995. It specifies a onetime signature scheme and a general
signature scheme. These systems provide asymmetric authentication
without using large integer mathematics and can achieve a high
security level. They are suitable for compact implementations, are
relatively simple to implement, and naturally resist sidechannel
attacks. Unlike most other signature systems, hashbased signatures
would still be secure even if it proves feasible for an attacker to
build a quantum computer.
This document is a product of the Crypto Forum Research Group (CFRG)
in the IRTF.
Status of This Memo
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This InternetDraft will expire on July 11, 2019.
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Copyright Notice
Copyright (c) 2019 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. CFRG Note on PostQuantum Cryptography . . . . . . . . . 5
1.2. Intellectual Property . . . . . . . . . . . . . . . . . . 6
1.2.1. Disclaimer . . . . . . . . . . . . . . . . . . . . . 6
1.3. Conventions Used In This Document . . . . . . . . . . . . 6
2. Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1. Data Types . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1. Operators . . . . . . . . . . . . . . . . . . . . . . 7
3.1.2. Functions . . . . . . . . . . . . . . . . . . . . . . 8
3.1.3. Strings of wbit elements . . . . . . . . . . . . . . 8
3.2. Typecodes . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3. Notation and Formats . . . . . . . . . . . . . . . . . . 9
4. LMOTS OneTime Signatures . . . . . . . . . . . . . . . . . 12
4.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . 12
4.2. Private Key . . . . . . . . . . . . . . . . . . . . . . . 14
4.3. Public Key . . . . . . . . . . . . . . . . . . . . . . . 15
4.4. Checksum . . . . . . . . . . . . . . . . . . . . . . . . 15
4.5. Signature Generation . . . . . . . . . . . . . . . . . . 16
4.6. Signature Verification . . . . . . . . . . . . . . . . . 17
5. LeightonMicali Signatures . . . . . . . . . . . . . . . . . 19
5.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . 20
5.2. LMS Private Key . . . . . . . . . . . . . . . . . . . . . 21
5.3. LMS Public Key . . . . . . . . . . . . . . . . . . . . . 21
5.4. LMS Signature . . . . . . . . . . . . . . . . . . . . . . 22
5.4.1. LMS Signature Generation . . . . . . . . . . . . . . 24
5.4.2. LMS Signature Verification . . . . . . . . . . . . . 24
6. Hierarchical signatures . . . . . . . . . . . . . . . . . . . 26
6.1. Key Generation . . . . . . . . . . . . . . . . . . . . . 29
6.2. Signature Generation . . . . . . . . . . . . . . . . . . 30
6.3. Signature Verification . . . . . . . . . . . . . . . . . 32
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6.4. Parameter Set Recommendations . . . . . . . . . . . . . . 32
7. Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.1. Security String . . . . . . . . . . . . . . . . . . . . . 35
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 36
9. Security Considerations . . . . . . . . . . . . . . . . . . . 38
9.1. Hash Formats . . . . . . . . . . . . . . . . . . . . . . 39
9.2. Stateful signature algorithm . . . . . . . . . . . . . . 40
9.3. Security of LMOTS Checksum . . . . . . . . . . . . . . . 41
10. Comparison with other work . . . . . . . . . . . . . . . . . 42
11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 42
12. References . . . . . . . . . . . . . . . . . . . . . . . . . 42
12.1. Normative References . . . . . . . . . . . . . . . . . . 42
12.2. Informative References . . . . . . . . . . . . . . . . . 43
Appendix A. Pseudorandom Key Generation . . . . . . . . . . . . 44
Appendix B. LMOTS Parameter Options . . . . . . . . . . . . . . 45
Appendix C. An iterative algorithm for computing an LMS public
key . . . . . . . . . . . . . . . . . . . . . . . . 46
Appendix D. Method for deriving authentication path for a
signature . . . . . . . . . . . . . . . . . . . . . 47
Appendix E. Example Implementation . . . . . . . . . . . . . . . 48
Appendix F. Test Cases . . . . . . . . . . . . . . . . . . . . . 48
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 60
1. Introduction
Onetime signature systems, and general purpose signature systems
built out of onetime signature systems, have been known since 1979
[Merkle79], were well studied in the 1990s [USPTO5432852], and have
benefited from renewed attention in the last decade. The
characteristics of these signature systems are small private and
public keys and fast signature generation and verification, but large
signatures and moderately slow key generation (in comparison with RSA
and ECDSA). Private keys can be made very small by appropriate key
generation, for example, as described in Appendix A. In recent years
there has been interest in these systems because of their post
quantum security and their suitability for compact verifier
implementations.
This note describes the Leighton and Micali adaptation [USPTO5432852]
of the original LamportDiffieWinternitzMerkle onetime signature
system [Merkle79] [C:Merkle87][C:Merkle89a][C:Merkle89b] and general
signature system [Merkle79] with enough specificity to ensure
interoperability between implementations.
A signature system provides asymmetric message authentication. The
key generation algorithm produces a public/private key pair. A
message is signed by a private key, producing a signature, and a
message/signature pair can be verified by a public key. A OneTime
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Signature (OTS) system can be used to sign one message securely, but
will become insecure if more than one is signed with the same public/
private key pair. An Ntime signature system can be used to sign N
or fewer messages securely. A Merkle tree signature scheme is an
Ntime signature system that uses an OTS system as a component.
In the Merkle scheme, a binary tree of height h is used to hold 2^h
OTS key pairs. Each interior node of the tree holds a value which is
the hash of the values of its two children nodes. The public key of
the tree is the value of the root node (a recursive hash of the OTS
public keys), while the private key of the tree is the collection of
all the OTS private keys, together with the index of the next OTS
private key to sign the next message with.
In this note we describe the LeightonMicali Signature (LMS) system,
which is a variant of the Merkle scheme, and a Hierarchical Signature
System (HSS) built on top of it that can efficiently scale to larger
numbers of signatures. In order to support signing a large number of
messages on resource constrained systems, the Merkle tree can be
subdivided into a number of smaller trees. Only the bottommost tree
is used to sign messages, while trees above that are used to sign the
public keys of their children. For example, in the simplest case
with 2 levels with both levels consisting of height h trees, the root
tree is used to sign 2^h trees with 2^h OTS key pairs, and each
second level tree has 2^h OTS key pairs, for a total of 2^(2h) bottom
level key pairs, and so can sign 2^(2h) messages. The advantage of
this scheme is that only the active trees need to be instantiated,
which saves both time (for key generation) and space (for key
storage). On the other hand, using a multilevel signature scheme
inceases the size of the signature, as well as the signature
verification time.
This note is structured as follows. Notes on postquantum
cryptography are discussed in Section 1.1. Intellectual Property
issues are discussed in Section 1.2. The notation used within this
note is defined in Section 3, and the public formats are described in
Section 3.3. The LMOTS signature system is described in Section 4,
and the LMS and HSS Ntime signature systems are described in
Section 5 and Section 6, respectively. Sufficient detail is provided
to ensure interoperability. The rationale for the design decisions
is given in Section 7. The IANA registry for these signature systems
is described in Section 8. Security considerations are presented in
Section 9. Comparison with another hash based signature algorithm
(XMSS) is in Section 10.
This document represents the rough consensus of the CFRG.
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1.1. CFRG Note on PostQuantum Cryptography
All postquantum algorithms documented by the Crypto Forum Research
Group (CFRG) are today considered ready for experimentation and
further engineering development (e.g., to establish the impact of
performance and sizes on IETF protocols). However, at the time of
writing, we do not have significant deployment experience with such
algorithms.
Many of these algorithms come with specific restrictions, e.g.,
change of classical interface or less cryptanalysis of proposed
parameters than established schemes. CFRG has consensus that all
documents describing postquantum technologies include the above
paragraph and a clear additional warning about any specific
restrictions, especially as those might affect use or deployment of
the specific scheme. That guidance may be changed over time via
document updates.
Additionally, for LMS:
CFRG consensus is that we are confident in the cryptographic security
of the signature schemes described in this document against quantum
computers, given the current state of the research community's
knowledge about quantum algorithms. Indeed, we are confident that
the security of a significant part of the Internet could be made
dependent on the signature schemes defined in this document, if
developers take care of the following.
In contrast to traditional signature schemes, the signature schemes
described in this document are stateful, meaning the secret key
changes over time. If a secret key state is used twice, no
cryptographic security guarantees remain. In consequence, it becomes
feasible to forge a signature on a new message. This is a new
property that most developers will not be familiar with and requires
careful handling of secret keys. Developers should not use the
schemes described here except in systems that prevent the reuse of
secret key states.
Note that the fact that the schemes described in this document are
stateful also implies that classical APIs for digital signatures
cannot be used without modification. The API MUST be able to handle
a secret key state; in particular, this means that the API MUST allow
to return an updated secret key state.
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1.2. Intellectual Property
This draft is based on U.S. patent 5,432,852, which was issued over
twenty years ago and is thus expired.
1.2.1. Disclaimer
This document is not intended as legal advice. Readers are advised
to consult with their own legal advisers if they would like a legal
interpretation of their rights.
The IETF policies and processes regarding intellectual property and
patents are outlined in [RFC3979] and [RFC4879] and at
https://datatracker.ietf.org/ipr/about.
1.3. Conventions Used In This Document
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. Interface
The LMS signing algorithm is stateful; it modifies and updates the
private key as a side effect of generating a signature. Once a
particular value of the private key is used to sign one message, it
MUST NOT be used to sign another.
The key generation algorithm takes as input an indication of the
parameters for the signature system. If it is successful, it returns
both a private key and a public key. Otherwise, it returns an
indication of failure.
The signing algorithm takes as input the message to be signed and the
current value of the private key. If successful, it returns a
signature and the next value of the private key, if there is such a
value. After the private key of an Ntime signature system has
signed N messages, the signing algorithm returns the signature and an
indication that there is no next value of the private key that can be
used for signing. If unsuccessful, it returns an indication of
failure.
The verification algorithm takes as input the public key, a message,
and a signature, and returns an indication of whether or not the
signature and message pair is valid.
A message/signature pair is valid if the signature was returned by
the signing algorithm upon input of the message and the private key
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corresponding to the public key; otherwise, the signature and message
pair is not valid with probability very close to one.
3. Notation
3.1. Data Types
Bytes and byte strings are the fundamental data types. A single byte
is denoted as a pair of hexadecimal digits with a leading "0x". A
byte string is an ordered sequence of zero or more bytes and is
denoted as an ordered sequence of hexadecimal characters with a
leading "0x". For example, 0xe534f0 is a byte string with a length
of three. An array of byte strings is an ordered set, indexed
starting at zero, in which all strings have the same length.
Unsigned integers are converted into byte strings by representing
them in network byte order. To make the number of bytes in the
representation explicit, we define the functions u8str(X), u16str(X),
and u32str(X), which take a nonnegative integer X as input and
return one, two, and four byte strings, respectively. We also make
use of the function strTou32(S), which takes a fourbyte string S as
input and returns a nonnegative integer; the identity
u32str(strTou32(S)) = S holds for any fourbyte string S.
3.1.1. Operators
When a and b are real numbers, mathematical operators are defined as
follows:
^ : a ^ b denotes the result of a raised to the power of b
* : a * b denotes the product of a multiplied by b
/ : a / b denotes the quotient of a divided by b
% : a % b denotes the remainder of the integer division of a by b
(with a and b being restricted to integers in this case)
+ : a + b denotes the sum of a and b
 : a  b denotes the difference of a and b
AND : a AND b denotes the bitwise AND of the two nonnegative
integers a and b (represented in binary notation)
The standard order of operations is used when evaluating arithmetic
expressions.
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When B is a byte and i is an integer, then B >> i denotes the logical
rightshift operation by i bit positions. Similarly, B << i denotes
the logical leftshift operation.
If S and T are byte strings, then S  T denotes the concatenation of
S and T. If S and T are equal length byte strings, then S AND T
denotes the bitwise logical and operation.
The ith element in an array A is denoted as A[i].
3.1.2. Functions
If r is a nonnegative real number, then we define the following
functions:
ceil(r) : returns the smallest integer greater than or equal to r
floor(r) : returns the largest integer less than or equal to r
lg(r) : returns the base2 logarithm of r
3.1.3. Strings of wbit elements
If S is a byte string, then byte(S, i) denotes its ith byte, where
the index starts at 0 at the left. Hence, byte(S, 0) is the leftmost
byte of S, byte(S, 1) is the second byte at the left and (assuming S
is n bytes long) byte(S, n1) is the rightmost byte of S. In
addition, bytes(S, i, j) denotes the range of bytes from the ith to
the jth byte, inclusive. For example, if S = 0x02040608, then
byte(S, 0) is 0x02 and bytes(S, 1, 2) is 0x0406.
A byte string can be considered to be a string of wbit unsigned
integers; the correspondence is defined by the function coef(S, i, w)
as follows:
If S is a string, i is a positive integer, and w is a member of the
set { 1, 2, 4, 8 }, then coef(S, i, w) is the ith, wbit value, if S
is interpreted as a sequence of wbit values. That is,
coef(S, i, w) = (2^w  1) AND
( byte(S, floor(i * w / 8)) >>
(8  (w * (i % (8 / w)) + w)) )
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For example, if S is the string 0x1234, then coef(S, 7, 1) is 0 and
coef(S, 0, 4) is 1.
S (represented as bits)
+++++++++++++++++
 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0
+++++++++++++++++
^

coef(S, 7, 1)
S (represented as fourbit values)
+++++
 1  2  3  4 
+++++
^

coef(S, 0, 4)
The return value of coef is an unsigned integer. If i is larger than
the number of wbit values in S, then coef(S, i, w) is undefined, and
an attempt to compute that value MUST raise an error.
3.2. Typecodes
A typecode is an unsigned integer that is associated with a
particular data format. The format of the LMOTS, LMS, and HSS
signatures and public keys all begin with a typecode that indicates
the precise details used in that format. These typecodes are
represented as fourbyte unsigned integers in network byte order;
equivalently, they are XDR enumerations (see Section 3.3).
3.3. Notation and Formats
The signature and public key formats are formally defined using the
External Data Representation (XDR) [RFC4506] in order to provide an
unambiguous, machine readable definition. For clarity, we also
include a private key format as well, though consistency is not
needed for interoperability and an implementation MAY use any private
key format. Though XDR is used, these formats are simple and easy to
parse without any special tools. An illustration of the layout of
data in these objects is provided below. The definitions are as
follows:
/* onetime signatures */
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enum lmots_algorithm_type {
lmots_reserved = 0,
lmots_sha256_n32_w1 = 1,
lmots_sha256_n32_w2 = 2,
lmots_sha256_n32_w4 = 3,
lmots_sha256_n32_w8 = 4
};
typedef opaque bytestring32[32];
struct lmots_signature_n32_p265 {
bytestring32 C;
bytestring32 y[265];
};
struct lmots_signature_n32_p133 {
bytestring32 C;
bytestring32 y[133];
};
struct lmots_signature_n32_p67 {
bytestring32 C;
bytestring32 y[67];
};
struct lmots_signature_n32_p34 {
bytestring32 C;
bytestring32 y[34];
};
union lmots_signature switch (lmots_algorithm_type type) {
case lmots_sha256_n32_w1:
lmots_signature_n32_p265 sig_n32_p265;
case lmots_sha256_n32_w2:
lmots_signature_n32_p133 sig_n32_p133;
case lmots_sha256_n32_w4:
lmots_signature_n32_p67 sig_n32_p67;
case lmots_sha256_n32_w8:
lmots_signature_n32_p34 sig_n32_p34;
default:
void; /* error condition */
};
/* hash based signatures (hbs) */
enum lms_algorithm_type {
lms_reserved = 0,
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lms_sha256_n32_h5 = 5,
lms_sha256_n32_h10 = 6,
lms_sha256_n32_h15 = 7,
lms_sha256_n32_h20 = 8,
lms_sha256_n32_h25 = 9,
};
/* leightonmicali signatures (lms) */
union lms_path switch (lms_algorithm_type type) {
case lms_sha256_n32_h5:
bytestring32 path_n32_h5[5];
case lms_sha256_n32_h10:
bytestring32 path_n32_h10[10];
case lms_sha256_n32_h15:
bytestring32 path_n32_h15[15];
case lms_sha256_n32_h20:
bytestring32 path_n32_h20[20];
case lms_sha256_n32_h25:
bytestring32 path_n32_h25[25];
default:
void; /* error condition */
};
struct lms_signature {
unsigned int q;
lmots_signature lmots_sig;
lms_path nodes;
};
struct lms_key_n32 {
lmots_algorithm_type ots_alg_type;
opaque I[16];
opaque K[32];
};
union lms_public_key switch (lms_algorithm_type type) {
case lms_sha256_n32_h5:
case lms_sha256_n32_h10:
case lms_sha256_n32_h15:
case lms_sha256_n32_h20:
case lms_sha256_n32_h25:
lms_key_n32 z_n32;
default:
void; /* error condition */
};
/* hierarchical signature system (hss) */
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struct hss_public_key {
unsigned int L;
lms_public_key pub;
};
struct signed_public_key {
lms_signature sig;
lms_public_key pub;
}
struct hss_signature {
signed_public_key signed_keys<7>;
lms_signature sig_of_message;
};
4. LMOTS OneTime Signatures
This section defines LMOTS signatures. The signature is used to
validate the authenticity of a message by associating a secret
private key with a shared public key. These are onetime signatures;
each private key MUST be used at most one time to sign any given
message.
As part of the signing process, a digest of the original message is
computed using the cryptographic hash function H (see Section 4.1),
and the resulting digest is signed.
In order to facilitate its use in an Ntime signature system, the LM
OTS key generation, signing, and verification algorithms all take as
input parameters I and q. The parameter I is a 16 byte string, which
indicates which Merkle tree this LMOTS is used with. The parameter
q is a 32 bit integer which indicates the leaf of the Merkle tree
where the OTS public key appears. These parameters are used as part
of the security string, as listed in Section 7.1. When the LMOTS
signature system is used outside of an Ntime signature system, the
value I MAY be used to differentiate this one time signatures from
others; however the value q MUST be set to the allzero value.
4.1. Parameters
The signature system uses the parameters n and w, which are both
positive integers. The algorithm description also makes use of the
internal parameters p and ls, which are dependent on n and w. These
parameters are summarized as follows:
n : the number of bytes of the output of the hash function
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w : the width (in bits) of the Winternitz coefficients; that is,
the number of bits from the hash or checksum that is used with a
single Winternitz chain. It is a member of the set { 1, 2, 4, 8 }
p : the number of nbyte string elements that make up the LMOTS
signature. This is a function of n and w; the values for the
defined parameter sets are lited in Table 1; it can also be
computed by the algorithm given in Appendix B.
ls : the number of leftshift bits used in the checksum function
Cksm (defined in Section 4.4)
H : a secondpreimageresistant cryptographic hash function that
accepts byte strings of any length, and returns an nbyte string
For more background on the cryptographic security requirements on H,
see the Section 9.
The value of n is determined by the hash function selected for use as
part of the LMOTS algorithm; the choice of this value has a strong
effect on the security of the system. The parameter w determines the
length of the Winternitz chains computed as a part of the OTS
signature (which involve 2^w1 invocations of the hash function); it
has little effect on security. Increasing w will shorten the
signature, but at a cost of a larger computation to generate and
verify a signature. The values of p and ls are dependent on the
choices of the parameters n and w, as described in Appendix B. A
table illustrating various combinations of n, w, p and ls, along with
the resulting signature length, is provided in Table 1.
The value of w describes a space/time tradeoff; increasing the value
of w will cause the signature to shrink (by decreasing the value of
p) while increasing the amount of time needed to perform operations
with it (generate the public key, generate and verify the signature);
in general, the LMOTS signature is 4+n*(p+1) bytes long, and public
key generation will take p*(2^w1)+1 hash computations (and signature
generation and verification will take approximately half that on
average).
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++++++++
 Parameter Set Name  H  n  w  p  ls  sig_len 
++++++++
 LMOTS_SHA256_N32_W1  SHA256  32  1  265  7  8516 
       
 LMOTS_SHA256_N32_W2  SHA256  32  2  133  6  4292 
       
 LMOTS_SHA256_N32_W4  SHA256  32  4  67  4  2180 
       
 LMOTS_SHA256_N32_W8  SHA256  32  8  34  0  1124 
++++++++
Table 1
Here SHA256 denotes the SHA256 hash fucntion defined in NIST
standard [FIPS180].
4.2. Private Key
The format of the LMOTS private key is an internal matter to the
implementation, and this document does not attempt to define it. One
possibility is that the private key may consist of a typecode
indicating the particular LMOTS algorithm, an array x[] containing p
nbyte strings, and the 16byte string I and the 4 byte string q.
This private key MUST be used to sign (at most) one message. The
following algorithm shows pseudocode for generating a private key.
Algorithm 0: Generating a Private Key
1. retrieve the values of q and I (the 16byte identifier of the
LMS public/private keypair) from the LMS tree that this LMOTS
private key will be used with
2. set type to the typecode of the algorithm
3. set n and p according to the typecode and Table 1
4. compute the array x as follows:
for ( i = 0; i < p; i = i + 1 ) {
set x[i] to a uniformly random nbyte string
}
5. return u32str(type)  I  u32str(q)  x[0]  x[1]  ...
 x[p1]
An implementation MAY use a pseudorandom method to compute x[i], as
suggested in [Merkle79], page 46. The details of the pseudorandom
method do not affect interoperability, but the cryptographic strength
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MUST match that of the LMOTS algorithm. Appendix A provides an
example of a pseudorandom method for computing the LMOTS private
key.
4.3. Public Key
The LMOTS public key is generated from the private key by
iteratively applying the function H to each individual element of x,
for 2^w  1 iterations, then hashing all of the resulting values.
The public key is generated from the private key using the following
algorithm, or any equivalent process.
Algorithm 1: Generating a One Time Signature Public Key From a
Private Key
1. set type to the typecode of the algorithm
2. set the integers n, p, and w according to the typecode and Table 1
3. determine x, I and q from the private key
4. compute the string K as follows:
for ( i = 0; i < p; i = i + 1 ) {
tmp = x[i]
for ( j = 0; j < 2^w  1; j = j + 1 ) {
tmp = H(I  u32str(q)  u16str(i)  u8str(j)  tmp)
}
y[i] = tmp
}
K = H(I  u32str(q)  u16str(D_PBLC)  y[0]  ...  y[p1])
5. return u32str(type)  I  u32str(q)  K
where D_PBLC is the fixed two byte value 0x8080, which is used to
distinguish the last hash from every other hash in this system.
The public key is the value returned by Algorithm 1.
4.4. Checksum
A checksum is used to ensure that any forgery attempt that
manipulates the elements of an existing signature will be detected.
This checksum is needed because an attacker can freely advance any of
the Winternitz chains. That is, if this checksum were not present,
then an attacker who could find a hash that has every digit larger
than the valid hash could replace it (and adjust the Winternitz
chains). The security property that it provides is detailed in
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Section 9. The checksum function Cksm is defined as follows, where S
denotes the nbyte string that is input to that function, and the
value sum is a 16bit unsigned integer:
Algorithm 2: Checksum Calculation
sum = 0
for ( i = 0; i < (n*8/w); i = i + 1 ) {
sum = sum + (2^w  1)  coef(S, i, w)
}
return (sum << ls)
ls is the parameter that shifts the significant bits of the checksum
into the positions that will actually be used by the coef function
when encoding the digits of the checksum. The actual ls parameter is
a function of the n and w parameters; the values for the currently
defined parameter sets is shown in table 1. It is calculated by the
algorithm given in Appendix B.
Because of the leftshift operation, the rightmost bits of the result
of Cksm will often be zeros. Due to the value of p, these bits will
not be used during signature generation or verification.
4.5. Signature Generation
The LMOTS signature of a message is generated by first prepending
the LMS key identifier I, the LMS leaf identifier q, the value D_MESG
(0x8181) and the randomizer C to the message, then computing the
hash, and then concatenating the checksum of the hash to the hash
itself, then considering the resulting value as a sequence of wbit
values, and using each of the wbit values to determine the number of
times to apply the function H to the corresponding element of the
private key. The outputs of the function H are concatenated together
and returned as the signature. The pseudocode for this procedure is
shown below.
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Algorithm 3: Generating a One Time Signature From a Private Key and a
Message
1. set type to the typecode of the algorithm
2. set n, p, and w according to the typecode and Table 1
3. determine x, I and q from the private key
4. set C to a uniformly random nbyte string
5. compute the array y as follows:
Q = H(I  u32str(q)  u16str(D_MESG)  C  message)
for ( i = 0; i < p; i = i + 1 ) {
a = coef(Q  Cksm(Q), i, w)
tmp = x[i]
for ( j = 0; j < a; j = j + 1 ) {
tmp = H(I  u32str(q)  u16str(i)  u8str(j)  tmp)
}
y[i] = tmp
}
6. return u32str(type)  C  y[0]  ...  y[p1]
Note that this algorithm results in a signature whose elements are
intermediate values of the elements computed by the public key
algorithm in Section 4.3.
The signature is the string returned by Algorithm 3. Section 3.3
specifies the typecode and more formally defines the encoding and
decoding of the string.
4.6. Signature Verification
In order to verify a message with its signature (an array of nbyte
strings, denoted as y), the receiver must "complete" the chain of
iterations of H using the wbit coefficients of the string resulting
from the concatenation of the message hash and its checksum. This
computation should result in a value that matches the provided public
key.
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Algorithm 4a: Verifying a Signature and Message Using a Public Key
1. if the public key is not at least four bytes long,
return INVALID
2. parse pubtype, I, q, and K from the public key as follows:
a. pubtype = strTou32(first 4 bytes of public key)
b. set n according to the pubkey and Table 1; if the public key
is not exactly 24 + n bytes long, return INVALID
c. I = next 16 bytes of public key
d. q = strTou32(next 4 bytes of public key)
e. K = next n bytes of public key
3. compute the public key candidate Kc from the signature,
message, pubtype and the identifiers I and q obtained from the
public key, using Algorithm 4b. If Algorithm 4b returns
INVALID, then return INVALID.
4. if Kc is equal to K, return VALID; otherwise, return INVALID
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Algorithm 4b: Computing a Public Key Candidate Kc from a Signature,
Message, Signature Typecode pubtype, and identifiers I, q
1. if the signature is not at least four bytes long, return INVALID
2. parse sigtype, C, and y from the signature as follows:
a. sigtype = strTou32(first 4 bytes of signature)
b. if sigtype is not equal to pubtype, return INVALID
c. set n and p according to the pubtype and Table 1; if the
signature is not exactly 4 + n * (p+1) bytes long, return INVALID
d. C = next n bytes of signature
e. y[0] = next n bytes of signature
y[1] = next n bytes of signature
...
y[p1] = next n bytes of signature
3. compute the string Kc as follows
Q = H(I  u32str(q)  u16str(D_MESG)  C  message)
for ( i = 0; i < p; i = i + 1 ) {
a = coef(Q  Cksm(Q), i, w)
tmp = y[i]
for ( j = a; j < 2^w  1; j = j + 1 ) {
tmp = H(I  u32str(q)  u16str(i)  u8str(j)  tmp)
}
z[i] = tmp
}
Kc = H(I  u32str(q)  u16str(D_PBLC) 
z[0]  z[1]  ...  z[p1])
4. return Kc
5. LeightonMicali Signatures
The LeightonMicali Signature (LMS) method can sign a potentially
large but fixed number of messages. An LMS system uses two
cryptographic components: a onetime signature method and a hash
function. Each LMS public/private key pair is associated with a
perfect binary tree, each node of which contains an mbyte value,
where m is the output length of the hash function. Each leaf of the
tree contains the value of the public key of an LMOTS public/private
key pair. The value contained by the root of the tree is the LMS
public key. Each interior node is computed by applying the hash
function to the concatenation of the values of its children nodes.
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Each node of the tree is associated with a node number, an unsigned
integer that is denoted as node_num in the algorithms below, which is
computed as follows. The root node has node number 1; for each node
with node number N < 2^h (where h is the height of the tree), its
left child has node number 2*N, while its right child has node number
2*N+1. The result of this is that each node within the tree will
have a unique node number, and the leaves will have node numbers 2^h,
(2^h)+1, (2^h)+2, ..., (2^h)+(2^h)1. In general, the jth node at
level i has node number 2^i + j. The node number can conveniently be
computed when it is needed in the LMS algorithms, as described in
those algorithms.
5.1. Parameters
An LMS system has the following parameters:
h : the height of the tree, and
m : the number of bytes associated with each node.
H : a secondpreimageresistant cryptographic hash function that
accepts byte strings of any length, and returns an mbyte string.
There are 2^h leaves in the tree.
The overall strength of the LMS signatures is governed by the weaker
of the hash function used within the LMOTS and the hash function
used within the LMS system. In order to minimize the risk, these two
hash functions SHOULD be the same (so that an attacker could not take
advantage of the weaker hash function choice).
+++++
 Name  H  m  h 
+++++
 LMS_SHA256_M32_H5  SHA256  32  5 
    
 LMS_SHA256_M32_H10  SHA256  32  10 
    
 LMS_SHA256_M32_H15  SHA256  32  15 
    
 LMS_SHA256_M32_H20  SHA256  32  20 
    
 LMS_SHA256_M32_H25  SHA256  32  25 
+++++
Table 2
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5.2. LMS Private Key
The format of the LMS private key is an internal matter to the
implementation, and this document does not attempt to define it. One
possibility is that it may consist of an array OTS_PRIV[] of 2^h LM
OTS private keys, and the leaf number q of the next LMOTS private
key that has not yet been used. The qth element of OTS_PRIV[] is
generated using Algorithm 0 with the identifiers I, q. The leaf
number q is initialized to zero when the LMS private key is created.
The process is as follows:
Algorithm 5: Computing an LMS Private Key.
1. determine h and m from the typecode and Table 2.
2. set I to a uniformly random 16byte string
3. compute the array OTS_PRIV[] as follows:
for ( q = 0; q < 2^h; q = q + 1) {
OTS_PRIV[q] = LMOTS private key with identifiers I, q
}
4. q = 0
An LMS private key MAY be generated pseudorandomly from a secret
value, in which case the secret value MUST be at least m bytes long,
be uniformly random, and MUST NOT be used for any other purpose than
the generation of the LMS private key. The details of how this
process is done do not affect interoperability; that is, the public
key verification operation is independent of these details.
Appendix A provides an example of a pseudorandom method for computing
an LMS private key.
The signature generation logic uses q as the next leaf to use, hence
step 4 starts it off at the leftmost one. Because the signature
proces increments q after the signature operation, the first
signature will have q=0.
5.3. LMS Public Key
An LMS public key is defined as follows, where we denote the public
key final hash value (namely, the K value computed in Algorithm 1)
associated with the ith LMOTS private key as OTS_PUB_HASH[i], with
i ranging from 0 to (2^h)1. Each instance of an LMS public/private
key pair is associated with a balanced binary tree, and the nodes of
that tree are indexed from 1 to 2^(h+1)1. Each node is associated
with an mbyte string, and the string for the rth node is denoted as
T[r] and is defined as
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if r >= 2^h:
H(Iu32str(r)u16str(D_LEAF)OTS_PUB_HASH[r2^h])
else
H(Iu32str(r)u16str(D_INTR)T[2*r]T[2*r+1])
where D_LEAF is the fixed two byte value 0x8282, and D_INTR is the
fixed two byte value 0x8383, both of which are used to distinguish
this hash from every other hash in this system.
When we have r >= 2^h, then we are processing a leaf node (and thus
hashing only a single LMOTS public key). When we have r < 2^h, then
we are processing an internal node, that is, a node with two child
nodes that we need to combine.
The LMS public key is the string
u32str(type)  u32str(otstype)  I  T[1]
Section 3.3 specifies the format of the type variable. The value
otstype is the parameter set for the LMOTS public/private keypairs
used. The value I is the private key identifier, and is the value
used for all computations for the same LMS tree. The value T[1] can
be computed via recursive application of the above equation, or by
any equivalent method. An iterative procedure is outlined in
Appendix C.
5.4. LMS Signature
An LMS signature consists of
the number q of the leaf associated with the LMOTS signature, as
a fourbyte unsigned integer in network byte order,
an LMOTS signature,
a typecode indicating the particular LMS algorithm,
an array of h mbyte values that is associated with the path
through the tree from the leaf associated with the LMOTS
signature to the root.
Symbolically, the signature can be represented as
u32str(q)  lmots_signature  u32str(type) 
path[0]  path[1]  path[2]  ...  path[h1]
Section 3.3 specifies the typecode and more formally defines the
format. The array for a tree with height h will have h values and
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contains the values of the siblings of (that is, is adjacent to) the
nodes on the path from the leaf to the root, where the sibling to
node A is the other node which shares node A's parent. In the
signature, 0 is counted from the bottom level of the tree, and so
path[0] is the value of the node adjacent to leaf node q; path[1] is
the second level node that is adjacent to leaf node q's parent, and
so up the tree until we get to path[h1], which is the value of the
nexttothetop level node that leaf node q does not reside in.
Below is a simple example of the authentication path for h=3 and q=2.
The leaf marked OTS is the one time signature which is used to sign
the actual message. The nodes on the path from the OTS public key to
the root are marked with a *, while the nodes that are used within
the path array are marked with a **. The values in the path array
are those nodes which are siblings of the nodes on the path; path[0]
is the leaf** node that is adjacent to the OTS public key (which is
the start of the path); path[1] is the T[4]** node which is the
sibling of the second node T[5]* on the path, and path[2] is the
T[3]** node which is the sibling of the third node T[2]* on the path.
Root


 
T[2]* T[3]**
 
 
   
T[4]** T[5]* T[6] T[7]
   
   
       
leaf leaf OTS leaf** leaf leaf leaf leaf
The idea behind this authentication path is that it allows us to
validate the OTS hash with using h path array values and hash
computations. What the verifier does is recompute the hashes up the
path; first, he hashes the given OTS and path[0] value, giving a
tentative T[5]' value. Then, he hashes his path[1] and tentative
T[5]' value to get a tentative T[2]' value. Then, he hashes that and
the path[2] value to get a tentative Root' value. If that value is
the known public key of the Merkle tree, then we can assume that the
value T[2]' he got was the correct T[2] value in the original tree,
and so the T[5]' value he got was the correct T[5] value in the
original tree, and so the OTS public key is the same as in the
original, and hence is correct.
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5.4.1. LMS Signature Generation
To compute the LMS signature of a message with an LMS private key,
the signer first computes the LMOTS signature of the message using
the leaf number of the next unused LMOTS private key. The leaf
number q in the signature is set to the leaf number of the LMS
private key that was used in the signature. Before releasing the
signature, the leaf number q in the LMS private key MUST be
incremented, to prevent the LMOTS private key from being used again.
If the LMS private key is maintained in nonvolatile memory, then the
implementation MUST ensure that the incremented value has been stored
before releasing the signature. The issue this tries to prevent is a
scenario where a) we generate a signature, using one LMOTS private
key, and release it to the application, b) before we update the
nonvolatile memory, we crash, and c) we reboot, and generate a second
signature using the same LMOTS private key; with two different
signatures using the same LMOTS private key, someone could
potentially generate a forged signature of a third message.
The array of node values in the signature MAY be computed in any way.
There are many potential time/storage tradeoffs that can be applied.
The fastest alternative is to store all of the nodes of the tree and
set the array in the signature by copying them; pseudocode to do so
appears in Appendix D. The least storage intensive alternative is to
recompute all of the nodes for each signature. Note that the details
of this procedure are not important for interoperability; it is not
necessary to know any of these details in order to perform the
signature verification operation. The internal nodes of the tree
need not be kept secret, and thus a nodecaching scheme that stores
only internal nodes can sidestep the need for strong protections.
Several useful time/storage tradeoffs are described in the 'Small
Memory LM Schemes' section of [USPTO5432852].
5.4.2. LMS Signature Verification
An LMS signature is verified by first using the LMOTS signature
verification algorithm (Algorithm 4b) to compute the LMOTS public
key from the LMOTS signature and the message. The value of that
public key is then assigned to the associated leaf of the LMS tree,
then the root of the tree is computed from the leaf value and the
array path[] as described in Algorithm 6 below. If the root value
matches the public key, then the signature is valid; otherwise, the
signature fails.
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Algorithm 6: LMS Signature Verification
1. if the public key is not at least eight bytes long, return
INVALID
2. parse pubtype, I, and T[1] from the public key as follows:
a. pubtype = strTou32(first 4 bytes of public key)
b. ots_typecode = strTou32(next 4 bytes of public key)
c. set m according to pubtype, based on Table 2
d. if the public key is not exactly 24 + m bytes
long, return INVALID
e. I = next 16 bytes of the public key
f. T[1] = next m bytes of the public key
3. compute the LMS Public Key Candidate Tc from the signature,
message, identifier, pubtype and ots_typecode using Algorithm 6a.
4. if Tc is equal to T[1], return VALID; otherwise, return INVALID
Algorithm 6a: Computing an LMS Public Key Candidate from a Signature,
Message, Identifier, and algorithm typecode
1. if the signature is not at least eight bytes long, return INVALID
2. parse sigtype, q, lmots_signature, and path from the signature as
follows:
a. q = strTou32(first 4 bytes of signature)
b. otssigtype = strTou32(next 4 bytes of signature)
c. if otssigtype is not the OTS typecode from the public key,
return INVALID
d. set n, p according to otssigtype and Table 1; if the
signature is not at least 12 + n * (p + 1) bytes long,
return INVALID
e. lmots_signature = bytes 4 through 7 + n * (p + 1)
of signature
f. sigtype = strTou32(bytes 8 + n * (p + 1)) through
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11 + n * (p + 1) of signature)
f. if sigtype is not the LM typecode from the public key,
return INVALID
g. set m, h according to sigtype and Table 2
h. if q >= 2^h or the signature is not exactly
12 + n * (p + 1) + m * h bytes long,
return INVALID
i. set path as follows:
path[0] = next m bytes of signature
path[1] = next m bytes of signature
...
path[h1] = next m bytes of signature
3. Kc = candidate public key computed by applying Algorithm 4b
to the signature lmots_signature, the message, and the
identifiers I, q
4. compute the candidate LMS root value Tc as follows:
node_num = 2^h + q
tmp = H(I  u32str(node_num)  u16str(D_LEAF)  Kc)
i = 0
while (node_num > 1) {
if (node_num is odd):
tmp = H(Iu32str(node_num/2)u16str(D_INTR)path[i]tmp)
else:
tmp = H(Iu32str(node_num/2)u16str(D_INTR)tmppath[i])
node_num = node_num/2
i = i + 1
}
Tc = tmp
5. return Tc
6. Hierarchical signatures
In scenarios where it is necessary to minimize the time taken by the
public key generation process, a Hierarchical Ntime Signature System
(HSS) can be used. This hierarchical scheme, which we describe in
this section, uses the LMS scheme as a component. In HSS, we have a
sequence of L LMS trees, where the public key for the first LMS tree
is included in the public key of the HSS system, and where each LMS
private key signs the next LMS public key, and where the last LMS
private key signs the actual message. For example, if we have a
three level hierarchy (L=3), then to sign a message, we would have:
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The first LMS private key (level 0) signs a level 1 LMS public
key.
The second LMS private key (level 1) signs a level 2 LMS public
key.
The third LMS private key (level 2) signs the message.
The root of the level 0 LMS tree is contained in the HSS public key.
To verify the LMS signature, we would verify all the signatures:
We would verify that the level 1 LMS public key is correctly
signed by the level 0 signature.
We would verify that the level 2 LMS public key is correctly
signed by the level 1 signature.
We would verify that the message is correctly signed by the level
2 signature.
We would accept the HSS signature only if all the signatures
validated.
During the signature generation process, we sign messages with the
lowest (level L1) LMS tree. Once we have used all the leafs in that
tree to sign messages, we would discard it, generate a fresh LMS
tree, and sign it with the next (level L2) LMS tree (and when that
is used up, recursively generate and sign a fresh level L2 LMS
tree).
HSS, in essence, utilizes a tree of LMS trees. There is a single LMS
tree at level 0 (the root). Each LMS tree (actually, the private key
corresponding to the LMS tree) at level i is used to sign 2^h objects
(where h is the height of trees at level i). If i < L1, then each
object will be another LMS tree (actually, the public key) at level
i+1; if i = L1, we've reached the bottom of the HSS tree, and so
each object is a message from the application. The HSS public key
contains the public key of the LMS tree at the root, and an HSS
signature is associated with a path from the root of the HSS tree to
the leaf.
Compared to LMS, HSS has a much reduced public key generation time,
as only the root tree needs to be generated prior to the distribution
of the HSS public key. For example, a L=3 tree (with h=10 at each
level) would have 1 level 0 LMS tree, 2^10 level 1 LMS trees (with
each such level 1 public key signed by one of the 1024 level 0 OTS
public keys), and 2^20 level 2 LMS trees. Only 1024 OTS public keys
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need to be computed to generate the HSS public key (as you need to
compute only the level 0 LMS tree to compute that value; you can, of
course, decide to compute the initial level 1 and level 2 LMS trees).
And, the 2^20 level 2 LMS trees can jointly sign a total of over a
billion messages. In contrast, a single LMS tree that could sign a
billion messages would require a billion OTS public keys to be
computed first (even if h=30 were allowed in a supported parameter
set).
Each LMS tree within the hierarchy is associated with a distinct LMS
public key, private key, signature, and identifier. The number of
levels is denoted L, and is between one and eight, inclusive. The
following notation is used, where i is an integer between 0 and L1
inclusive, and the root of the hierarchy is level 0:
prv[i] is the current LMS private key of the ith level.
pub[i] is the current LMS public key of the ith level, as
described in Section 5.3.
sig[i] is the LMS signature of public key pub[i+1] generated using
the private key prv[i].
It is expected that the above arrays are maintained for the course of
the HSS key. The contents of the prv[] array MUST be kept private;
the pub[] and sig[] array may be revealed, should the implementation
find that convienent.
In this section, we say that an Ntime private key is exhausted when
it has generated N signatures, and thus it can no longer be used for
signing.
For i > 0, the values prv[i], pub[i] and (for all values of i) sig[i]
will be updated over time, as private keys are exhausted, and
replaced by newer keys.
When these keys pairs are updated (or initially generated before the
first message is signed), then the LMS key generation processes
outlined in sections Section 5.2 and Section 5.3 are performed. If
the generated key pairs are for level i of the HSS hierarchy, then we
store the public key in pub[i] and the private key in prv[i]. In
addition, if i > 0, then we sign the generated public key with the
LMS private key at level i1, placing the signature into sig[i1].
When the LMS key pair are generated, the key pair and the
corresponding identifier MUST be generated indepently of all other
keypairs.
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HSS allows L=1, in which case the HSS public key and signature
formats are essentially the LMS public key and signature formats,
prepended by a fixed field. Since HSS with L=1 has very little
overhead compared to LMS, all implementations MUST support HSS in
order to maximize interoperability.
We specifically allow different LMS levels to use different parameter
sets. For example, the 0th LMS public key (the root) may use the
LMS_SHA256_M32_H15 parameter set, while the 1th public key may use
LMS_SHA256_M32_H10. There are practical reasons to allow this; for
one, the signer may decide to store parts of the 0th LMS tree (that
it needs to construct while computing the public key) to accelerate
later operations. As the 0th tree is never updated, these internal
nodes will never need to be recomputed. In addition, during the
signature generation operation, almost all the operations involved
with updating the authentication path occurs with the bottom (L1th)
LMS public key; hence it may be useful to make the tree that
implements that to be shorter.
A close reading of the HSS verification pseudocode would show that it
would allow the parameters of the nontop LMS public keys to change
over time; for example, the signer might initially have the 1th LMS
public key to use LMS_SHA256_M32_H10, but when that tree is
exhausted, the signer might replace it with LMS_SHA256_M32_H15 LMS
public key. While this would work with the example verification
pseudocode, the signer MUST NOT change the parameter sets for a
specific level. This prohibition is to support verifiers that may
keep state over the course of several signature verifications.
6.1. Key Generation
The public key of the HSS scheme consists of the number of levels L,
followed by pub[0], the public key of the top level.
The HSS private key consists of prv[0], ... , prv[L1], along with
the associated pub[0], ... pub[L1] and sig[0], ..., sig[L2] values.
As stated earlier, the values of the pub[] and sig[] arrays need not
be kept secret, but may be revealed. The value of pub[0] does not
change (and, except for the index q, the value of prv[0] need not
change), though the values of pub[i] and prv[i] are dynamic for i >
0, and are changed by the signature generation algorithm.
During the key generation, the public and private keys are
initialized. Here is some pseudocode that explains the key
generation logic
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Algorithm 7: Generating an HSS keypair
1. generate an LMS key pair, as specified in sections 5.2 and 5.3,
placing the private key into priv[0], and the public key into
pub[0]
2. for i = 1 to L1 do {
generate an LMS key pair, placing the private key into priv[i]
and the public key into pub[i]
sig[i1] = lms_signature( pub[i], priv[i1] )
}
3. return u32str(L)  pub[0] as the public key, and the priv[],
pub[] and sig[] arrays as the private key
In the above algorithm, each LMS public/private keypair generated
MUST be generated indepedently.
Note that the value of the public key does not depend on the
execution of step 2. As a result, an implementation may decide to
delay step 2 until later, for example, during the initial signature
generation operation.
6.2. Signature Generation
To sign a message using an HSS keypair, the following steps are
performed:
If prv[L1] is exhausted, then determine the smallest integer d
such that all of the private keys prv[d], prv[d+1], ... , prv[L1]
are exhausted. If d is equal to zero, then the HSS key pair is
exhausted, and it MUST NOT generate any more signatures.
Otherwise, the key pairs for levels d through L1 must be
regenerated during the signature generation process, as follows.
For i from d to L1, a new LMS public and private key pair with a
new identifier is generated, pub[i] and prv[i] are set to those
values, then the public key pub[i] is signed with prv[i1], and
sig[i1] is set to the resulting value.
The message is signed with prv[L1], and the value sig[L1] is set
to that result.
The value of the HSS signature is set as follows. We let
signed_pub_key denote an array of octet strings, where
signed_pub_key[i] = sig[i]  pub[i+1], for i between 0 and Nspk
1, inclusive, where Nspk = L1 denotes the number of signed public
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keys. Then the HSS signature is u32str(Nspk) 
signed_pub_key[0]  ...  signed_pub_key[Nspk1]  sig[Nspk].
Note that the number of signed_pub_key elements in the signature
is indicated by the value Nspk that appears in the initial four
bytes of the signature.
Here is some pseudocode of the above logic
Algorithm 8: Generating an HSS signature
1. If the messagesigning key prv[L1] is exhausted, regenerate that
key pair, together with any parent key pairs that might be
necessary.
If the root key pair is exhausted, then the HSS key pair is
exhausted and it MUST NOT generate any more signatures.
d = L
while (prv[d1].q == 2^(prv[d1].h)) {
d = d  1
if (d == 0)
return FAILURE
}
while (d < L) {
create lms keypair pub[d], prv[d]
sig[d1] = lms_signature( pub[d], prv[d1] )
d = d + 1
}
2. sign the message
sig[L1] = lms_signature( msg, prv[L1] )
3. Create the list of signed public keys
i = 0;
while (i < L1) {
signed_pub_key[i] = sig[i]  pub[i+1]
i = i + 1
}
4. return u32str(L1)  signed_pub_key[0]  ...
 signed_pub_key[L2]  sig[L1]
In the specific case of L=1, the format of an HSS signature is
u32str(0)  sig[0]
In the general case, the format of an HSS signature is
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u32str(Nspk)  signed_pub_key[0]  ...
 signed_pub_key[Nspk1]  sig[Nspk]
which is equivalent to
u32str(Nspk)  sig[0]  pub[1]  ...
 sig[Nspk1]  pub[Nspk]  sig[Nspk]
6.3. Signature Verification
To verify a signature S and message using the public key pub, the
following steps are performed:
The signature S is parsed into its components as follows:
Nspk = strTou32(first four bytes of S)
if Nspk+1 is not equal to the number of levels L in pub:
return INVALID
for (i = 0; i < Nspk; i = i + 1) {
siglist[i] = next LMS signature parsed from S
publist[i] = next LMS public key parsed from S
}
siglist[Nspk] = next LMS signature parsed from S
key = pub
for (i = 0; i < Nspk; i = i + 1) {
sig = siglist[i]
msg = publist[i]
if (lms_verify(msg, key, sig) != VALID):
return INVALID
key = msg
}
return lms_verify(message, key, siglist[Nspk])
Since the length of an LMS signature cannot be known without parsing
it, the HSS signature verification algorithm makes use of an LMS
signature parsing routine that takes as input a string consisting of
an LMS signature with an arbitrary string appended to it, and returns
both the LMS signature and the appended string. The latter is passed
on for further processing.
6.4. Parameter Set Recommendations
As for guidance as to the number of LMS level, and the size of each,
any discussion of performance is implementation specific. In
general, the sole drawback for a single LMS tree is the time it takes
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to generate the public key; as every LMOTS public key needs to be
generated, the time this takes can be substantial. For a two level
tree, only the top level LMS tree and the initial bottom level LMS
tree needs to be generated initially (before the first signature is
generated); this will in general be significantly quicker.
To give a general idea on the tradeoffs available, we include some
measurements taken with the github.com/cisco/hashsigs LMS
implementation, taken on a 3.3 GHz Xeon processor, with threading
enabled. We tried various parameter sets, all with W=8 (which
minimizes signature size, while increasing time). These are here to
give a guideline as to what's possible; for the computational time,
your mileage may vary, depending on the computing resources you have.
The machine these tests were performed on does not have the SHA256
extensions; you could possibly do significantly better.
+++++
 ParmSet  KeyGenTime  SigSize  KeyLifetime 
+++++
 15  6 sec  1616  30 seconds 
    
 20  3 min  1776  16 minutes 
    
 25  1.5 hour  1936  9 hours 
    
 15/10  6 sec  3172  9 hours 
    
 15/15  6 sec  3332  12 days 
    
 20/10  3 min  3332  12 days 
    
 20/15  3 min  3492  1 year 
    
 25/10  1.5 hour  3492  1 year 
    
 25/15  1.5 hour  3652  34 years 
+++++
Table 3
ParmSet: this is the height of the Merkle tree(s); parameter sets
listed as a single integer have L=1, and consist a single Merkle tree
of that height; parameter sets with L=2 are listed as x/y, with x
being the height of the top level Merkle tree, and y being the bottom
level.
KeyGenTime: the measured key generation time; that is, the time
needed to generate the public private key pair.
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SigSize: the size of a signature (in bytes)
KeyLifetime: the lifetime of a key, assuming we generated 1000
signatures per second. In practice, we're not likely to get anywhere
close to 1000 signatures per second sustained; if you have a more
appropriate figure for your scenario, this column is pretty easy to
recompute.
As for signature generation or verification times, those are
moderately insensitive to the above parameter settings (except for
the Winternitz setting, and the number of Merkle trees for
verification). Tests on the same machine (without multithreading)
gave approximately 4msec to sign a short message, 2.6msec to verify;
these tests used a two level ParmSet; a single level would
approximately halve the verification time. All times can be
significantly improved (by perhaps a factor of 8) by using a
parameter set with W=4; however that also about doubles the signature
size.
7. Rationale
The goal of this note is to describe the LMOTS, LMS and HSS
algorithms following the original references and present the modern
security analysis of those algorithms. Other signature methods are
out of scope and may be interesting followon work.
We adopt the techniques described by Leighton and Micali to mitigate
attacks that amortize their work over multiple invocations of the
hash function.
The values taken by the identifier I across different LMS public/
private key pairs are chosen randomly in order to improve security.
The analysis of this method in [Fluhrer17] shows that we do not need
uniqueness to ensure security; we do need to ensure that we don't
have a large number of private keys that use the same I value. By
randomly selecting 16 byte I values, the chance that, out of 2^64
private keys, 4 or more of them will use the same I value is
negligible (that is, has probability less than 2^128).
The reason 16 bytes I values were selected was to optimize the
Winternitz hash chain operation. With the current settings, the
value being hashed is exactly 55 bytes long (for a 32 byte hash
function), which SHA256 can hash in a single hash compression
operation. Other hash functions may be used in future
specifications; all the ones that we will be likely to support (SHA
512/256 and the various SHA3 hashes) would work well with a 16byte
I value.
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The signature and public key formats are designed so that they are
relatively easy to parse. Each format starts with a 32bit
enumeration value that indicates the details of the signature
algorithm and provides all of the information that is needed in order
to parse the format.
The Checksum Section 4.4 is calculated using a nonnegative integer
"sum", whose width was chosen to be an integer number of wbit fields
such that it is capable of holding the difference of the total
possible number of applications of the function H as defined in the
signing algorithm of Section 4.5 and the total actual number. In the
case that the number of times H is applied is 0, the sum is (2^w  1)
* (8*n/w). Thus for the purposes of this document, which describes
signature methods based on H = SHA256 (n = 32 bytes) and w = { 1, 2,
4, 8 }, the sum variable is a 16bit nonnegative integer for all
combinations of n and w. The calculation uses the parameter ls
defined in Section 4.1 and calculated in Appendix B, which indicates
the number of bits used in the leftshift operation.
7.1. Security String
To improve security against attacks that amortize their effort
against multiple invocations of the hash function, Leighton and
Micali introduce a "security string" that is distinct for each
invocation of that function. Whenever this process computes a hash,
the string being hashed will start with a string formed from the
below fields. These fields will appear in fixed locations in the
value we compute the hash of, and so we list where in the hash these
fields would be present. These fields that make up this string are:
I  a 16byte identifier for the LMS public/private key pair. It
MUST be chosen uniformly at random, or via a pseudorandom process,
at the time that a key pair is generated, in order to minimize the
probability that any specific value of I be used for a large
number of different LMS private keys. This is always bytes 015
of the value being hashed.
r  in the LMS Ntime signature scheme, the node number r
associated with a particular node of a hash tree is used as an
input to the hash used to compute that node. This value is
represented as a 32bit (four byte) unsigned integer in network
byte order. Either r or q (depending on the domain separation
parameter) will be bytes 1619 of the value being hashed.
q  in the LMS Ntime signature scheme, each LMOTS signature is
associated with the leaf of a hash tree, and q is set to the leaf
number. This ensures that a distinct value of q is used for each
distinct LMOTS public/private key pair. This value is
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represented as a 32bit (four byte) unsigned integer in network
byte order. Either r or q (depending on the domain separation
parameter) will be bytes 1619 of the value being hashed.
D  a domain separation parameter, which is a two byte identifier
that takes on different values in the different contexts in which
the hash function is invoked. D occurs in bytes 20, 21 of the
value being hashed and takes on the following values:
D_PBLC = 0x8080 when computing the hash of all of the iterates
in the LMOTS algorithm
D_MESG = 0x8181 when computing the hash of the message in the
LMOTS algorithms
D_LEAF = 0x8282 when computing the hash of the leaf of an LMS
tree
D_INTR = 0x8383 when computing the hash of an interior node of
an LMS tree
i  a value between 0 and 264; this is used in the LMOTS scheme,
when either computing the iterations of the Winternitz chain, or
when using the suggested LMOTS private key generation process.
It is represented as a 16bit (twobyte) unsigned integer in
network byte order. If present, it occurs at bytes 20, 21 of the
value being hashed.
j  in the LMOTS scheme, j is the iteration number used when the
private key element is being iteratively hashed. It is
represented as an 8bit (one byte) unsigned integer and is present
if i is a value between 0 and 264. If present, it occurs at bytes
22 to 21+n of the value being hashed.
C  an nbyte randomizer that is included with the message
whenever it is being hashed to improve security. C MUST be chosen
uniformly at random, or via a pseudorandom process. It is present
if D=D_MESG, and it occurs at bytes 22 to 21+n of the value being
hashed.
8. IANA Considerations
The Internet Assigned Numbers Authority (IANA) is requested to create
two registries: one for OTS signatures, which includes all of the LM
OTS signatures as defined in Section 4, and one for LeightonMicali
Signatures, as defined in Section 5.
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Additions to these registries require that a specification be
documented in an RFC or another permanent and readily available
reference in sufficient detail that interoperability between
independent implementations is possible. IANA MUST verify that all
applications for additions to these registries hve first been
reviewed by the IRTF Crypto Forum Research Group (CFRG).
Each entry in the registry contains the following elements:
a short name, such as "LMS_SHA256_M32_H10",
a positive number, and
a reference to a specification that completely defines the
signature method test cases that can be used to verify the
correctness of an implementation.
The numbers between 0xDDDDDDDD (decimal 3,722,304,989) and 0xFFFFFFFF
(decimal 4,294,967,295) inclusive, will not be assigned by IANA, and
are reserved for private use; no attempt will be made to prevent
multiple sites from using the same value in different (and
incompatible) ways [RFC2434].
The LMOTS registry is as follows.
++++
 Name  Reference  Numeric Identifier 
++++
 Reserved   0x00000000 
   
 LMOTS_SHA256_N32_W1  Section 4  0x00000001 
   
 LMOTS_SHA256_N32_W2  Section 4  0x00000002 
   
 LMOTS_SHA256_N32_W4  Section 4  0x00000003 
   
 LMOTS_SHA256_N32_W8  Section 4  0x00000004 
++++
Table 4
The LMS registry is as follows.
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++++
 Name  Reference  Numeric Identifier 
++++
 Reserved   0x0  0x4 
   
 LMS_SHA256_M32_H5  Section 5  0x00000005 
   
 LMS_SHA256_M32_H10  Section 5  0x00000006 
   
 LMS_SHA256_M32_H15  Section 5  0x00000007 
   
 LMS_SHA256_M32_H20  Section 5  0x00000008 
   
 LMS_SHA256_M32_H25  Section 5  0x00000009 
++++
Table 5
An IANA registration of a signature system does not constitute an
endorsement of that system or its security.
The LMOTS and the LMS registries currently occupy a disjoint set of
values. This coincidence is a historical accident; the correctness
of the system does not depend on this. IANA is not required to
maintain this situation.
9. Security Considerations
The hash function H MUST have second preimage resistance: it must be
computationally infeasible for an attacker that is given one message
M to be able to find a second message M' such that H(M) = H(M').
The security goal of a signature system is to prevent forgeries. A
successful forgery occurs when an attacker who does not know the
private key associated with a public key can find a message (distinct
from all previously signed ones) and signature that is valid with
that public key (that is, the Signature Verification algorithm
applied to that signature and message and public key will return
VALID). Such an attacker, in the strongest case, may have the
ability to forge valid signatures for an arbitrary number of other
messages.
LMS is provably secure in the random oracle model, where the hash
compression function is considered the random oracle, as shown by
[Fluhrer17]. Corollary 1 of that paper states:
If we have no more than 2^64 randomly chosen LMS private keys,
allow the attacker access to a signing oracle and a SHA256 hash
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compression oracle, and allow a maximum of 2^120 hash compression
computations, then the probability of an attacker being able to
generate a single forgery against any of those LMS keys is less
than 2^129.
Many of the objects within the public key and the signature start
with a typecode. A verifier MUST check each of these typecodes, and
a verification operation on a signature with an unknown type, or a
type that does not correspond to the type within the public key MUST
return INVALID. The expected length of a variablelength object can
be determined from its typecode, and if an object has a different
length, then any signature computed from the object is INVALID.
9.1. Hash Formats
The format of the inputs to the hash function H have the property
that each invocation of that function has an input that is repeated
by a small bounded number of other inputs (due to potential repeats
of the I value), and in particular, will vary somewhere in the first
23 bytes of the value being hashed. This property is important for a
proof of security in the random oracle model.
The formats used during key generation and signing (including the
recommended pseudorandom key generation procedure in Appendix A):
I  u32str(q)  u16str(i)  u8str(j)  tmp
I  u32str(q)  u16str(D_PBLC)  y[0]  ...  y[p1]
I  u32str(q)  u16str(D_MESG)  C  message
I  u32str(r)  u16str(D_LEAF)  OTS_PUB_HASH[r2^h]
I  u32str(r)  u16str(D_INTR)  T[2*r]  T[2*r+1]
I  u32str(q)  u16str(i)  u8str(0xff)  SEED
Each hash type listed is distinct; at locations 20, 21 of the value
being hashed, there exists either a fixed value D_PBLC, D_MESG,
D_LEAF, D_INTR, or a 16 bit value i. These fixed values are distinct
from each other, and are large (over 32768), while the 16 bit values
of i are small (currently no more than 265; possibly being slightly
larger if larger hash functions are supported); hence the range of
possible values of i will not collide any of the D_PBLC, D_MESG,
D_LEAF, D_INTR identifiers. The only other collision possibility is
the Winternitz chain hash colliding with the recommended pseudorandom
key generation process; here, at location 22 of the value being
hashed, the Winternitz chain function has the value u8str(j), where j
is a value between 0 and 254, while location 22 of the recommended
pseudorandom key generation process has value 255.
For the Winternitz chaining function, D_PBLC, and D_MESG, the value
of I  u32str(q) is distinct for each LMS leaf (or equivalently, for
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each q value). For the Winternitz chaining function, the value of
u16str(i)  u8str(j) is distinct for each invocation of H for a
given leaf. For D_PBLC and D_MESG, the input format is used only
once for each value of q, and thus distinctness is assured. The
formats for D_INTR and D_LEAF are used exactly once for each value of
r, which ensures their distinctness. For the recommended
pseudorandom key generation process, for a given value of I, q and j
are distinct for each invocation of H.
The value of I is chosen uniformly at random from the set of all 128
bit strings. If 2^64 public keys are generated (and hence 2^64
random I values), there is a nontrivial probability of a duplicate
(which would imply duplicate prefixes). However, there will be an
extremely high probability there will not be a fourway collision
(that is, any I value used for four distinct LMS keys; probability <
2^132), and hence the number of repeats for any specific prefix will
be limited to at most 3. This is shown (in [Fluhrer17]) to have only
a limited effect on the security of the system.
9.2. Stateful signature algorithm
The LMS signature system, like all Ntime signature systems, requires
that the signer maintain state across different invocations of the
signing algorithm, to ensure that none of the component onetime
signature systems are used more than once. This section calls out
some important practical considerations around this statefulness.
These issues are discussed in greater detail in [STMGMT].
In a typical computing environment, a private key will be stored in
nonvolatile media such as on a hard drive. Before it is used to
sign a message, it will be read into an application's Random Access
Memory (RAM). After a signature is generated, the value of the
private key will need to be updated by writing the new value of the
private key into nonvolatile storage. It is essential for security
that the application ensure that this value is actually written into
that storage, yet there may be one or more memory caches between it
and the application. Memory caching is commonly done in the file
system, and in a physical memory unit on the hard disk that is
dedicated to that purpose. To ensure that the updated value is
written to physical media, the application may need to take several
special steps. In a POSIX environment, for instance, the O_SYNC flag
(for the open() system call) will cause invocations of the write()
system call to block the calling process until the data has been
written to the underlying hardware. However, if that hardware has
its own memory cache, it must be separately dealt with using an
operating system or device specific tool such as hdparm to flush the
ondrive cache, or turn off write caching for that drive. Because
these details vary across different operating systems and devices,
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this note does not attempt to provide complete guidance; instead, we
call the implementer's attention to these issues.
When hierarchical signatures are used, an easy way to minimize the
private key synchronization issues is to have the private key for the
second level resident in RAM only, and never write that value into
nonvolatile memory. A new second level public/private key pair will
be generated whenever the application (re)starts; thus, failures such
as a power outage or application crash are automatically
accommodated. Implementations SHOULD use this approach wherever
possible.
9.3. Security of LMOTS Checksum
To show the security of LMOTS checksum, we consider the signature y
of a message with a private key x and let h = H(message) and
c = Cksm(H(message)) (see Section 4.5). To attempt a forgery, an
attacker may try to change the values of h and c. Let h' and c'
denote the values used in the forgery attempt. If for some integer j
in the range 0 to u, where u = ceil(8*n/w) is the size of the range
that the checksum value can cover, inclusive,
a' = coef(h', j, w),
a = coef(h, j, w), and
a' > a
then the attacker can compute F^a'(x[j]) from F^a(x[j]) = y[j] by
iteratively applying function F to the jth term of the signature an
additional (a'  a) times. However, as a result of the increased
number of hashing iterations, the checksum value c' will decrease
from its original value of c. Thus a valid signature's checksum will
have, for some number k in the range u to (p1), inclusive,
b' = coef(c', k, w),
b = coef(c, k, w), and
b' < b
Due to the oneway property of F, the attacker cannot easily compute
F^b'(x[k]) from F^b(x[k]) = y[k].
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10. Comparison with other work
The eXtended Merkle Signature Scheme (XMSS) [XMSS], [RFC8391] is
similar to HSS in several ways. Both are stateful hash based
signature schemes, and both use a hierarchical approach, with a
Merkle tree at each level of the hierarchy. XMSS signatures are
slightly shorter than HSS signatures, for equivalent security and an
equal number of signatures.
HSS has several advantages over XMSS. HSS operations are roughly
four times faster than the comparable XMSS ones, when SHA256 is used
as the underlying hash. This occurs because the hash operation done
as a part of the Winternitz iterations dominates performance, and
XMSS performs four compression function invocations (two for the PRF,
two for the F function) where HSS needs only perform one.
Additionally, HSS is somewhat simpler (as each hash invocation is
just a prefix followed by the data being hashed).
11. Acknowledgements
Thanks are due to Chirag Shroff, Andreas Huelsing, Burt Kaliski, Eric
Osterweil, Ahmed Kosba, Russ Housley, Philip Lafrance, Alexander
Truskovsky, Mark Peruzel for constructive suggestions and valuable
detailed review. We especially acknowledge Jerry Solinas, Laurie
Law, and Kevin Igoe, who pointed out the security benefits of the
approach of Leighton and Micali [USPTO5432852] and Jonathan Katz, who
gave us security guidance, and Bruno Couillard and Jim Goodman for an
especially thorough review.
12. References
12.1. Normative References
[FIPS180] National Institute of Standards and Technology, "Secure
Hash Standard (SHS)", FIPS 1804, March 2012.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RFC2434] Narten, T. and H. Alvestrand, "Guidelines for Writing an
IANA Considerations Section in RFCs", RFC 2434,
DOI 10.17487/RFC2434, October 1998,
.
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[RFC3979] Bradner, S., Ed., "Intellectual Property Rights in IETF
Technology", RFC 3979, DOI 10.17487/RFC3979, March 2005,
.
[RFC4506] Eisler, M., Ed., "XDR: External Data Representation
Standard", STD 67, RFC 4506, DOI 10.17487/RFC4506, May
2006, .
[RFC4879] Narten, T., "Clarification of the Third Party Disclosure
Procedure in RFC 3979", RFC 4879, DOI 10.17487/RFC4879,
April 2007, .
[USPTO5432852]
Leighton, T. and S. Micali, "Large provably fast and
secure digital signature schemes from secure hash
functions", U.S. Patent 5,432,852, July 1995.
12.2. Informative References
[C:Merkle87]
Merkle, R., "A Digital Signature Based on a Conventional
Encryption Function", Lecture Notes in Computer
Science crypto87vol, 1988.
[C:Merkle89a]
Merkle, R., "A Certified Digital Signature", Lecture Notes
in Computer Science crypto89vol, 1990.
[C:Merkle89b]
Merkle, R., "One Way Hash Functions and DES", Lecture
Notes in Computer Science crypto89vol, 1990.
[Fluhrer17]
Fluhrer, S., "Further analysis of a proposed hashbased
signature standard",
EPrint https://eprint.iacr.org/2017/553.pdf, 2017.
[Grover96]
Grover, L., "A fast quantum mechanical algorithm for
database search", 28th ACM Symposium on the Theory of
Computing p. 212, 1996.
[Katz16] Katz, J., "Analysis of a proposed hashbased signature
standard", Security Standardization Research (SSR)
Conference
https://www.cs.umd.edu/~jkatz/papers/HashBasedSigs
SSR16.pdf, 2016.
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[Merkle79]
Merkle, R., "Secrecy, Authentication, and Public Key
Systems", Stanford University Information Systems
Laboratory Technical Report 19791, 1979.
[RFC8391] Huelsing, A., Butin, D., Gazdag, S., Rijneveld, J., and A.
Mohaisen, "XMSS: eXtended Merkle Signature Scheme",
RFC 8391, DOI 10.17487/RFC8391, May 2018,
.
[STMGMT] McGrew, D., Fluhrer, S., Kampanakis, P., Gazdag, S.,
Butin, D., and J. Buchmann, "State Management for Hash
based Signatures.", Security Standardization Resarch (SSR)
Conference 224., 2016.
[XMSS] Buchmann, J., Dahmen, E., and Andreas Huelsing, "XMSSa
practical forward secure signature scheme based on minimal
security assumptions.", International Workshop on Post
Quantum Cryptography Springer Berlin., 2011.
Appendix A. Pseudorandom Key Generation
An implementation MAY use the following pseudorandom process for
generating an LMS private key.
SEED is an mbyte value that is generated uniformly at random at
the start of the process,
I is LMS key pair identifier,
q denotes the LMS leaf number of an LMOTS private key,
x_q denotes the x array of private elements in the LMOTS private
key with leaf number q,
i is the index of the private key element, and
H is the hash function used in LMOTS.
The elements of the LMOTS private keys are computed as:
x_q[i] = H(I  u32str(q)  u16str(i)  u8str(0xff)  SEED).
This process stretches the mbyte random value SEED into a (much
larger) set of pseudorandom values, using a unique counter in each
invocation of H. The format of the inputs to H are chosen so that
they are distinct from all other uses of H in LMS and LMOTS. A
careful reader will note that this is similar to the hash we perform
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when iterating through the Winternitz chain; however in that chain,
the iteration index will vary between 0 and 254 maximum (for W=8),
while the corresponding value in this formula is 255. This algorithm
is included in the proof of security in [Fluhrer17] and hence this
method is safe when used within the LMS system; however any other
cryptographical secure method of generating private keys would also
be safe.
Appendix B. LMOTS Parameter Options
The LMOTS one time signature method uses several internal
parameters, which are a function of the selected parameter set.
These internal parameters set:
p  This is the number of independent Winternitz chains used in
the signature; it will be the number of wbit digits needed to
hold the nbit hash (u in the below equations), along with the
number of digits needed to hold the checksum (v in the below
equations)
ls  This is the size of the shift needed to move the checksum so
that it appears in the checksum digits
ls is needed because, while we express the checksum internally as a
16 bit value, we don't always express all 16 bits in the signature;
for example, if w=4, we might use only the top 12 bits. Because we
read the checksum in network order, this means that, without the
shift, we'll use the higher order bits (which may be always 0), and
omit the lower order bits (where the checksum value actually
resides). This shift is here to ensure that the parts of the
checksum we need to express (for security) actually contribute to the
signature; when multiple such shifts are possible, we take the
minimal value.
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The parameters ls, and p are computed as follows:
u = ceil(8*n/w)
v = ceil((floor(lg((2^w  1) * u)) + 1) / w)
ls = 16  (v * w)
p = u + v
Here u and v represent the number of wbit fields required to contain
the hash of the message and the checksum byte strings, respectively.
And as the value of p is the number of wbit elements of
( H(message)  Cksm(H(message)) ), it is also equivalently the
number of byte strings that form the private key and the number of
byte strings in the signature. The value 16 in the ls computation of
ls corresponds to the 16 bits value used for the sum variable in
Algorithm 2 in Section 4.4
A table illustrating various combinations of n and w with the
associated values of u, v, ls, and p is provided in Table 6.
+++++++
 Hash  Winternitz  wbit  wbit  Left  Total 
 Length  Parameter  Elements  Elements  Shift  Number of 
 in  (w)  in Hash  in  (ls)  wbit 
 Bytes   (u)  Checksum   Elements 
 (n)    (v)   (p) 
+++++++
 32  1  256  9  7  265 
      
 32  2  128  5  6  133 
      
 32  4  64  3  4  67 
      
 32  8  32  2  0  34 
+++++++
Table 6
Appendix C. An iterative algorithm for computing an LMS public key
The LMS public key can be computed using the following algorithm or
any equivalent method. The algorithm uses a stack of hashes for
data. It also makes use of a hash function with the typical
init/update/final interface to hash functions; the result of the
invocations hash_init(), hash_update(N[1]), hash_update(N[2]), ... ,
hash_update(N[n]), v = hash_final(), in that order, is identical to
that of the invocation of H(N[1]  N[2]  ...  N[n]).
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Generating an LMS Public Key from an LMS Private Key
for ( i = 0; i < 2^h; i = i + 1 ) {
r = i + num_lmots_keys;
temp = H(I  u32str(r)  u16str(D_LEAF)  OTS_PUB_HASH[i])
j = i;
while (j % 2 == 1) {
r = (r  1)/2;
j = (j1) / 2;
left_side = pop(data stack);
temp = H(I  u32str(r)  u16str(D_INTR)  left_side  temp)
}
push temp onto the data stack
}
public_key = pop(data stack)
Note that this pseudocode expects that all 2^h leaves of the tree
have equal depth; that is, num_lmots_keys to be a power of 2. The
maximum depth of the stack will be h1 elements, that is, a total of
(h1)*n bytes; for the currently defined parameter sets, this will
never be more than 768 bytes of data.
Appendix D. Method for deriving authentication path for a signature
The LMS signature consists of u32str(q)  lmots_signature 
u32str(type)  path[0]  path[1]  ...  path[h1]. This
appendix shows one method of constructing this signature, assuming
that the implementation has stored the T[] array that was used to
construct the public key. Note that this is not the only possible
method; other methods exist which don't assume that you have the
entire T[] array in memory. To construct a signature, you perform
the following algorithm:
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Generating an LMS Signature
1. set type to the typecode of the LMS algorithm
2. extract h from the typecode according to table 2
3. create the LMOTS signature for the message:
ots_signature = lmots_sign(message, LMS_PRIV[q])
4. compute the array path as follows:
i = 0
r = 2^h + q
while (i < h) {
temp = (r / 2^i) xor 1
path[i] = T[temp]
i = i + 1
}
5. S = u32str(q)  ots_signature  u32str(type) 
path[0]  path[1]  ...  path[h1]
6. q = q + 1
7. return S
where 'xor' is the bitwise exclusiveor operation, and / is integer
division (that is, rounded down to an integer value)
Appendix E. Example Implementation
An example implementation can be found online at
https://github.com/cisco/hashsigs.
Appendix F. Test Cases
This section provides test cases that can be used to verify or debug
an implementation. This data is formatted with the name of the
elements on the left, and the value of the elements on the right, in
hexadecimal. The concatenation of all of the values within a public
key or signature produces that public key or signature, and values
that do not fit within a single line are listed across successive
lines.
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Test Case 1 Public Key

HSS public key
levels 00000002

LMS type 00000005 # LM_SHA256_M32_H5
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
I 61a5d57d37f5e46bfb7520806b07a1b8
K 50650e3b31fe4a773ea29a07f09cf2ea
30e579f0df58ef8e298da0434cb2b878


Test Case 1 Message

Message 54686520706f77657273206e6f742064 The powers not d
656c65676174656420746f2074686520 elegated to the 
556e6974656420537461746573206279 United States by
2074686520436f6e737469747574696f  the Constitutio
6e2c206e6f722070726f686962697465 n, nor prohibite
6420627920697420746f207468652053 d by it to the S
74617465732c20617265207265736572 tates, are reser
76656420746f20746865205374617465 ved to the State
7320726573706563746976656c792c20 s respectively, 
6f7220746f207468652070656f706c65 or to the people
2e0a ..

Test Case 1 Signature

HSS signature
Nspk 00000001
sig[0]:

LMS signature
q 00000005

LMOTS signature
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
C d32b56671d7eb98833c49b433c272586
bc4a1c8a8970528ffa04b966f9426eb9
y[0] 965a25bfd37f196b9073f3d4a232feb6
9128ec45146f86292f9dff9610a7bf95
y[1] a64c7f60f6261a62043f86c70324b770
7f5b4a8a6e19c114c7be866d488778a0
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y[2] e05fd5c6509a6e61d559cf1a77a970de
927d60c70d3de31a7fa0100994e162a2
y[3] 582e8ff1b10cd99d4e8e413ef469559f
7d7ed12c838342f9b9c96b83a4943d16
y[4] 81d84b15357ff48ca579f19f5e71f184
66f2bbef4bf660c2518eb20de2f66e3b
y[5] 14784269d7d876f5d35d3fbfc7039a46
2c716bb9f6891a7f41ad133e9e1f6d95
y[6] 60b960e7777c52f060492f2d7c660e14
71e07e72655562035abc9a701b473ecb
y[7] c3943c6b9c4f2405a3cb8bf8a691ca51
d3f6ad2f428bab6f3a30f55dd9625563
y[8] f0a75ee390e385e3ae0b906961ecf41a
e073a0590c2eb6204f44831c26dd768c
y[9] 35b167b28ce8dc988a3748255230cef9
9ebf14e730632f27414489808afab1d1
y[10] e783ed04516de012498682212b078105
79b250365941bcc98142da13609e9768
y[11] aaf65de7620dabec29eb82a17fde35af
15ad238c73f81bdb8dec2fc0e7f93270
y[12] 1099762b37f43c4a3c20010a3d72e2f6
06be108d310e639f09ce7286800d9ef8
y[13] a1a40281cc5a7ea98d2adc7c7400c2fe
5a101552df4e3cccfd0cbf2ddf5dc677
y[14] 9cbbc68fee0c3efe4ec22b83a2caa3e4
8e0809a0a750b73ccdcf3c79e6580c15
y[15] 4f8a58f7f24335eec5c5eb5e0cf01dcf
4439424095fceb077f66ded5bec73b27
y[16] c5b9f64a2a9af2f07c05e99e5cf80f00
252e39db32f6c19674f190c9fbc506d8
y[17] 26857713afd2ca6bb85cd8c107347552
f30575a5417816ab4db3f603f2df56fb
y[18] c413e7d0acd8bdd81352b2471fc1bc4f
1ef296fea1220403466b1afe78b94f7e
y[19] cf7cc62fb92be14f18c2192384ebceaf
8801afdf947f698ce9c6ceb696ed70e9
y[20] e87b0144417e8d7baf25eb5f70f09f01
6fc925b4db048ab8d8cb2a661ce3b57a
y[21] da67571f5dd546fc22cb1f97e0ebd1a6
5926b1234fd04f171cf469c76b884cf3
y[22] 115cce6f792cc84e36da58960c5f1d76
0f32c12faef477e94c92eb75625b6a37
y[23] 1efc72d60ca5e908b3a7dd69fef02491
50e3eebdfed39cbdc3ce9704882a2072
y[24] c75e13527b7a581a556168783dc1e975
45e31865ddc46b3c957835da252bb732
y[25] 8d3ee2062445dfb85ef8c35f8e1f3371
af34023cef626e0af1e0bc017351aae2
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y[26] ab8f5c612ead0b729a1d059d02bfe18e
fa971b7300e882360a93b025ff97e9e0
y[27] eec0f3f3f13039a17f88b0cf808f4884
31606cb13f9241f40f44e537d302c64a
y[28] 4f1f4ab949b9feefadcb71ab50ef27d6
d6ca8510f150c85fb525bf25703df720
y[29] 9b6066f09c37280d59128d2f0f637c7d
7d7fad4ed1c1ea04e628d221e3d8db77
y[30] b7c878c9411cafc5071a34a00f4cf077
38912753dfce48f07576f0d4f94f42c6
y[31] d76f7ce973e9367095ba7e9a3649b7f4
61d9f9ac1332a4d1044c96aefee67676
y[32] 401b64457c54d65fef6500c59cdfb69a
f7b6dddfcb0f086278dd8ad0686078df
y[33] b0f3f79cd893d314168648499898fbc0
ced5f95b74e8ff14d735cdea968bee74

LMS type 00000005 # LM_SHA256_M32_H5
path[0] d8b8112f9200a5e50c4a262165bd342c
d800b8496810bc716277435ac376728d
path[1] 129ac6eda839a6f357b5a04387c5ce97
382a78f2a4372917eefcbf93f63bb591
path[2] 12f5dbe400bd49e4501e859f885bf073
6e90a509b30a26bfac8c17b5991c157e
path[3] b5971115aa39efd8d564a6b90282c316
8af2d30ef89d51bf14654510a12b8a14
path[4] 4cca1848cf7da59cc2b3d9d0692dd2a2
0ba3863480e25b1b85ee860c62bf5136

LMS public key
LMS type 00000005 # LM_SHA256_M32_H5
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
I d2f14ff6346af964569f7d6cb880a1b6
K 6c5004917da6eafe4d9ef6c6407b3db0
e5485b122d9ebe15cda93cfec582d7ab

final_signature:

LMS signature
q 0000000a

LMOTS signature
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
C 0703c491e7558b35011ece3592eaa5da
4d918786771233e8353bc4f62323185c
y[0] 95cae05b899e35dffd71705470620998
8ebfdf6e37960bb5c38d7657e8bffeef
y[1] 9bc042da4b4525650485c66d0ce19b31
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7587c6ba4bffcc428e25d08931e72dfb
y[2] 6a120c5612344258b85efdb7db1db9e1
865a73caf96557eb39ed3e3f426933ac
y[3] 9eeddb03a1d2374af7bf771855774562
37f9de2d60113c23f846df26fa942008
y[4] a698994c0827d90e86d43e0df7f4bfcd
b09b86a373b98288b7094ad81a0185ac
y[5] 100e4f2c5fc38c003c1ab6fea479eb2f
5ebe48f584d7159b8ada03586e65ad9c
y[6] 969f6aecbfe44cf356888a7b15a3ff07
4f771760b26f9c04884ee1faa329fbf4
y[7] e61af23aee7fa5d4d9a5dfcf43c4c26c
e8aea2ce8a2990d7ba7b57108b47dabf
y[8] beadb2b25b3cacc1ac0cef346cbb90fb
044beee4fac2603a442bdf7e507243b7
y[9] 319c9944b1586e899d431c7f91bcccc8
690dbf59b28386b2315f3d36ef2eaa3c
y[10] f30b2b51f48b71b003dfb08249484201
043f65f5a3ef6bbd61ddfee81aca9ce6
y[11] 0081262a00000480dcbc9a3da6fbef5c
1c0a55e48a0e729f9184fcb1407c3152
y[12] 9db268f6fe50032a363c9801306837fa
fabdf957fd97eafc80dbd165e435d0e2
y[13] dfd836a28b354023924b6fb7e48bc0b3
ed95eea64c2d402f4d734c8dc26f3ac5
y[14] 91825daef01eae3c38e3328d00a77dc6
57034f287ccb0f0e1c9a7cbdc828f627
y[15] 205e4737b84b58376551d44c12c3c215
c812a0970789c83de51d6ad787271963
y[16] 327f0a5fbb6b5907dec02c9a90934af5
a1c63b72c82653605d1dcce51596b3c2
y[17] b45696689f2eb382007497557692caac
4d57b5de9f5569bc2ad0137fd47fb47e
y[18] 664fcb6db4971f5b3e07aceda9ac130e
9f38182de994cff192ec0e82fd6d4cb7
y[19] f3fe00812589b7a7ce51544045643301
6b84a59bec6619a1c6c0b37dd1450ed4
y[20] f2d8b584410ceda8025f5d2d8dd0d217
6fc1cf2cc06fa8c82bed4d944e71339e
y[21] ce780fd025bd41ec34ebff9d4270a322
4e019fcb444474d482fd2dbe75efb203
y[22] 89cc10cd600abb54c47ede93e08c114e
db04117d714dc1d525e11bed8756192f
y[23] 929d15462b939ff3f52f2252da2ed64d
8fae88818b1efa2c7b08c8794fb1b214
y[24] aa233db3162833141ea4383f1a6f120b
e1db82ce3630b3429114463157a64e91
y[25] 234d475e2f79cbf05e4db6a9407d72c6
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bff7d1198b5c4d6aad2831db61274993
y[26] 715a0182c7dc8089e32c8531deed4f74
31c07c02195eba2ef91efb5613c37af7
y[27] ae0c066babc69369700e1dd26eddc0d2
16c781d56e4ce47e3303fa73007ff7b9
y[28] 49ef23be2aa4dbf25206fe45c20dd888
395b2526391a724996a44156beac8082
y[29] 12858792bf8e74cba49dee5e8812e019
da87454bff9e847ed83db07af3137430
y[30] 82f880a278f682c2bd0ad6887cb59f65
2e155987d61bbf6a88d36ee93b6072e6
y[31] 656d9ccbaae3d655852e38deb3a2dcf8
058dc9fb6f2ab3d3b3539eb77b248a66
y[32] 1091d05eb6e2f297774fe6053598457c
c61908318de4b826f0fc86d4bb117d33
y[33] e865aa805009cc2918d9c2f840c4da43
a703ad9f5b5806163d7161696b5a0adc

LMS type 00000005 # LM_SHA256_M32_H5
path[0] d5c0d1bebb06048ed6fe2ef2c6cef305
b3ed633941ebc8b3bec9738754cddd60
path[1] e1920ada52f43d055b5031cee6192520
d6a5115514851ce7fd448d4a39fae2ab
path[2] 2335b525f484e9b40d6a4a969394843b
dcf6d14c48e8015e08ab92662c05c6e9
path[3] f90b65a7a6201689999f32bfd368e5e3
ec9cb70ac7b8399003f175c40885081a
path[4] 09ab3034911fe125631051df0408b394
6b0bde790911e8978ba07dd56c73e7ee
Test Case 2 Private Key

(note: procedure in Appendix A is used)
SEED 000102030405060708090a0b0c0d0e0f
101112131415161718191a1b1c1d1e1f
I d08fabd4a2091ff0a8cb4ed834e74534


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Test Case 2 Public Key

HSS public key
levels 00000002

LMS type 00000006 # LM_SHA256_M32_H10
LMOTS type 00000003 # LMOTS_SHA256_N32_W4
I d08fabd4a2091ff0a8cb4ed834e74534
K 32a58885cd9ba0431235466bff9651c6
c92124404d45fa53cf161c28f1ad5a8e


Test Case 2 Message

Message 54686520656e756d65726174696f6e20 The enumeration
696e2074686520436f6e737469747574 in the Constitut
696f6e2c206f66206365727461696e20 ion, of certain
7269676874732c207368616c6c206e6f rights, shall no
7420626520636f6e7374727565642074 t be construed t
6f2064656e79206f7220646973706172 o deny or dispar
616765206f7468657273207265746169 age others retai
6e6564206279207468652070656f706c ned by the peopl
652e0a e..

Test Case 2 Signature

HSS signature
Nspk 00000001
sig[0]:

LMS signature
q 00000003

LMOTS signature
LMOTS type 00000003 # LMOTS_SHA256_N32_W4
C 3d46bee8660f8f215d3f96408a7a64cf
1c4da02b63a55f62c666ef5707a914ce
y[0] 0674e8cb7a55f0c48d484f31f3aa4af9
719a74f22cf823b94431d01c926e2a76
y[1] bb71226d279700ec81c9e95fb11a0d10
d065279a5796e265ae17737c44eb8c59
y[2] 4508e126a9a7870bf4360820bdeb9a01
d9693779e416828e75bddd7d8c70d50a
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y[3] 0ac8ba39810909d445f44cb5bb58de73
7e60cb4345302786ef2c6b14af212ca1
y[4] 9edeaa3bfcfe8baa6621ce88480df237
1dd37add732c9de4ea2ce0dffa53c926
y[5] 49a18d39a50788f4652987f226a1d481
68205df6ae7c58e049a25d4907edc1aa
y[6] 90da8aa5e5f7671773e941d805536021
5c6b60dd35463cf2240a9c06d694e9cb
y[7] 54e7b1e1bf494d0d1a28c0d31acc7516
1f4f485dfd3cb9578e836ec2dc722f37
y[8] ed30872e07f2b8bd0374eb57d22c614e
09150f6c0d8774a39a6e168211035dc5
y[9] 2988ab46eaca9ec597fb18b4936e66ef
2f0df26e8d1e34da28cbb3af75231372
y[10] 0c7b345434f72d65314328bbb030d0f0
f6d5e47b28ea91008fb11b05017705a8
y[11] be3b2adb83c60a54f9d1d1b2f476f9e3
93eb5695203d2ba6ad815e6a111ea293
y[12] dcc21033f9453d49c8e5a6387f588b1e
a4f706217c151e05f55a6eb7997be09d
y[13] 56a326a32f9cba1fbe1c07bb49fa04ce
cf9df1a1b815483c75d7a27cc88ad1b1
y[14] 238e5ea986b53e087045723ce16187ed
a22e33b2c70709e53251025abde89396
y[15] 45fc8c0693e97763928f00b2e3c75af3
942d8ddaee81b59a6f1f67efda0ef81d
y[16] 11873b59137f67800b35e81b01563d18
7c4a1575a1acb92d087b517a8833383f
y[17] 05d357ef4678de0c57ff9f1b2da61dfd
e5d88318bcdde4d9061cc75c2de3cd47
y[18] 40dd7739ca3ef66f1930026f47d9ebaa
713b07176f76f953e1c2e7f8f271a6ca
y[19] 375dbfb83d719b1635a7d8a138919579
44b1c29bb101913e166e11bd5f34186f
y[20] a6c0a555c9026b256a6860f4866bd6d0
b5bf90627086c6149133f8282ce6c9b3
y[21] 622442443d5eca959d6c14ca8389d12c
4068b503e4e3c39b635bea245d9d05a2
y[22] 558f249c9661c0427d2e489ca5b5dde2
20a90333f4862aec793223c781997da9
y[23] 8266c12c50ea28b2c438e7a379eb106e
ca0c7fd6006e9bf612f3ea0a454ba3bd
y[24] b76e8027992e60de01e9094fddeb3349
883914fb17a9621ab929d970d101e45f
y[25] 8278c14b032bcab02bd15692d21b6c5c
204abbf077d465553bd6eda645e6c306
y[26] 5d33b10d518a61e15ed0f092c3222628
1a29c8a0f50cde0a8c66236e29c2f310
McGrew, et al. Expires July 11, 2019 [Page 55]
InternetDraft HashBased Signatures January 2019
y[27] a375cebda1dc6bb9a1a01dae6c7aba8e
bedc6371a7d52aacb955f83bd6e4f84d
y[28] 2949dcc198fb77c7e5cdf6040b0f84fa
f82808bf985577f0a2acf2ec7ed7c0b0
y[29] ae8a270e951743ff23e0b2dd12e9c3c8
28fb5598a22461af94d568f29240ba28
y[30] 20c4591f71c088f96e095dd98beae456
579ebbba36f6d9ca2613d1c26eee4d8c
y[31] 73217ac5962b5f3147b492e8831597fd
89b64aa7fde82e1974d2f6779504dc21
y[32] 435eb3109350756b9fdabe1c6f368081
bd40b27ebcb9819a75d7df8bb07bb05d
y[33] b1bab705a4b7e37125186339464ad8fa
aa4f052cc1272919fde3e025bb64aa8e
y[34] 0eb1fcbfcc25acb5f718ce4f7c2182fb
393a1814b0e942490e52d3bca817b2b2
y[35] 6e90d4c9b0cc38608a6cef5eb153af08
58acc867c9922aed43bb67d7b33acc51
y[36] 9313d28d41a5c6fe6cf3595dd5ee63f0
a4c4065a083590b275788bee7ad875a7
y[37] f88dd73720708c6c6c0ecf1f43bbaada
e6f208557fdc07bd4ed91f88ce4c0de8
y[38] 42761c70c186bfdafafc444834bd3418
be4253a71eaf41d718753ad07754ca3e
y[39] ffd5960b0336981795721426803599ed
5b2b7516920efcbe32ada4bcf6c73bd2
y[40] 9e3fa152d9adeca36020fdeeee1b7395
21d3ea8c0da497003df1513897b0f547
y[41] 94a873670b8d93bcca2ae47e64424b74
23e1f078d9554bb5232cc6de8aae9b83
y[42] fa5b9510beb39ccf4b4e1d9c0f19d5e1
7f58e5b8705d9a6837a7d9bf99cd1338
y[43] 7af256a8491671f1f2f22af253bcff54
b673199bdb7d05d81064ef05f80f0153
y[44] d0be7919684b23da8d42ff3effdb7ca0
985033f389181f47659138003d712b5e
y[45] c0a614d31cc7487f52de8664916af79c
98456b2c94a8038083db55391e347586
y[46] 2250274a1de2584fec975fb09536792c
fbfcf6192856cc76eb5b13dc4709e2f7
y[47] 301ddff26ec1b23de2d188c999166c74
e1e14bbc15f457cf4e471ae13dcbdd9c
y[48] 50f4d646fc6278e8fe7eb6cb5c94100f
a870187380b777ed19d7868fd8ca7ceb
y[49] 7fa7d5cc861c5bdac98e7495eb0a2cee
c1924ae979f44c5390ebedddc65d6ec1
y[50] 1287d978b8df064219bc5679f7d7b264
a76ff272b2ac9f2f7cfc9fdcfb6a5142
McGrew, et al. Expires July 11, 2019 [Page 56]
InternetDraft HashBased Signatures January 2019
y[51] 8240027afd9d52a79b647c90c2709e06
0ed70f87299dd798d68f4fadd3da6c51
y[52] d839f851f98f67840b964ebe73f8cec4
1572538ec6bc131034ca2894eb736b3b
y[53] da93d9f5f6fa6f6c0f03ce43362b8414
940355fb54d3dfdd03633ae108f3de3e
y[54] bc85a3ff51efeea3bc2cf27e1658f178
9ee612c83d0f5fd56f7cd071930e2946
y[55] beeecaa04dccea9f97786001475e0294
bc2852f62eb5d39bb9fbeef75916efe4
y[56] 4a662ecae37ede27e9d6eadfdeb8f8b2
b2dbccbf96fa6dbaf7321fb0e701f4d4
y[57] 29c2f4dcd153a2742574126e5eaccc77
686acf6e3ee48f423766e0fc466810a9
y[58] 05ff5453ec99897b56bc55dd49b99114
2f65043f2d744eeb935ba7f4ef23cf80
y[59] cc5a8a335d3619d781e7454826df720e
ec82e06034c44699b5f0c44a8787752e
y[60] 057fa3419b5bb0e25d30981e41cb1361
322dba8f69931cf42fad3f3bce6ded5b
y[61] 8bfc3d20a2148861b2afc14562ddd27f
12897abf0685288dcc5c4982f8260268
y[62] 46a24bf77e383c7aacab1ab692b29ed8
c018a65f3dc2b87ff619a633c41b4fad
y[63] b1c78725c1f8f922f6009787b1964247
df0136b1bc614ab575c59a16d089917b
y[64] d4a8b6f04d95c581279a139be09fcf6e
98a470a0bceca191fce476f9370021cb
y[65] c05518a7efd35d89d8577c990a5e1996
1ba16203c959c91829ba7497cffcbb4b
y[66] 294546454fa5388a23a22e805a5ca35f
956598848bda678615fec28afd5da61a

LMS type 00000006 # LM_SHA256_M32_H10
path[0] b326493313053ced3876db9d23714818
1b7173bc7d042cefb4dbe94d2e58cd21
path[1] a769db4657a103279ba8ef3a629ca84e
e836172a9c50e51f45581741cf808315
path[2] 0b491cb4ecbbabec128e7c81a46e62a6
7b57640a0a78be1cbf7dd9d419a10cd8
path[3] 686d16621a80816bfdb5bdc56211d72c
a70b81f1117d129529a7570cf79cf52a
path[4] 7028a48538ecdd3b38d3d5d62d262465
95c4fb73a525a5ed2c30524ebb1d8cc8
path[5] 2e0c19bc4977c6898ff95fd3d310b0ba
e71696cef93c6a552456bf96e9d075e3
path[6] 83bb7543c675842bafbfc7cdb88483b3
276c29d4f0a341c2d406e40d4653b7e4
McGrew, et al. Expires July 11, 2019 [Page 57]
InternetDraft HashBased Signatures January 2019
path[7] d045851acf6a0a0ea9c710b805cced46
35ee8c107362f0fc8d80c14d0ac49c51
path[8] 6703d26d14752f34c1c0d2c4247581c1
8c2cf4de48e9ce949be7c888e9caebe4
path[9] a415e291fd107d21dc1f084b11582082
49f28f4f7c7e931ba7b3bd0d824a4570

LMS public key
LMS type 00000005 # LM_SHA256_M32_H5
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
I 215f83b7ccb9acbcd08db97b0d04dc2b
K a1cd035833e0e90059603f26e07ad2aa
d152338e7a5e5984bcd5f7bb4eba40b7

final_signature:

LMS signature
q 00000004

LMOTS signature
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
C 0eb1ed54a2460d512388cad533138d24
0534e97b1e82d33bd927d201dfc24ebb
y[0] 11b3649023696f85150b189e50c00e98
850ac343a77b3638319c347d7310269d
y[1] 3b7714fa406b8c35b021d54d4fdada7b
9ce5d4ba5b06719e72aaf58c5aae7aca
y[2] 057aa0e2e74e7dcfd17a0823429db629
65b7d563c57b4cec942cc865e29c1dad
y[3] 83cac8b4d61aacc457f336e6a10b6632
3f5887bf3523dfcadee158503bfaa89d
y[4] c6bf59daa82afd2b5ebb2a9ca6572a60
67cee7c327e9039b3b6ea6a1edc7fdc3
y[5] df927aade10c1c9f2d5ff446450d2a39
98d0f9f6202b5e07c3f97d2458c69d3c
y[6] 8190643978d7a7f4d64e97e3f1c4a08a
7c5bc03fd55682c017e2907eab07e5bb
y[7] 2f190143475a6043d5e6d5263471f4ee
cf6e2575fbc6ff37edfa249d6cda1a09
y[8] f797fd5a3cd53a066700f45863f04b6c
8a58cfd341241e002d0d2c0217472bf1
y[9] 8b636ae547c1771368d9f317835c9b0e
f430b3df4034f6af00d0da44f4af7800
y[10] bc7a5cf8a5abdb12dc718b559b74cab9
090e33cc58a955300981c420c4da8ffd
y[11] 67df540890a062fe40dba8b2c1c548ce
d22473219c534911d48ccaabfb71bc71
y[12] 862f4a24ebd376d288fd4e6fb06ed870
McGrew, et al. Expires July 11, 2019 [Page 58]
InternetDraft HashBased Signatures January 2019
5787c5fedc813cd2697e5b1aac1ced45
y[13] 767b14ce88409eaebb601a93559aae89
3e143d1c395bc326da821d79a9ed41dc
y[14] fbe549147f71c092f4f3ac522b5cc572
90706650487bae9bb5671ecc9ccc2ce5
y[15] 1ead87ac01985268521222fb9057df7e
d41810b5ef0d4f7cc67368c90f573b1a
y[16] c2ce956c365ed38e893ce7b2fae15d36
85a3df2fa3d4cc098fa57dd60d2c9754
y[17] a8ade980ad0f93f6787075c3f680a2ba
1936a8c61d1af52ab7e21f416be09d2a
y[18] 8d64c3d3d8582968c2839902229f85ae
e297e717c094c8df4a23bb5db658dd37
y[19] 7bf0f4ff3ffd8fba5e383a48574802ed
545bbe7a6b4753533353d73706067640
y[20] 135a7ce517279cd683039747d218647c
86e097b0daa2872d54b8f3e508598762
y[21] 9547b830d8118161b65079fe7bc59a99
e9c3c7380e3e70b7138fe5d9be255150
y[22] 2b698d09ae193972f27d40f38dea264a
0126e637d74ae4c92a6249fa103436d3
y[23] eb0d4029ac712bfc7a5eacbdd7518d6d
4fe903a5ae65527cd65bb0d4e9925ca2
y[24] 4fd7214dc617c150544e423f450c99ce
51ac8005d33acd74f1bed3b17b7266a4
y[25] a3bb86da7eba80b101e15cb79de9a207
852cf91249ef480619ff2af8cabca831
y[26] 25d1faa94cbb0a03a906f683b3f47a97
c871fd513e510a7a25f283b196075778
y[27] 496152a91c2bf9da76ebe089f4654877
f2d586ae7149c406e663eadeb2b5c7e8
y[28] 2429b9e8cb4834c83464f079995332e4
b3c8f5a72bb4b8c6f74b0d45dc6c1f79
y[29] 952c0b7420df525e37c15377b5f09843
19c3993921e5ccd97e097592064530d3
y[30] 3de3afad5733cbe7703c5296263f7734
2efbf5a04755b0b3c997c4328463e84c
y[31] aa2de3ffdcd297baaaacd7ae646e44b5
c0f16044df38fabd296a47b3a838a913
y[32] 982fb2e370c078edb042c84db34ce36b
46ccb76460a690cc86c302457dd1cde1
y[33] 97ec8075e82b393d542075134e2a17ee
70a5e187075d03ae3c853cff60729ba4

LMS type 00000005 # LM_SHA256_M32_H5
path[0] 4de1f6965bdabc676c5a4dc7c35f97f8
2cb0e31c68d04f1dad96314ff09e6b3d
path[1] e96aeee300d1f68bf1bca9fc58e40323
McGrew, et al. Expires July 11, 2019 [Page 59]
InternetDraft HashBased Signatures January 2019
36cd819aaf578744e50d1357a0e42867
path[2] 04d341aa0a337b19fe4bc43c2e79964d
4f351089f2e0e41c7c43ae0d49e7f404
path[3] b0f75be80ea3af098c9752420a8ac0ea
2bbb1f4eeba05238aef0d8ce63f0c6e5
path[4] e4041d95398a6f7f3e0ee97cc1591849
d4ed236338b147abde9f51ef9fd4e1c1
Authors' Addresses
David McGrew
Cisco Systems
13600 Dulles Technology Drive
Herndon, VA 20171
USA
Email: mcgrew@cisco.com
Michael Curcio
Cisco Systems
70252 Kit Creek Road
Research Triangle Park, NC 277094987
USA
Email: micurcio@cisco.com
Scott Fluhrer
Cisco Systems
170 West Tasman Drive
San Jose, CA
USA
Email: sfluhrer@cisco.com
McGrew, et al. Expires July 11, 2019 [Page 60]