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Hash-Based SignaturesCisco Systems13600 Dulles Technology DriveHerndon20171VAUSAmcgrew@cisco.comCisco Systems7025-2 Kit Creek RoadResearch Triangle Park27709-4987NCUSAmicurcio@cisco.comCisco Systems170 West Tasman DriveSan JoseCAUSAsfluhrer@cisco.com
IRTF
Crypto Forum Research Group
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 one-time 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 side-channel attacks. Unlike
most other signature systems, hash-based signatures would still
be secure even if it proves feasible for an attacker to build a
quantum computer.
One-time signature systems, and general purpose signature systems
built out of one-time signature systems, have been known since 1979
, were well studied in the 1990s , 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 relatively slow key generation.
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 of the original
Lamport-Diffie-Winternitz-Merkle one-time signature system
and general
signature system 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 One-Time
Signature (OTS) system can be used to sign at most one message
securely, but cannot securely sign more than one. An N-time signature
system can be used to sign N or fewer messages securely. A Merkle
tree signature scheme is an N-time signature system that uses an OTS
system as a component.
In this note we describe the Leighton-Micali 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. We denote the one-time signature scheme
incorporate in LMS as LM-OTS. This note is structured as follows.
Notation is introduced in . The LM-OTS
signature system is described in , and the LMS
and HSS N-time signature systems are described in
and , respectively.
Sufficient detail is provided to ensure interoperability.
The IANA registry for these signature
systems is described in . Security
considerations are presented in .
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 .
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 N-time 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 are valid.
A message/signature pair are valid if the signature was returned by
the signing algorithm upon input of the message and the private key
corresponding to the public key; otherwise, the signature and
message pair are not valid with probability very close to one.
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 non-negative 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 four byte string S as input
and returns a non-negative integer; the identity u32str(strTou32(S)) =
S holds for any four-byte string S.
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+ : a + b denotes the sum of a and b- : a - b denotes the difference of a and b
The standard order of operations is used when evaluating arithmetic expressions.
When B is a byte and i is an integer, then B >> i denotes the logical
right-shift operation.
Similarly, B << i denotes the logical left-shift 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 i^th element in an array A is denoted as A[i].
If S is a byte string, then byte(S, i) denotes its i^th byte, where
byte(S, 0) is the leftmost byte. In addition, bytes(S, i, j) denotes the
range of bytes from the i^th to the j^th 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 w-bit unsigned
integers; the correspondence is defined by the function coef(S, i, w) as follows:
The return value of coef is an unsigned integer.
If i is larger than the number of w-bit values in S, then
coef(S, i, w) is undefined, and an attempt to compute
that value should raise an error.
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. The following fields can appear in a security string:
I - an identifier for the LMS public/private key pair. The
length of this value varies based on the LMS parameter set and it
MUST be chosen uniformly at random, or via a pseudorandom process,
at the time that a key pair is generated, in order to ensure that
it will be distinct from the identifier of any other LMS private
key with probability close to one.
D - a domain separation parameter, which is a single byte that
takes on different values in the different algorithms in which
H is invoked. D takes on the following values:
D_ITER = 0x00 in the iterations of the LM-OTS algorithms
D_PBLC = 0x01 when computing the hash of all of the
iterates in the LM-OTS algorithm
D_MESG = 0x02 when computing the hash of the message in
the LM-OTS algorithms
D_LEAF = 0x03 when computing the hash of the leaf of an LMS tree
D_INTR = 0x04 when computing the hash of an interior node
of an LMS tree
D_PRG = 0x05 in the recommended pseudorandom process for
generating LMS private keys
C - an n-byte 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.
r - in the LMS N-time 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 32-bit (four byte) unsigned integer in network
byte order.
q - in the LMS N-time signature scheme, each LM-OTS 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 LM-OTS public/private key pair. This value is
represented as a 32-bit (four byte) unsigned integer in network
byte order.
i - in the LM-OTS scheme, i is the index of
the private key element upon which H is being applied. It is
represented as a 16-bit (two byte) unsigned integer in network
byte order.
j - in the LM-OTS scheme, j is the iteration
number used when the private key element is being iteratively
hashed. It is represented as an 8-bit (one byte) unsigned
integer.
If r is a non-negative real number, then we define the following functions:
ceil(r) : returns the smallest integer larger than rfloor(r) : returns the largest integer smaller than rlg(r) : returns the base-2 logarithm of r
A typecode is an unsigned integer that is associated with a particular
data format. The format of the LM-OTS, 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 four-byte unsigned integers in network byte order; equivalently,
they are XDR enumerations (see ).
This section defines LM-OTS 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 one-time 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 ), and the resulting digest is signed.
In order to facilitate its use in an N-time signature system, the
LM-OTS key generation, signing, and verification algorithms all take
as input a diversification parameter q. When the LM-OTS signature
system is used outside of an N-time signature system, this value
SHOULD be set to the all-zero value.
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 functionw : the width (number of bits) of the Winternitz coefficients; it is a member of the set { 1, 2, 4, 8 }p : the number of n-byte string elements that make up the LM-OTS signaturels : the number of left-shift bits used in the checksum function Cksm (defined in ).
H : a second-preimage-resistant cryptographic hash function that accepts byte strings of any length, and returns an n-byte string.
For more background on the cryptographic security requirements on H, see
the .
The value of n is determined by the functions selected for use as part
of the LM-OTS 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^w-1 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 . A table illustrating various
combinations of n, w, p, and ls is provided in .
To fully describe a LM-OTS signature method, the parameters n and
w, the length LenS of the security string S, as well as the
function H, MUST be specified. This section defines
several LM-OTS methods, each of which is identified by a
name. The values for p and ls are provided as a convenience.
NameHnwLenSplsLMOTS_SHA256_N32_W1SHA256321682657LMOTS_SHA256_N32_W2SHA256322681336LMOTS_SHA256_N32_W4SHA25632468674LMOTS_SHA256_N32_W8SHA25632868340
Here SHA256 denotes the NIST standard hash function .
The LM-OTS private key consists of a typecode indicating the
particular LM-OTS algorithm, an array x[] containing p n-byte strings,
and a LenS-byte security string S. This private key MUST be used to
sign (at most) one message. The following algorithm shows pseudocode for
generating a private key.
An implementation MAY use a pseudorandom method to compute x[i], as
suggested in , page 46. The details of the
pseudorandom method do not affect interoperability, but the
cryptographic strength MUST match that of the LM-OTS algorithm.
provides an example of a pseudorandom method
for computing LM-OTS private key.
The LM-OTS 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.
The public key is the value returned by Algorithm 1.
A checksum is used to ensure that any forgery attempt that manipulates
the elements of an existing signature will be detected. The security
property that it provides is detailed in .
The checksum function Cksm is defined as follows, where S denotes
the n-byte string that is input to that function, and the value
sum is a 16-bit unsigned integer:
The LM-OTS signature of a message is generated by first prepending the
randomizer C and the security string S to the message, then appending
D_MESG to the resulting string then computing its hash, concatenating
the checksum of the hash to the hash itself, then considering the
resulting value as a sequence of w-bit values, and using each of the
w-bit 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.
The signature is the string returned by Algorithm 3. specifies the typecode and more formally
defines the encoding and decoding of the string.
In order to verify a message with its signature (an array of n-byte
strings, denoted as y), the receiver must "complete" the chain of
iterations of H using the w-bit 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.
The Leighton Micali Signature (LMS) method can sign a potentially large
but fixed number of messages. An LMS system uses two cryptographic
components: a one-time 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 m-byte value. Each leaf of the tree
contains the value of the public key of an LM-OTS 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.
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, 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 j^th node at level L has node
number 2^L + j. The node number can conveniently be computed when
it is needed in the LMS algorithms, as described in those
algorithms.
An LMS system has the following parameters:
h : the height (number of levels - 1) in the tree, and
m : the number of bytes associated with each node.
H : a second-preimage-resistant cryptographic hash function that accepts byte strings of any length, and
returns an m-byte string. H SHOULD be the same as in , but MAY be different.
There are 2^h leaves in the tree. The hash function used within the
LMS system MUST be the same as the hash function used within the
LM-OTS system used to generate the leaves. This is required because
both use the same I value, and hence must have the same length of I
value (and the length of the I value is dependent on the hash
function).
NameHmhLMS_SHA256_M32_H5SHA256325LMS_SHA256_M32_H10SHA2563210LMS_SHA256_M32_H15SHA2563215LMS_SHA256_M32_H20SHA2563220LMS_SHA256_M32_H24SHA2563225
An LMS private key consists of an array OTS_PRIV[] of 2^h LM-OTS
private keys, and the leaf number q of the next LM-OTS private key
that has not yet been used. The q^th element of OTS_PRIV[] is
generated using Algorithm 0 with the security string S = I || q.
The leaf number q is initialized to zero when the LMS private key is
created. The process is as follows:
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.
provides an example of a pseudorandom method
for computing an LMS private key.
An LMS public key is defined as follows, where we denote the public
key associated with the i^th LM-OTS private key as OTS_PUB[i],
with i ranging from 0 to (2^h)-1. Each instance of an LMS
public/private key pair is associated with a perfect binary tree,
and the nodes of that tree are indexed from 1 to 2^(h+1)-1. Each
node is associated with an m-byte string, and the string for the r^th
node is denoted as T[r] and is defined as
The LMS public key is the string u32str(type) || I || T[1]. specifies the format of the type variable.
The value I is the private key identifier (whose length is denoted by
the parameter set), 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 .
An LMS signature consists of
a typecode indicating the particular LMS algorithm,
the number q of the leaf associated with the LM-OTS signature,
as a four-byte unsigned integer in network byte order,
an LM-OTS signature, and
an array of h m-byte values that is associated with the path
through the tree from the leaf associated with the LM-OTS
signature to the root.
Symbolically, the signature can be represented as u32str(q) ||
ots_signature || u32str(type) || path[0] || path[1] || ... ||
path[h-1]. specifies the typecode and
more formally defines the format. The array of values contains
the siblings of the nodes on the path from the leaf to the root but does not contain the nodes on the path themselves. The
array for a tree with height h will have h values. The first
value is the sibling of the leaf, the next value is the sibling of the parent of the leaf, and so on up the path to the root.
To compute the LMS signature of a message with an LMS private key,
the signer first computes the LM-OTS signature of the message
using the leaf number of the next unused LM-OTS 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 LM-OTS 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 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. 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 node-caching 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 .
An LMS signature is verified by first using the LM-OTS signature
verification algorithm (Algorithm 4b) to compute the LM-OTS public key
from the LM-OTS 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.
In scenarios where it is necessary to minimize the time taken by the
public key generation process, a Hierarchical N-time Signature System
(HSS) can be used. Leighton and Micali describe a scheme in which an
LMS public key is used to sign a second LMS public key, which is then
distributed along with the signatures generated with the second public
key . This hierarchical scheme, which we
describe in this section, uses an LMS scheme as a component.
HSS, in essence, utilizes a tree of LMS trees, in which 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
one of its leaves. 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.
Each level of 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 L-1 inclusive,
and the root of the hierarchy is level 0:
prv[i] is the LMS private key of the i^th level,
pub[i] is the LMS public key of the i^th level (which includes
the identifier I as well as the key value K),
sig[i] is the LMS signature of the i^th level,
In this section, we say that an N-time private key is exhausted when
it has generated N signatures, and thus it can no longer be used for
signing.
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.
When an HSS key pair is generated, the key pair for each level
MUST have its own identifier.
To generate an HSS private and public key pair, new LMS private and
public keys are generated for prv[i] and pub[i] for i=0, ... , L-1.
These key pairs, and their identifiers, MUST be generated
independently. All of the information of the leaf level L-1,
including the private key, MUST NOT be stored in nonvolatile memory.
Letting Nnv denote the lowest level for which prv[Nnv] is stored
in nonvolatile memory, there are Nnv nonvolatile levels, and
L-Nnv volatile levels. For security, Nnv should be as close
to one as possible (see ).
The public key of the HSS scheme is 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[L-1]. The values
pub[0] and prv[0] do not change, though the values of pub[i] and
prv[i] are dynamic for i > 0, and are changed by the signature
generation algorithm.
To sign a message using the private key prv, the following
steps are performed:
If prv[L-1] is exhausted, then determine the smallest integer d
such that all of the private keys prv[d], prv[d+1], ... , prv[L-1]
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 L-1 must be
regenerated during the signature generation process, as follows.
For i from d to L-1, 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[i-1], and
sig[i-1] is set to the resulting value.
The message is signed with prv[L-1], and the value sig[L-1] 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 = L-1 denotes the number of
signed public keys. Then the HSS signature is u32str(Nspk) ||
signed_pub_key[0] || ... || signed_pub_key[Nspk-1] || 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.
In the specific case of L=1, the format of an HSS signature is
In the general case, the format of an HSS signature is
which is equivalent to
To verify a signature sig and message using the public key pub, the
following steps are performed:
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.
Many of the objects 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
variable-length object can be determined from its typecode, and if an
object has a different length, then any signature computed from the
object is INVALID.
The goal of this note is to describe the LM-OTS and LMS 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 follow-on 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 required to be distinct in order to
improve security. That distinctness ensures the uniqueness of the
inputs to H across all of those public/private key pair instances,
which is important for provable security in the random oracle model.
The length of I is set at 31 or 64 bytes so that randomly chosen values of
I will be distinct with probability at least 1 - 1/2^128 as long as
there are 2^60 or fewer instances of LMS public/private key pairs.
The sizes of the parameters in the security string are such that the
hashes computed by both LM and LM-OTS start with a fixed 64-byte I
value. The reason this size was selected was to allow an
implementation to compute the intermediate hash state after processing
I once (similar to the well-known optimization for HMAC), and hence
the majority of hashes computed during LM-OTS processing can be
performed using a single hash compression operation when using
SHA-256. Other hash functions, which may be used in future
specifications, can use a similar strategy, as long as I is long
enough that it is very unlikely to repeat if chosen uniformly at
random.
The signature and public key formats are designed so that they are
relatively easy to parse. Each format starts with a 32-bit
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 is calculated using a
non-negative integer "sum", whose width was chosen to be an integer
number of w-bit 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 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 16-bit
non-negative integer for all combinations of n and w. The calculation
uses the parameter ls defined in and
calculated in , which indicates the
number of bits used in the left-shift operation.
This is the fifth version of this draft. It has the
following changes from previous versions:Version 05
Clarified the L=1 specific case.
Extended the parameter sets to include an H=25 option
A large number of corrections and clarifications
Added a comparison to XMSS and SPHINCS, and citations to
those algorithms and to the recent Security Standardization
Research 2016 publications on the security of LMS and on the
state management in hash-based signatures.
Version 04
Specified that, in the HSS method, the I value was computed
from the I value of the parent LM tree. Previous versions
had the I value extracted from the public key (which meant
that all LM trees of a particular level and public key used
the same I value)
Changed the length of the I field based on the parameter set.
As noted in the Rationale section, this allows an
implementation to compute SHA256 n=32 based parameter sets
significantly faster.
Modified the XDR of an HSS signature not to use an array
of LM signatures; LM signatures are variable length, and
XDR doesn't support arrays of variable length structures.
Changed the LMS registry to be in a consistent order with the
LM-OTS parameter sets. Also, added LMS parameter sets with
height 15 trees
Previous versions
In Algorithms 3 and 4, the message was moved from the initial
position of the input to the function H to the final position, in
the computation of the intermediate variable Q. This was done to
improve security by preventing an attacker that can find a
collision in H from taking advantage of that fact via the forward
chaining property of Merkle-Damgard.
The Hierarchical Signature Scheme was generalized slightly so
that it can use more than two levels.
Several points of confusion were corrected; these had resulted from
incomplete or inconsistent changes from the Merkle approach of the
earlier draft to the Leighton-Micali approach.
This section is to be removed by the RFC editor upon publication.
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 3, and one for Leighton-Micali
Signatures, as defined in Section 4. 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.
Each entry in the registry contains the following elements:
a short name, such as "LMS_SHA256_M32_H10", a positive number, anda reference to a specification that completely defines the
signature method test cases that can be used to verify the
correctness of an implementation.
Requests to add an entry to the registry MUST include the name and the
reference. The number is assigned by IANA. Submitters
SHOULD have their requests reviewed by the IRTF Crypto Forum Research
Group (CFRG) at cfrg@ietf.org. Interested applicants that are
unfamiliar with IANA processes should visit http://www.iana.org.
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
.
The LM-OTS registry is as follows.
NameReferenceNumeric Identifier LMOTS_SHA256_N32_W1 0x00000001 LMOTS_SHA256_N32_W2 0x00000002 LMOTS_SHA256_N32_W4 0x00000003 LMOTS_SHA256_N32_W8 0x00000004
The LMS registry is as follows.
NameReferenceNumeric Identifier LMS_SHA256_M32_H5 0x00000005 LMS_SHA256_M32_H10 0x00000006 LMS_SHA256_M32_H15 0x00000007 LMS_SHA256_M32_H20 0x00000008 LMS_SHA256_M32_H25 0x00000009
An IANA registration of a signature system does not constitute an
endorsement of that system or its security.
This draft is based on U.S. patent 5,432,852, which issued over twenty
years ago and is thus expired.
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 and
and at
https://datatracker.ietf.org/ipr/about.
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 and
signature that are 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, as shown by Katz
. From Theorem 2 of
that reference:
For any adversary attacking the LMS scheme and making at most q
hash queries, the probability the adversary forges a signature is
at most 3*q/2^(8*n).
Here n is the number of bytes in the output of the hash function (as
defined in ). The security of all of the
the algorithms and parameter sets defined in this note is roughly 128
bits, even assuming that there are quantum computers that can compute
the input to an arbitrary function with computational cost equivalent
to the square root of the size of the domain of that function .
The format of the inputs to the hash function H have the property that
each invocation of that function has an input that is distinct from
all others, with very high probability. This property is important for a
proof of security in the random oracle model. The formats used during
key generation and signing are
Because the suffixes D_ITER, D_PBLC, D_LEAF, D_INTR, and D_PRG are
distinct, the input formats ending with different suffixes are all
distinct. It remains to show the distinctness of the inputs for each
suffix.
The values of I and C are chosen uniformly at random from the set of
all n*8 bit strings. For n=32, it is highly likely that each value of
I and C will be distinct, even when 2^96 such values are chosen.
For D_ITER, D_PBLC, and D_MESG, the value of S = I || u32str(q) is
distinct for each LMS leaf (or equivalently, for each q value). For
D_ITER, 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 S, 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 D_PRG, for a
given value of I, q and j are distinct for each invocation of H (note
that x_q[0] = SEED when j=0).
The LMS signature system, like all N-time signature systems,
requires that the signer maintain state across different invocations
of the signing algorithm, to ensure that none of the component
one-time signature systems are used more than once. This section
calls out some important practical considerations around this
statefulness.
In a typical computing environment, a private key will be stored in
non-volatile 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 non-volatile 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 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 on-drive
cache, or turn off write caching for that drive. Because these
details vary across different operating systems and devices, 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 non-volatile 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.
To show the security of LM-OTS 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 ). 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 over), 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 j^th 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 (p-1), inclusive,
b' = coef(c', k, w),
b = coef(c, k, w), and
b' < b
Due to the one-way property of F, the attacker cannot easily compute F^b'(x[k])
from F^b(x[k]) = y[k].
The eXtended Merkle Signature Scheme (XMSS) 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, because the hash operation dominates any measure of
performance, and XMSS performs four compression function invocations
(two for the PRF, two for the F function) where HSS need only perform
one. Additionally, HSS is somewhat simpler, and it admits a single-level
tree in a simple way (as described in ).
Another advantage of HSS is the fact that it can use a stateless
hash-based signature scheme in its non-volatile levels, while
continuing to use LMS in its volatile levels, and thus realize a
hybrid stateless/stateful scheme as described in . While we conjecture that hybrid schemes will offer
lower computation times and signature sizes than purely stateless
schemes, the details are outside the scope of this note. HSS is
therefore amenable to future extensions that will enable it to be used
in environments in which a purely stateful scheme would be too
brittle.
SPHINCS is a purely stateless hash based
signature scheme. While that property benefits security, its
signature sizes and generation times are an order of magnitude (or
more) larger than those of HSS, making it more difficult to adopt in
some practical scenarios.
Thanks are due to Chirag Shroff, Andreas Huelsing, Burt Kaliski, Eric
Osterweil, Ahmed Kosba, Russ Housley and Philip Lafrance 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 and Jonathan Katz, who gave us security
guidance.
&rfc2119;
&rfc2434;
&rfc3979;
&rfc4506;
&rfc4879;
Secure Hash Standard (SHS)National Institute of Standards and TechnologyLarge provably fast and secure digital signature schemes from secure hash functionsAnalysis of a proposed hash-based signature standardXMSS-a practical forward secure signature scheme based on minimal security assumptions.State Management for Hash-based Signatures.SPHINCS: Practical Stateless Hash-Based Signatures.A fast quantum mechanical algorithm for database searchA Certified Digital SignatureOne Way Hash Functions and DESA Digital Signature Based on a Conventional Encryption FunctionSecrecy, Authentication, and Public Key Systems
An implementation MAY use the following pseudorandom process
for generating an LMS private key.
SEED is an m-byte 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 LM-OTS private key,
x_q denotes the x array of private elements in the LM-OTS private
key with leaf number q,
j is an index of the private key element,
D_PRG is a diversification constant, and
H is the hash function used in LM-OTS.
The elements of the LM-OTS private keys are computed as:
This process stretches the m-byte 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 LM-OTS.
A table illustrating various combinations of n and w with the associated values of
u, v, ls, and p is provided in
.
Hash Length in Bytes (n)Winternitz Parameter (w)w-bit Elements in Hash (u)w-bit Elements in Checksum (v)Left Shift (ls)Total Number of w-bit Elements (p)161128881371626448681643234351681620183212569726532212856133324643467328322034
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]).
An example implementation can be found online at
http://github.com/davidmcgrew/hash-sigs/.
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.