# W-OTS#


Author: Herman Schoenfeld
Version: 1.0
Date: 2020-07-20
Copyright: (c) Sphere 10 Software Pty Ltd



### Abstract

A very simple modification to the standard W-OTS scheme is presented called W-OTS# that achieves a security enhancement similar to W-OTS+[1] but without the overhead of hashing a randomization vector in every round of the chaining function. The idea proffered by W-OTS# is to simply thwart Birthday-attacks[2] altogether by signing an HMAC of the message-digest (keyed with cryptographically random salt) rather than the message-digest itself. The signer thwarts a birthday attack by virtue of requiring that the attacker guess the salt bits in addition to the message-digest bits during the collision scanning process. By choosing a salt length matching the message-digest length, the security of W-OTS# reduces to that of the cryptographic hash function. This essentially doubles the security level of W-OTS and facilitates the use of shorter hash functions which provide shorter and faster signatures for same security. For example, WOTS# 128-bit signatures have commensurate security to standard W-OTS 256-bit signatures yet are roughly half the size and twice as fast. It is proposed that Blake2b-128 and Winternitz parameter w=4 (i.e. base-16 digits) be adopted as the default parameter set for the W-OTS# scheme.

# 1. Birthday Attack

A birthday attack involves an attacker forging a signature for a "malicious" message M by reusing a signature for an "agreed" message m . In this class of attack, the attacker has preknowledge of a message m that the victim is willing and intending to sign in the future.

The attacker creates variations of m as {m_1..m_k} any of which will also be deemed "valid" and signed by the victim. Whilst the victim considers each message m_i "identical", their hash digests are unique. This can be achieved by simply varying nonces or whitespace within m to create this set.

The attacker simultaneously generates variations of a "malicious" message M as the set {M1..M_l} and stops until a collision H(m_i) = H(M_j) is found (where H is the hash function used in the scheme).

NOTE the probability of finding such collisions is far more likely than a standard brute-force attack by virtue of the Birthday problem[2][3]. When a collision-pair (m_i, M_j) is found, the attacker asks the victim to sign valid m_i giving s = Sign(m_i, key) = SignDigest(H(m_i), key). The attacker then proceeds to forge a signature for invalid M_i by simply re-using s , as follows:


1: S = Sign(M_j, key)
2:   = SignDigest(H(M_j), key)
3:   = SignDigest(H(m_i), key)
4:   = s



Unbeknownst to the victim, by signing m_i , they have also signed M_j.

# 2. W-OTS & W-OTS+

The Winternitz scheme is a well-documented[4][5] scheme whose description is beyond the scope of this document. However, of relevance is the relationship between the W-OTS "security parameter" n (the bit-length of H ) and it's "security level" which is generally n/2 . This follows from the fact that if a brute-force attack on H requires 2^n hash rounds then a birthday attack requires 2^(n/2)[2] hash rounds. By eliminating the birthday attack, and assuming no such other class of attacks exist for H , the security level of the scheme is restored back to that of a brute-force attack on H which is n.

W-OTS+ achieves a similar security enhancement through obfuscation of pre-images in the hashing chains, however they are performed during the chaining function which adds an overhead (significant in some implementations). W-OTS# is similar to W-OTS+ in this regard except it only obfuscates the message-digest once via an HMAC (keyed with the salt) and uses the standard W-OTS chaining function, which is faster than W-OTS+. Despite the concatenation of the salt to the signature, the overall signature size decreases by virtue of selecting a shorter hash function H .

# 3. W-OTS#

The W-OTS# construction is identical to a standard W-OTS construction for Winternitz parameter w and cryptographic hash function H. The security parameter n is inferred from the the bitlength of H.

In W-OTS, a message-digest md is computed as md=H(message) . During signing, digits of base 2^w are read from md and signed in a Winternitz chain. In W-OTS#, the message-digest md is replaced with the "sig-mac" smac defined as:

### 3.1 Signature Message Authentication Code (SMAC)


1: smac = SMAC(m, salt)
2:      = HMAC(H(m), salt)
3:      = H(Salt || H(Salt || H(m)))



The salt is concatenated to the signature and used to compute smac during verification.

NOTE the checksum digits are calculated and signed identically as per W-OTS but derived from smac not md.

## 3.2 Salt

The Salt is generated by the signer using cryptographic random number generator. The length of the Salt is n bits which is the minimum value required to nullify a birthday attack (proven below). The salt is defined as:


1: Salt = {0,1}^n   (i.e. n cryptographically random bits)



### 3.2.2 Proof

1. A birthday-collision is expected after 1.25 * SQRT(U)[2] hashing rounds where U is maximum hashing rounds ever required (non-repeating).
2. In W-OTS, U=2^n where n is the security parameter (bits-length of H) and thus (1) becomes 1.25 * 2^(n/2) .
3. In W-OTS#, adding a d -bit salt hardens a birthday-collision to A = 1.25 * 2^((n+d)/2) rounds. This follows from the fact that an attacker must scan for collision (HMAC(H(m_i), Salt), HMAC(H(M_j), Salt)) which involves d more bits (whereas in W-OTS they just scan for (H(m_i), H(M_j)) ).
4. A brute-force attack on H requires B = 2^n hashing rounds[2].
5. We need to choose d such A = B, since we only need to harden a birthday attack to match that of a brute-force attack. Hardening beyond is redundant since the security level of the scheme is only as strong as the weakest attack vector.
6. Evaluating (5) gives d = 2 ln(0.8)/ln(0.2) + n = 0.2773 + n which is approximately n
7. Thus choosing d=n is sufficient to thwart birthday-attack. QED.

# References

1. Hülsing, A. "W-OTS+ -Shorter Signatures for Hash-Based Signature Schemes". 2013. Url: https://eprint.iacr.org/2017/965.pdf. Accessed: 2020-07-22.
2. Wikipedia. "Birthday Attack". Url: https://en.wikipedia.org/wiki/Birthday_attack. Accessed: 2020-07-22
3. Wikipedia. "Birthday Problem". Url: https://en.wikipedia.org/wiki/Birthday_problem. Accessed: 2020-07-22
4. Ralph Merkle. "Secrecy, authentication and public key systems / A certified digital signature". Ph.D. dissertation, Dept. of Electrical Engineering, Stanford University, 1979. Url: http://www.merkle.com/papers/Certified1979.pdf
5. Sphere 10 Software. "Winternitz One-Time Signature Scheme (W-OTS)". URL: https://sphere10.com/articles/cryptography/pqc/wots.

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