Schnorr signature

In cryptography, a Schnorr signature is a digital signature produced by the Schnorr signature algorithm. Its security is based on the intractability of certain discrete logarithm problems. The Schnorr signature is considered the simplest digital signature scheme to be provably secure in a random oracle model. It is efficient and generates short signatures. It is covered by U.S. Patent 4,995,082, which expired in February 2008.

Algorithm

Choosing parameters

• All users of the signature scheme agree on a group, $G$, of prime order, $q$, with generator, $g$, in which the discrete log problem is hard. Typically a Schnorr group is used.
• All users agree on a cryptographic hash function $H: \{0,1\}^* \rightarrow \mathbb{Z}_q$.

Notation

In the following,

• Exponentiation stands for repeated application of the group operation
• Juxtaposition stands for multiplication on the set of congruence classes or application of the group operation (as applicable)
• Subtraction stands for subtraction on set of equivalence groups
• $M \in \{0,1\}^*$, the set of finite bit strings
• $s, e, e_v \in \mathbb{Z}_q$, the set of congruence classes modulo $q$
• $x, k \in \mathbb{Z}_q^\times$, the multiplicative group of integers modulo $q$ (for prime $q$, $\mathbb{Z}_q^\times = \mathbb{Z}_q \setminus \overline{0}_q$)
• $y, r, r_v \in G$.

Key generation

• Choose a private signing key, $x$, from the allowed set.
• The public verification key is $y = g^x$.

Signing

To sign a message, $M$:

• Choose a random $k$ from the allowed set.
• Let $r = g^k$.
• Let $e = H(M \parallel r)$, where $\parallel$ denotes concatenation and $r$ is represented as a bit string.
• Let $s = k - xe$.

The signature is the pair, $(s, e)$.

Note that $s, e \in \mathbb{Z}_q$; if $q < 2^{160}$, then the signature representation can fit into 40 bytes.

Verifying

• Let $r_v = g^s y^e$
• Let $e_v = H(M \parallel r_v)$

If $e_v = e$ then the signature is verified.

Proof of correctness

It is relatively easy to see that $e_v = e$ if the signed message equals the verified message:

$r_v = g^s y^e = g^{k - xe} g^{xe} = g^k = r$, and hence $e_v = H(M \parallel r_v) = H(M \parallel r) = e$.

Public elements: $G$, $g$, $q$, $y$, $s$, $e$, $r$. Private elements: $k$, $x$.

This shows only that a correctly signed message will verify correctly; many other properties are required for a secure signature algorithm.

Security argument

The signature scheme was constructed by applying the Fiat–Shamir transform to Schnorr's identification protocol. Therefore, (per Fiat and Shamir's arguments), it is secure if $H$ is modeled as a random oracle.

Its security can also be argued in the generic group model, under the assumption that $H$ is "random-prefix preimage resistant" and "random-prefix second-preimage resistant". In particular, $H$ does not need to be collision resistant.

In 2012, Seurin provided an exact proof of the Schnorr signature scheme. In particular, Seurin shows that the security proof using the Forking lemma is the best possible result for any signature schemes based on one-way group homomorphisms including Schnorr-Type signatures and the Guillou-Quisquater signature schemes. Namely, under the ROMDL assumption, any algebraic reduction must lose a factor $f({\epsilon}_F)q_h$ in its time-to-success ratio, where $f \le 1$ is a function that remains close to 1 as long as "${\epsilon}_F$ is noticeably smaller than 1", where ${\epsilon}_F$ is the probability of forging an error making at most $q_h$ queries to the random oracle.

See also

• DSA
• ElGamal signature scheme
• EdDSA

References

• C.P. Schnorr, Efficient identification and signatures for smart cards, in G. Brassard, ed. Advances in Cryptology—Crypto '89, 239-252, Springer-Verlag, 1990. Lecture Notes in Computer Science, nr 435
• Claus-Peter Schnorr, Efficient Signature Generation by Smart Cards, J. Cryptology 4(3), pp161–174 (1991) (PS).
• Menezes, Alfred J. et al. Handbook of Applied Cryptography CRC Press. 1996.

External links

• Schnorr IETF draft