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. It is considered the simplest digital signature scheme to be provably secure in a random oracle model^{[citation needed]}. 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 $G$ with generator $g$ of prime order $q$ 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
• $x, k \in \mathbb{Z}_q^*$, the set of congruence classes modulo $q$ excluding $\overline{0}_q$
• $s, e, e_v \in \mathbb{Z}_q$, the set of congruence classes modulo $q$
• $y, r, r_v \in G$.

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 $e = H(M || r)$, where || 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 $e_v = H(M || r_v)$

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

Demonstration 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 $e_v = H(M || r_v) = H(M || r) = e$.

The other direction goes through using the fact that $k<q$ and the assumption that the hash function is collision-resistant.

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

See also

• DSA
• ElGamal signature scheme

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.