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Schnorr signature

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Title: Schnorr signature  
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Subject: Claus P. Schnorr, Digital signature, EdDSA, Proof of knowledge, Benaloh cryptosystem
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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[1] digital signature scheme to be provably secure in a random oracle model.[2] It is efficient and generates short signatures. It is covered by U.S. Patent 4,995,082, which expired in February 2008.

Contents

  • Algorithm 1
    • Choosing parameters 1.1
    • Notation 1.2
    • Key generation 1.3
    • Signing 1.4
    • Verifying 1.5
    • Proof of correctness 1.6
    • Security argument 1.7
  • See also 2
  • References 3
  • External links 4

Algorithm

Choosing parameters

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 \| 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.

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 \| r_v) = H(M \| r) = e.

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

Security argument

The signature scheme was constructed by applying the Fiat–Shamir transform[3] to Schnorr's identification protocol.[4] 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".[5] In particular, H does not need to be collision resistant.

In 2012, Seurin[2] provided an exact proof of the Schnorr signature scheme. In particular, Seurin shows that the security proof using the

  • Schnorr IETF draft

External links

  • 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.
  1. ^ Savu, Laura (2012). "SIGNCRYPTION SCHEME BASED ON SCHNORR DIGITAL SIGNATURE" (PDF). arXiv.org. 
  2. ^ a b Seurin, Yannick (2012-01-12). "On the Exact Security of Schnorr-Type Signatures in the Random Oracle Model" (PDF). Cryptology ePrint Archive. International Association for Cryptologic Research. Retrieved 2014-08-11. 
  3. ^ Fiat; Shamir (1986). "How To Prove Yourself: Practical Solutions to Identification and Signature Problems" (PDF). Proceedings of CRYPTO '86. 
  4. ^ Schnorr (1989). "Efficient Identification and Signatures for Smart Cards" (PDF). Proceedings of CRYPTO '89. 
  5. ^ Neven, Smart, Warinschi. "Hash Function Requirements for Schnorr Signatures". IBM Research. Retrieved 19 July 2012. 

References

See also
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