The Schnorr signature scheme [6] is derived from Schnorr's identification protocol using the Fiat– Shamir heuristic [2]. The resulting digital signature scheme is related to the Digital Signature Standard (DSS). As in DSS, the system works in a subgroup of the group \(\mathbb{Z}_{p}^{\ast}\) for some prime number p. The resulting signatures have the same length as DSS signatures. The signature scheme works as follows:
Key Generation. Same as in the DSS system. Given two security parameters \(\tau, \lambda \in \mathbb{Z}\) (\(\tau > \lambda\)) as input do the following:
- 1.
Generate a random λ-bit prime q.
- 2.
Generate a random τ-bit prime prime p such that q divides \(p-1\).
- 3.
Pick an element \(g \in \mathbb{Z}_{p}^{\ast}\) of order q.
- 4.
Pick a random integer \(\alpha \in [1,q]\) and compute \(y = g^\alpha \in \mathbb{Z}_{p}^{\ast}\).
- 5.
Let H be a hash function \(H:\{0,1\}^* \to \mathbb{Z}_q\).
The resulting public key is \((p,q,g,y,H)\). The private key is \((p,q,g,\alpha,H)\).
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References
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Fiat, Amos and Adi Shamir (1986). “How to prove yourself: Practical solutions to identification and signature problems.” Advances in Cryptology—CRYPTO'86, Lecture Notes in Computer Science, vol. 263, ed. Andrew M. Odlyzko. Springer-Verlag, Berlin, 186–194.
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Ohta, Kazuo and Tatsuaki Okamoto (1998). “On concrete security treatment of signatures derived from identification.” Advances in Cryptology—CRYPTO'98, Lecture Notes in Computer Science, vol. 1462, ed. H. Krawczyk. Springer-Verlag, Berlin, 354–369.
Schnorr, C. (1991). “Efficient signature generation by smart cards.” Journal of Cryptology, 4 (3), 161–174.
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Boneh, D. (2005). Schnorr Digital Signature Scheme. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_369
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