Fiat–Shamir heuristic

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In cryptography, the Fiat–Shamir heuristic is a technique for taking an interactive proof of knowledge and creating a digital signature based on it. This way, some fact (for example, knowledge of a certain secret number) can be publicly proven without revealing underlying information. The technique is due to Amos Fiat and Adi Shamir (1986).[1] For the method to work, the original interactive proof must have the property of being public-coin, i.e. verifier's random coins are made public throughout the proof protocol.

The heuristic was originally presented without a proof of security; later, Pointcheval and Stern[2] proved its security against chosen message attacks in the random oracle model, that is, assuming random oracles exist. This result was generalized to the quantum-accessible random oracle (QROM) by Don, Fehr, Majenz and Schaffner[3], and concurrently by Liu and Zhandry[4]. In the case that random oracles do not exist, the Fiat–Shamir heuristic has been proven insecure by Shafi Goldwasser and Yael Tauman Kalai.[5] The Fiat–Shamir heuristic thus demonstrates a major application of random oracles. More generally, the Fiat–Shamir heuristic may also be viewed as converting a public-coin interactive proof of knowledge into a non-interactive proof of knowledge. If the interactive proof is used as an identification tool, then the non-interactive version can be used directly as a digital signature by using the message as part of the input to the random oracle.

Example[edit]

For the algorithm specified below, readers should be familiar with the laws of modular arithmetic, especially with multiplicative groups of integers modulo n with prime n.

Here is an interactive proof of knowledge of a discrete logarithm.[6]

  1. Peggy wants to prove to Victor the verifier that she knows : the discrete logarithm of to the base (mod n).
  2. She picks a random , computes and sends to Victor.
  3. Victor picks a random and sends it to Peggy.
  4. Peggy computes and returns to Victor.
  5. He checks whether . This holds because .

Fiat–Shamir heuristic allows to replace the interactive step 3 with a non-interactive random oracle access. In practice, we can use a cryptographic hash function instead.[7]

  1. Peggy wants to prove that she knows : the discrete logarithm of to the base .
  2. She picks a random and computes .
  3. Peggy computes , where is a cryptographic hash function.
  4. She computes . The resulting proof is the pair . As is an exponent of , it is calculated modulo , not modulo .
  5. Anyone can check whether .

If the hash value used below does not depend on the (public) value of y, the security of the scheme is weakened, as a malicious prover can then select a certain value x so that the product cx is known.[8]

Extension of this method[edit]

As long as a fixed random generator can be constructed with the data known to both parties, then any interactive protocol can be transformed into a non-interactive one.[citation needed]

See also[edit]

References[edit]

  1. ^ Fiat, Amos; Shamir, Adi (1987). "How To Prove Yourself: Practical Solutions to Identification and Signature Problems". Advances in Cryptology — CRYPTO' 86. Lecture Notes in Computer Science. Springer Berlin Heidelberg. 263: 186–194. doi:10.1007/3-540-47721-7_12. ISBN 978-3-540-18047-0.
  2. ^ Pointcheval, David; Stern, Jacques (1996). "Security Proofs for Signature Schemes". Advances in Cryptology — EUROCRYPT '96. Lecture Notes in Computer Science. Springer Berlin Heidelberg. 1070: 387–398. doi:10.1007/3-540-68339-9_33. ISBN 978-3-540-61186-8.
  3. ^ Don, Jelle; Fehr, Serge; Majenz, Christian; Schaffner, Christian (2019). "Security of the Fiat-Shamir Transformation in the Quantum Random-Oracle Model". Advances in Cryptology — CRYPTO '19. Lecture Notes in Computer Science. Springer Cham. 11693: 356–383. arXiv:1902.07556. Bibcode:2019arXiv190207556D. doi:10.1007/978-3-030-26951-7_13. ISBN 978-3-030-26950-0.
  4. ^ Liu, Qipeng; Zhandry, Mark (2019). "Revisiting Post-quantum Fiat-Shamir". Advances in Cryptology — CRYPTO '19. Lecture Notes in Computer Science. Springer Cham. 11693: 326–355. doi:10.1007/978-3-030-26951-7_12. ISBN 978-3-030-26950-0.
  5. ^ Goldwasser, S.; Kalai, Y. T. (October 2003). "On the (In)security of the Fiat-Shamir paradigm". 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.: 102–113. doi:10.1109/SFCS.2003.1238185. ISBN 0-7695-2040-5.
  6. ^ Camenisch, Jan; Stadler, Markus (1997). "Proof Systems for General Statements about Discrete Logarithms" (PDF). Dept. Of Computer Science, ETH Zurich.
  7. ^ Bellare, Mihir; Rogaway, Phillip (1995). "Random Oracles are Practical: A Paradigm for Designing Efficient Protocols". ACM Press: 62–73. CiteSeerX 10.1.1.50.3345. Cite journal requires |journal= (help)
  8. ^ Bernhard, David; Pereira, Olivier; Warinschi, Bogdan. "How not to Prove Yourself: Pitfalls of the Fiat-Shamir Heuristic and Applications to Helios" (PDF). In Wang, Xiaoyun; Sako, Kazue (eds.). Advances in Cryptology – ASIACRYPT 2012. pp. 626–643.