Fiat-Shamir Heuristic

The Fiat–Shamir heuristic is a technique for taking an Interactive Proof of Knowledge and creating a digital signature based on it.

Example

Interactive Version

We have a prime \(n\).

Here is an interactive proof of knowledge of a discrete logarithm.<ref>Template:Cite journal</ref>

1. Peggy wants to prove to Victor the verifier that she knows \(x\): the discrete logarithm of \(y = g^x\) to the base \(g\) (mod n).
2. She picks a random \(v\in \Z^*_q\), computes \(t = g^v\) and sends \(t\) to Victor.
3. Victor picks a random \(c\in \Z^*_q\) and sends it to Peggy.
4. Peggy computes \(r = v - cx\) and returns \(r\) to Victor.
5. He checks whether \(t \equiv g^ry^c\). This holds because \(g^ry^c = g^{v - cx}g^{xc} = g^v = t\).

Non-Interactive Version

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.<ref>Template:Cite journal</ref>

1. Peggy wants to prove that she knows \(x\): the discrete logarithm of \(y = g^x\) to the base \(g\).
2. She picks a random \(v\in\Z^*_q\) and computes \(t = g^v\).
3. Peggy computes \(c = H(g,y,t)\), where \(H()\) is a cryptographic hash function.
4. She computes \(r = v - cx\). The resulting proof is the pair \((t,r)\). As \(r\) is an exponent of \(g\), it is calculated modulo \(q-1\), not modulo \(q\).
5. Anyone can check whether \(t \equiv g^ry^c\).

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.<ref>Template:Cite book</ref>

Relevant papers

It was first introduced in a paper by Amos Fiat and Adi Shamir (1986).<ref>Template:Cite journal</ref>.

Assumptions

1. The first assumption is that the original interactive proof must be public coin
2. The second assumption is the existence of Random Oracles i.e. the heuristic only holds in the Random Oracle Model

Security

The heuristic was originally presented without a proof of security; later, Pointcheval and Stern<ref>Template:Cite journal</ref> 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<ref>Template:Cite journal</ref>, and concurrently by Liu and Zhandry<ref>Template:Cite journal</ref>. In the case that random oracles do not exist, the Fiat–Shamir heuristic has been proven insecure by Shafi Goldwasser and Yael Tauman Kalai.<ref>Template:Cite journal</ref> 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.

References

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