The Fiat–Shamir Transformation in a Quantum World

Abstract

Abstract. The Fiat-Shamir transformation is a famous technique to turn identification schemes into signature schemes. The derived scheme is provably secure in the random-oracle model against classical adversaries. Still, the technique has also been suggested to be used in connection with quantum-immune identification schemes, in order to get quantum-immune signature schemes. However, a recent paper by Boneh et al. (Asiacrypt 2011) has raised the issue that results in the random-oracle model may not be immediately applicable to quantum adversaries, because such adversaries should be allowed to query the random oracle in superposition. It has been unclear if the Fiat-Shamir technique is still secure in this quantum oracle model (QROM).Here, we discuss that giving proofs for the Fiat-Shamir transformation in the QROM is presumably hard. We show that there cannot be black-box extractors, as long as the underlying quantum-immune identification scheme is secure against active adversaries and the first message of the prover is independent of its witness. Most schemes are of this type. We then discuss that for some schemes one may be able to resurrect the Fiat-Shamir result in the QROM by modifying the underlying protocol first. We discuss in particular a version of the Lyubashevsky scheme which is provably secure in the QROM.

Practical Lattice-Based Digital Signature Schemes

“…Although making the DSSs less efficient, schemes by Gentry et al [2008] and Lyubashevsky [2012] are respectively shown by Boneh and Zhandry [2013] and Dagdelen et al [2013] to be secure to such an adversary, creating the quantum random oracle model. This could also motivate an important area for future research, such as proving security for more DSSs to a quantum adversary or possibly creating a generic technique that could turn a DSS secure in the random oracle model to one secure in the quantum random oracle model.…”

Quantum Attacks on Classical Proof Systems: The Hardness of Quantum Rewinding

“…Let X 2 be the set of all N 2 size-two subsets of X. In addition to (14), we are also interested in the following two representations of W defined by its action on the sets Y × X and Y × X 2 , respectively: (σ, π) : (y, x) → (π(y), σ π(y) (x)),…”

“…They also do not allow the extractor to use a purified (i.e., unitary) adversary to avoid measurements that introduce randomness. 3 The unforgeability proof from [14] is already almost a proof of the proof of knowledge property. And the techniques from [33] can probably be applied to show that the protocol form [14] is zero-knowledge.…”

Isogeny-Based Quantum-Resistant Undeniable Signatures

“…Isogeny-based cryptosystems to date have dealt primarily with encryption, with the exception of the entity authentication protocol of [13, §3.1]. We remark that, although entity authentication in the classical setting enables data authentication via the Fiat-Shamir transformation [14], the Fiat-Shamir transformation fails against a quantum adversary [10]. This work, together with Sun et al's construction of strong designated verifier signatures [30], provides some evidence that isogenies can also be used as the basis for signatures and data authentication in the post-quantum setting.…”