The Hidden Subgroup Problem and MKTP

The Hidden Subgroup Problem and Non-interactive Perfect Zero-Knowledge Proofs

“…Consider a family of groups B = {B n } n≥1 such that: the elements of B n are uniquely represented by words of length poly (n); inverse, product, and identity testing operations of each B n are computed in poly (n) time, denoting by e the identity element of B n . The formal definition of HSP that we consider is the one proposed by [Sdroievski et al 2019], in which we are given a positive integer n (in unary) and a boolean circuit C f that takes encodings of elements of a group G ⊆ B n as input and returns an output of m bits, being m a positive integer. We assume that…”

“…To verify it, we check if f (h) = f (e). It is shown in [Sdroievski et al 2019] a perfect zero-knowledge protocol with honest verifier for dHSP, establishing that dHSP ∈ HVPZK. Furthermore, [Sdroievski et al 2019] showed that if we know the size of the group G, as it is the case for permutation groups [Seress 2003], then there is a polynomial Karp reduction from dHSP to the Entropy Approximation Problem (EA), a complete promise problem for NISZK.…”

“…The most commonly found definition in the literature for HSP is the one introduced by [Babai and Szemerédi 1984] in a black-box context. Determining whether a hidden subgroup is the trivial subgroup or not is a wellstudied definition for a decision version of HSP [Ettinger et al 2004, Hayashi et al 2008, Sdroievski et al 2019 and it is the one used in this paper, although it is not the only decision version (technical details and further definitions are discussed in the sequel). This decision version (dHSP) was shown to be in the zero-knowledge class HVPZK and, if the size of the group is known, it also belongs to the class NISZK [Sdroievski et al 2019].…”

Intractable Group-theoretic Problems Around Zero-knowledge Proofs

“…They showed that the permutation group problems Coset Intersection, Double Coset Membership, Group Conjugacy are in perfect zero-knowledge class. Sdroievski et al showed that the hidden subgroup problem has a statistical zero-knowledge proof [22].…”