Kouichi Sakurai

On the Complexity of Constant Round ZKIP of Possession of Knowledge

In this paper, we show that if a relation R has a three move blackbox simulation zero-knowledge interactive proof system of possession of knowledge, then there exists a probabilistic polynomial time algorithm that on input x ∈ {0,1}*, outputs y such that (x, y) ∈ R with overwhelming probability if x ∈ dom R, and outputs “⊥” with probability 1 if x ∉ dom R. In the present paper, we also show that without any unproven assumption, there exists a four move blackbox simulation perfect zero-knowledge interactive proof system of possession of the prime factorization, which is optimal in the light of the round complexity.

Any Language in IP Has a Divertable ZKIP

A notion of “divertible” zero-knowledge interactive proof systems was introduced by Okamoto and Ohta, and they showed that for any commutative random self-reducible relation, there exists a divertible (perfect) zero-knowledge interactive proof system of possession of information. In addition, Burmester and Desmedt proved that for any language L ∈ \(\mathcal{N}\mathcal{P}\), there exists a divertible zero-knowledge interactive proof system for the language L under the assumption that probabilistic encryption homomorphisms exist. In this paper, we classify the notion of divertible into three types, i.e., perfectly divertible, almost perfectly divertible, and computationally divertible, and investigate which complexity class of languages has a perfectly (almost perfectly) (computationally) divertible zero-knowledge interactive proof system. The main results in this paper are: (1) there exists a perfectly divertible perfect zero-knowledge interactive proof system for graph non-isomorphism (GNI) without any unproven assumption; and (2) for any language L having an interactive proof system, there exists a computationally divertible computational zero-knowledge interactive proof system for the language L under the assumption that probabilistic encryption homomorphisms exist.

On the complexity of hyperelliptic discrete logarithm problem

We give a characterization for the intractability of hyperelliptic discrete logarithm problem from a viewpoint of computational complexity theory. It is shown that the language of which complexity is equivalent to that of the hyperelliptic discrete logarithm problem is in NP ∩ co-AM, and that especially for elliptic curves, the corresponding language is in NP ∩ co-NP. It should be noted here that the language of which complexity is equivalent to that of the discrete logarithm problem defined over the multiplicative group of a finite field is also characterized as in NP ∩ co-NP.

4 Move Perfect ZKIP of Knowledge with No Assumption

This paper presents a 4-move perfect ZKIP of knowledge with no cryptographic assumption for the random self reducible problems [TW87] whose domain is NP∩BPP. The certified discrete log problem is such an example. (Finding a witness is more difficult than the language membership problem.) A largely simplified 4-move ZKIP for the Hamilton Circuit problem is also shown. In our ZKIP, a trapdoor coin flipping protocol is introduced to generate a challenge bit. P and V cooperatively generate a random bit in a coin flipping protocol. In a trapdoor coin flipping protocol, V who knows the trapdoor can create the view which he can later reveal in two possible ways: both as head and as tail.

On the Discrepancy between Serial and Parallel of Zero-Knowledge Protocols

In this paper, we investigate the discrepancy between a serial version and a parallel version of zero-knowledge protocols, and clarify the information “leaked” in the parallel version, which is not zero-knowledge unlike the case of the serial version. We consider two sides: one negative and the other positive in the parallel version of zero-knowledge protocols, especially of the Fiat-Shamir scheme.

How intractable is the discrete logarithm for a general finite group

GDL is the discrete logarithm prablem for a general finite group G. This paper gives a characterization for the intractability of GDL from the viewpoint of computational complexity theory. It is shown that GDL ∈ NP ∩ co-AM, assuming that G is in NP ∩ co-NP, and that the group law operation of G can be cxecuted in a polynomial time of the element size. Furthermore, as a natural probabilistic extension, the complexity of GDL is investigated under the assumption that the group law operation is executed in an expected polynomial time of the element size. In this case, it is shown that GDL ∈ MA ∩ co-AM if G ∈ NP ∩ co-NP. Finally, we show that GDL is less intractable than NP-complete problems unless the polynomial time hierarchy collapses to the second level.

On Bit Correlations Among Preimages of Many to One One-Way Functions

This paper presents a new measure of the complexity of many to one functions. We study bit correlations among the preimages of an element of the range of many to one one-way functions. Especially, we investigate the correlation among the least significant bit of the preimages of 2 to 1 one-way functions based on algebraic problems such as the factorization and the discrete logarithm.