Volume 7, Issue 1

Abstract. One of the fundamental problems of multi-party computation is Oblivious Polynomial Evaluation. In that problem, that was introduced by Naor and Pinkas, Alice has a polynomial and Bob has a point . The goal is to allow Bob to compute so that Alice remains oblivious of and Bob of , apart from what is implied by and . We introduce the multivariate version of this problem, where x and are vectors, and offer an efficient secure protocol. In addition, we discuss several applications that may be solved efficiently using oblivious multivariate polynomial evaluation, such as private linear algebraic computations and private support vector machines (SVM).

Abstract. A self-pairing is a pairing computation where both inputs are the same group element. Self-pairings are used in some cryptographic schemes and protocols. In this paper, we show how to compute the Tate–Lichtenbaum pairing on a curve more efficiently than the general case. The speedup is obtained by using a simpler final exponentiation. We also discuss how to use this pairing in cryptographic applications.

Abstract. The connection between exponential sums and algebraic varieties has been known for at least six decades. Recently, Lisoněk exploited it to reformulate the Charpin–Gong characterization of a large class of hyper-bent functions in terms of numbers of points on hyperelliptic curves. As a consequence, he obtained a polynomial time and space algorithm for certain subclasses of functions in the Charpin–Gong family. In this paper, we settle a more general framework, together with detailed proofs, for such an approach and show that it applies naturally to a distinct family of functions proposed by Mesnager. Doing so, a polynomial time and space test for the hyper-bentness of functions in this family is obtained as well. Nonetheless, a straightforward application of such results does not provide a satisfactory criterion for explicit generation of functions in the Mesnager family. To address this issue, we show how to obtain a more efficient test leading to a substantial practical gain. We finally elaborate on an open problem about hyperelliptic curves related to a family of Boolean functions studied by Charpin and Gong.

Abstract. As one of the most fundamental problems in multivariate public key cryptosystems (MPKC), Isomorphism of Polynomial (IP) induces an equivalence relation on the polynomials systems. The enumeration problem associated to IP consists of counting the number of equivalence classes and the cardinality of each class. The two problems correspond exactly to the study of the total number of different cryptographic schemes and the size of “equivalent keys”. In this paper we give an algorithm using a divide-and-conquer method to count the number of equivalence classes according to the linear equivalence relation induced by the IP1S problem when . Then by giving the complexity gain of this algorithm compared with the exhaustive search algorithm and the experimental results, we show the high efficiency of this new algorithm.