A Polynomial Representation of the Diffie-Hellman Mapping (original) (raw)

Polynomial representations of the Diffie-Hellman mapping

Bulletin of the Australian Mathematical Society, 2001

We obtain lower bounds on the degrees of polynomials representing the Diffie-Hellman mapping (gx, gy) → gxy, where g is a primitive root of a finite field q of q elements. These bounds are exponential in terms of log q. In particular, these results can be used to obtain lower bounds on the parallel arithmetic complexity of breaking the Diffie-Hellman cryptosystem. The method is based on bounds of numbers of solutions of some polynomial equations.

A note on the interpolation of the Diffie-Hellman mapping

Bulletin of the Australian Mathematical Society, 2001

We obtain lower bounds on the degrees of polynomials representing the Diffie-Hellman mapping f (γx, γy) = γxy, where γ is a nonzero element of Fq of order d, x runs through a subset of [0,d – 1], and y runs through a set of consecutive integers.

On the interpolation of bivariate polynomials related to the Diffie-Hellman mapping

Bulletin of the Australian Mathematical Society, 2004

We obtain lower bounds on degree and weight of bivariate polynomials representing the Diffie-Hellman mapping for finite fields and the Diffie-Hellman mapping for elliptic curves over finite fields. This complements and improves several earlier results. We also consider some closely related bivariate mappings called P-Diffie-Hellman mappings introduced by the first author. We show that the existence of a low degree polynomial representing a P-Diffie-Hellman mapping would lead to an efficient algorithm for solving the Diffie-Hellman problem. Motivated by this result we prove lower bounds on weight and degree of such interpolation polynomials, as well. P -d h ( 7 I , 7 y ) = 7 i ' ( l ' ! ' ) , for a bivariate polynomial P of small degree D > 2 with respect to d. (See also for the univariate analogue.) If D is small then these investigations are motivated by an efficient

Lower bounds on weight and degree of bivariate polynomials related to the Diffie-Hellman mapping

We obtain lower bounds on degree and weight of bivariate polynomials representing the Diffie-Hellman mapping for finite fields and the Diffie-Hellman mapping for elliptic curves over finite fields. This complements and improves several earlier results. We also consider some closely related bivariate mappings called P -Diffie-Hellman mappings introduced by the first author. We show that the existence of a low degree polynomial representing a P -Diffie-Hellman mapping would lead to an efficient algorithm for solving the Diffie-Hellman problem. Motivated by this result we prove lower bounds on weight and degree of such interpolation polynomials, as well.

Polynomial Interpolation of Cryptographic Functions Related to the Diffie-Hellman Problem

2003

Recently, the first author introduced some cryptographic functions closely related to the Diffie-Hellman problem called P-Diffie-Hellman functions. We show that the existence of a low-degree polynomial representing a P-Diffie-Hellman function on a large set would lead to an efficient algorithm for solving the Diffie-Hellman problem. Motivated by this result we prove lower bounds on the degree of such interpolation polynomials. Analogously, we introduce a class of functions related to the discrete logarithm and show similar reduction and interpolation results.

Polynomial approximation of bilinear Diffie–Hellman maps

Finite Fields and Their Applications, 2008

The problem of computing Bilinear-Diffie-Hellman maps is considered. It is shown that the problem of computing the map is equivalent to computing a diagonal version of it. Various lower bounds on the degree of any polynomial that interpolates this diagonal version of the map are found that shows that such an interpolation will involve a polynomial of large degree, relative to the size of the set on which it interpolates.

On the Index of Diffie-Hellman Mapping

2020

Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that f(x)x^-r is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie-Hellman mapping d(γ^a)=γ^a^2, a=0,1,…,n-1, and show that any mapping of small index coincides with d only on a small subset of G. Moreover, we prove similar results for the bivariate Diffie-Hellman mapping D(γ^a,γ^b)=γ^ab, a,b=0,1,…,n-1. In the special case that G is a subgroup of the multiplicative group of a finite field we present improvements.

On the index of the Diffie–Hellman mapping

arXiv: Combinatorics, 2020

Let gamma\gammagamma be a generator of a cyclic group GGG of order nnn. The least index of a self-mapping fff of GGG is the index of the largest subgroup UUU of GGG such that f(x)x−rf(x)x^{-r}f(x)x−r is constant on each coset of UUU for some positive integer~$r$. We determine the index of the univariate Diffie-Hellman mapping d(gammaa)=gammaa2d(\gamma^a)=\gamma^{a^2}d(gammaa)=gammaa2, a=0,1,ldots,n−1a=0,1,\ldots,n-1a=0,1,ldots,n−1, and show that any mapping of small index coincides with~$d$ only on a small subset of GGG. Moreover, we prove similar results for the bivariate Diffie-Hellman mapping D(gammaa,gammab)=gammaabD(\gamma^a,\gamma^b)=\gamma^{ab}D(gammaa,gammab)=gammaab, a,b=0,1,ldots,n−1a,b=0,1,\ldots,n-1a,b=0,1,ldots,n−1. In the special case that GGG is a subgroup of the multiplicative group of a finite field we present improvements.

Interpolation of the Elliptic Curve Diffie-Hellman Mapping

Lecture Notes in Computer Science, 2003

We prove lower bounds on the degree of polynomials interpolating the Diffie-Hellman mapping for elliptic curves over finite fields and some related mappings including the discrete logarithm. Our results support the assumption that the elliptic curve Diffie-Hellman key exchange and related cryptosystems are secure.

A note on complete polynomials over finite fields and their applications in cryptography

Finite Fields and Their Applications, 2014

A recursive construction of complete mappings over finite fields is provided in this work. These permutation polynomials, characterized by the property that both f (x) ∈ F q [x] and its associated mapping f (x) + x are permutations, have an important application in cryptography in the construction of bent-negabent functions which actually leads to some new classes of these functions. Furthermore, we also provide a recursive construction of mappings over finite fields of odd characteristic, having an interesting property that both f (x) and f (x + c) + f (x) are permutations for every c ∈ F q. Both the multivariate and univariate representations are treated and some results concerning fixed points and the cycle structure of these permutations are given. Finally, we utilize our main result for the construction of so-called negabent functions and bent functions over finite fields.