Polynomial representations of the Diffie-Hellman mapping (original) (raw)
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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.
A Polynomial Representation of the Diffie-Hellman Mapping
Applicable Algebra in Engineering, Communication and Computing, 2002
Let F q be the finite field of order q and γ be an element of F q of order d. The construction of an explicit polynomial f (X) ∈ F q [X] of degree ≤ d − 1 with the property f γ i = γ i 2 for 0 ≤ i ≤ d − 1 is described. In particular the exact degree and sparsity of f are determined.
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
Discrete Applied Mathematics, 2006
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.
On the complexity of the discrete logarithm and Diffie–Hellman problems
Journal of Complexity, 2004
The discrete logarithm problem plays a central role in cryptographic protocols and computational number theory. To establish the exact complexity, not only of the discrete logarithm problem but also of its relatives, the Diffie-Hellman (DH) problem and the decision DH problem, is of some importance. These problems can be set in a variety of groups, and in some of these they can assume different characteristics. This work considers the bit complexity of the DH and the decision DH problems. It was previously shown by Boneh and Venkatesan that it is as hard to compute Oð ffiffi ffi n p Þ of the most significant bits of the DH function, as it is to compute the whole function, implying that if the DH function is difficult then so is computing this number of bits of it. The main result of this paper is to show that if the decision DH problem is hard then computing the two most significant bits of the DH function is hard. To place the result in perspective a brief overview of relevant recent advances on related problems is given.
Comparison of the complexity of Diffie–Hellman and discrete logarithm problems
Journal of Computer Virology and Hacking Techniques, 2020
The article presents an algorithm for solving the discrete logarithm problem with an oracle, solving the Diffie-Hellman problem. Certified the discrete logarithm problem is considered. The Diffie-Hellman oracle works with elements of the original group, but with new group operations that are compositions of the Diffie-Hellman oracle. In particular, a universal (generic) algorithm can be substituted as the Diffie-Hellman oracle. The result is improved since 1996-the degree of logarithm in the estimation of the complexity of the algorithm presented is reduced to one. Of course, this does not affect the property of polynomial reduction of the considered problems to each other but excludes from the evaluation in a sense unnecessary terms.
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 bit security of the Diffie-Hellman key
Applicable Algebra in Engineering, Communication and Computing, 2006
Let IF p be a finite field of p elements, where p is prime. The bit security of the Diffie-Hellman function over subgroups of IF * p and of an elliptic curve over IF p , is considered. It is shown that if the Decision Diffie-Hellman problem is hard in these groups, then the two most significant bits of the Diffie-Hellman function are secure. Under the weaker assumption of the computational (rather than decisional) hardness of the Diffie-Hellman problems, only about (log p) 1/2 bits are known to be secure.
Linear Complexity and Polynomial Degree of a Function Over a Finite Field
Finite Fields with Applications to Coding Theory, Cryptography and Related Areas, 2002
We compare the complexities of the polynomial representation and the periodic sequence representation of a function over a finite field in the complexity measures degree and linear complexity. We prove a sharp inequality describing the relation between degree and linear complexity. These investigations are motivated by results on some cryptographic functions. In particular, as an application of the above mentioned inequality we prove new lower bounds on the linear complexity of sequences related to the Diffie-Hellman mapping.