scholarly journals A new method for solving the elliptic curve discrete logarithm problem

2021 ◽  
Vol volume 12, issue 2 ◽  
Author(s):  
Ansari Abdullah ◽  
Ayan Mahalanobis ◽  
Vivek M. Mallick
Keyword(s):  
Elliptic Curve ◽  
Paradigm Shift ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
New Method ◽  
Cryptographic Primitive ◽  
Schur Complements

The elliptic curve discrete logarithm problem is considered a secure cryptographic primitive. The purpose of this paper is to propose a paradigm shift in attacking the elliptic curve discrete logarithm problem. In this paper, we will argue that initial minors are a viable way to solve this problem. This paper will present necessary algorithms for this attack. We have written a code to verify the conjecture of initial minors using Schur complements. We were able to solve the problem for groups of order up to $2^{50}$. Comment: 13 pages; revised for publication

scholarly journals On the first fall degree of summation polynomials

2019 ◽  
Vol 13 (3-4) ◽  
pp. 229-237
Author(s):  
Stavros Kousidis ◽  
Andreas Wiemers
Keyword(s):  
Elliptic Curve ◽  
Gröbner Basis ◽  
Discrete Logarithm ◽  
Polynomial Systems ◽  
Discrete Logarithm Problem ◽  
Grobner Basis ◽  
Weil Descent ◽  
Degree Bound

Abstract We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

Problems in Cryptography and Cryptanalysis

2018 ◽  
pp. 23-39
Author(s):  
Kannan Balasubramanian ◽  
Rajakani M.
Keyword(s):  
Elliptic Curve ◽  
Optimization Problems ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Random Number Generators ◽  
Factorization Problem ◽  
Diffie Hellman ◽  
Integer Factorization Problem ◽  
Diffie Hellman Key Exchange ◽  
Membership Problems

The integer factorization problem used in the RSA cryptosystem, the discrete logarithm problem used in Diffie-Hellman Key Exchange protocol and the Elliptic Curve Discrete Logarithm problem used in Elliptic Curve Cryptography are traditionally considered the difficult problems and used extensively in the design of cryptographic algorithms. We provide a number of other computationally difficult problems in the areas of Cryptography and Cryptanalysis. A class of problems called the Search problems, Group membership problems, and the Discrete Optimization problems are examples of such problems. A number of computationally difficult problems in Cryptanalysis have also been identified including the Cryptanalysis of Block ciphers, Pseudo-Random Number Generators and Hash functions.

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