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Ayan Mahalanobis

Publications and source records attributed to Ayan Mahalanobis.

At least 19 recordsLinked to original sources
arXiv2026

A Guess and Determine Attack on the Elliptic Curve Discrete Logarithm Problem

This paper is a continuation of our earlier work, in which we described a Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The Las Vegas algorithm reduces the elliptic curve discrete logarithm problem to finding a zero minor in a matrix. Using the intersection poset of a hyperplane arrangement, we develop an algorithm to find a zero minor in a rectangular matrix. Our methods are elementary. We discuss the complexity and success probability of our algorithm and provide implementation details. Finding a zero minor in a matrix is also of independent interest.

cs.CR↗
arXiv2025

Symplectic Elimination

We develop the symplectic elimnation algorithm. This algorithm using simple row operations reduce a symplectic matrix to a diagonal matrix. This algorithm gives rise to a decomposition of an arbitrary matrix into a product of a symplectic matrix and a reduced matrix. This decomposition is similar to the SR decomposition studied for a long time, which is analogous to the QR decomposition.

math.GR↗
arXiv2023

Minors solve the elliptic curve discrete logarithm problem

The elliptic curve discrete logarithm problem is of fundamental importance in public-key cryptography. It is in use for a long time. Moreover, it is an interesting challenge in computational mathematics. Its solution is supposed to provide interesting research directions. In this paper, we explore ways to solve the elliptic curve discrete logarithm problem. Our results are mostly computational. However, it seems, the methods that we develop and directions that we pursue can provide a potent attack on this problem. This work follows our earlier work, where we tried to solve this problem by finding a zero minor in a matrix over the same finite field on which the elliptic curve is defined. This paper is self-contained.

cs.CR↗
arXiv2023

A McEliece cryptosystem using permutation codes

This paper is an attempt to build a new public-key cryptosystem; similar to the McEliece cryptosystem, using permutation error-correcting codes. We study a public-key cryptosystem built using two permutation error-correcting codes. We show that these cryptosystems are insecure. However, the general framework in these cryptosystems can use any permutation error-correcting code and is interesting. We present an enhanced McEliece cryptosystem which subsumes McEliece cryptosystem based on linear error correcting codes.

cs.IT↗
arXiv2021

Niederreiter cryptosystems using quasi-cyclic codes that resist quantum Fourier sampling

McEliece and Niederreiter cryptosystems are robust and versatile cryptosystems. These cryptosystems work with many linear error-correcting codes. They are popular these days because they can be quantum-secure. In this paper, we study the Niederreiter cryptosystem using non-binary quasi-cyclic codes. We prove, if these quasi-cyclic codes satisfy certain conditions, the corresponding Niederreiter cryptosystem is resistant to the hidden subgroup problem using weak quantum Fourier sampling. Though our work uses the weak Fourier sampling, we argue that its conclusions should remain valid for the strong Fourier sampling as well.

cs.CR↗
arXiv2021

A new method for solving the elliptic curve discrete logarithm problem

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 $2^{50}$.

cs.CR↗
arXiv2020

A remark on MAKE -- a Matrix Action Key Exchange

In a recent paper [arXiv:2009.00716], Rahman and Shpilrain proposed a new key-exchange protocol MAKE based on external semidirect product of groups. The purpose of this paper is to show that the key exchange protocol is insecure. We were able to break their challenge problem in under a second.

cs.CR↗
arXiv2020

Algorithms in Linear Algebraic Groups

This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decomposition with respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups.

math.GR↗
arXiv2018

A Las Vegas algorithm to solve the elliptic curve discrete logarithm problem

In this paper, we describe a new Las Vegas algorithm to solve the elliptic curve discrete logarithm problem. The algorithm depends on a property of the group of rational points of an elliptic curve and is thus not a generic algorithm. The algorithm that we describe has some similarities with the most powerful index-calculus algorithm for the discrete logarithm problem over a finite field.

cs.CR↗
arXiv2017

Gaussian Elimination in Symplectic and split orthogonal groups

This paper studies an algorithm similar to that of Gaussian elimination in symplectic and orthogonal groups. We discuss two applications of this algorithm in computational group theory. One computes the spinor norm and the other computes the double coset decomposition with respect to Siegel maximal parabolic subgroup.

math.GR↗
arXiv2017

A Probabilistic Baby-Step Giant-Step Algorithm

In this paper, a new algorithm to solve the discrete logarithm problem is presented which is similar to the usual baby-step giant-step algorithm. Our algorithm exploits the order of the discrete logarithm in the multiplicative group of a finite field. Using randomization with parallelized collision search, our algorithm indicates some weakness in NIST curves over prime fields which are considered to be the most conservative and safest curves among all NIST curves.

cs.CR↗
arXiv2016

On improvements of the $r$-adding walk in a finite field of characteristic 2

It is currently known from the work of Shoup and Nechaev that a generic algorithm to solve the discrete logarithm problem in a group of prime order must have complexity at least $k\sqrt{N}$ where $N$ is the order of the group. In many collision search algorithms this complexity is achieved. So with generic algorithms one can only hope to make the $k$ smaller. This $k$ depends on the complexity of the iterative step in the generic algorithms. The $\sqrt{N}$ comes from the fact there is about $\sqrt{N}$ iterations before a collision. So if we can find ways that can reduce the amount of work in one iteration then that is of great interest and probably the only possible modification of a generic algorithm. The modified $r$-adding walk allegedly does just that. It claims to reduce the amount of work done in one iteration of the original $r$-adding walk. In this paper we study this modified $r$-adding walk, we critically analyze it and we compare it with the original $r$-adding walk.

cs.CR↗
arXiv2014

MOR Cryptosystem and classical Chevalley groups in odd characteristic

In this paper we study the MOR cryptosystem using finite classical Chevalley groups over a finite field of odd characteristic. In the process we develop an algorithm for these Chevalley groups in the same spirit as the row-column operation for special linear group. We focus our study on orthogonal and symplectic groups. We find the hardness of the proposed MOR cryptosystem for these groups.

math.GR↗
arXiv2013

The MOR cryptosystem and finite $p$-groups

The ElGamal cryptosystem is the most widely used public key cryptosystem. It uses the discrete logarithm problem as the cryptographic primitive. The MOR cryptosystem is a similar cryptosystem. It uses the discrete logarithm problem in the automorphism group as the cryptographic primitive. In this paper, we study the MOR cryptosystem for finite $p$-groups. The study is complete for $p^\prime$-automorphisms. For $p$-automorphisms there are some interesting open problems.

math.GR↗
arXiv2012

The ElGamal cryptosystem over circulant matrices

In this paper we study extensively the discrete logarithm problem in the group of non-singular circulant matrices. The emphasis of this study was to find the exact parameters for the group of circulant matrices for a secure implementation. We tabulate these parameters. We also compare the discrete logarithm problem in the group of circulant matrices with the discrete logarithm problem in finite fields and with the discrete logarithm problem in the group of rational points of an elliptic curve.

cs.CR↗