Mayank Deora | Searcharxiv

arXiv2025

Efficient algorithm for linear diophantine equations in two variables

Solving linear diophantine equations in two variables have applications in computer science and mathematics. In this paper, we revisit an algorithm for solving linear diophantine equations in two variables, which we refer as DEA-R algorithm. The DEA-R algorithm always incurs equal or less number of recursions or recursive calls as compared to extended euclidean algorithm. With the objective of taking advantage of the less number of recursive calls , we propose an optimized version of the DEA-R algorithm as DEA-OPTD. In the recursive function calls in DEA-OPTD, we propose a sequence of more efficient computations. We do a theoretical comparison of the execution times of DEA-OPTD algorithm and DEA-R algorithm to find any possible bound on the value of $c$ for DEA-OPTD being better than DEA-R. We implement and compare an iterative version of DEA-OPTD (DEA-OPTDI) with two versions of a widely used algorithm on an specific input setting. In this comparison, we find out that our algorithm outperforms on the other algorithm against atleast 96% of the inputs.

cs.DS↗

arXiv2024

An average case efficient algorithm for solving two-variable linear Diophantine equations

Solving two-variable linear Diophantine equations has applications in many cryptographic protocols such as RSA and Elliptic curve cryptography. The Extended Euclid's algorithm is a well known algorithm to solve these equations. We revisit two algorithms to solve two-variable linear Diophantine equations. We write the iterative version of one of the revisited algorithms. For another, we do a fine-grained analysis of the number of recursive calls and arrive at a periodic function that represents the number of recursive calls. We find the period and use it to derive multiple upper bounds on the average number of recursive calls. The upper bounds conclude that the average number of recursive calls in the analyzed algorithm is a constant term improvement over the Extended Euclid's algorithm. We propose an iterative variant of the algorithm. We implement this algorithm and find that the average number of iterations by our algorithm is less than that of two existing algorithms. We compare the number of iterations in this algorithm and in Extended Euclid's algorithm. In the comparison, we find that for 100\% of the solvable instances of inputs, the iterations are less than that in Extended Euclid's algorithm.

cs.CR↗