Finding optimal addition chains using a genetic algorithm approach
Nareli Cruz-Cortés,Francisco Rodríguez-Henríquez,Raúl Juárez-Morales,Carlos A. Coello Coello +3 more
- 15 Dec 2005
- pp 208-215
TL;DR: This contribution explores the usage of a Genetic Algorithm (GA) approach for the problem of finding optimal (shortest) addition chains and reports several promising experimental results that suggest that evolutionary algorithms may be a viable alternative to solve this illustrious problem in a quasi optimal fashion.
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Abstract: Since most public key cryptosystem primitives require the computation of modular exponentiation as their main building block, the problem of performing modular exponentiation efficiently has received considerable attention over the years. It is known that optimal (shortest) addition chains are the key mathematical concept for accomplishing modular exponentiations optimally. However, finding an optimal addition chain of length r is an NP-hard problem whose search space size is comparable to r !. In this contribution we explore the usage of a Genetic Algorithm (GA) approach for the problem of finding optimal (shortest) addition chains. We explain our GA strategy in detail reporting several promising experimental results that suggest that evolutionary algorithms may be a viable alternative to solve this illustrious problem in a quasi optimal fashion.
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Figures
![Table 1. Accumulated addition chain lengths for exponents e ∈ [1, 1000]](/figures/table-1-accumulated-addition-chain-lengths-for-exponents-e-1-3r3pkq0s.webp)
Citations
Addition chain heuristics
Jurjen Bos,Matthijs J. Coster +1 more
- 01 Jan 1989
TL;DR: A precomputation method is presented that reduces the number of multiplications for the computation of a given power in RSA, which consists of computing powers modulo a large number n.
109
A Genetic Algorithm with repair and local search mechanisms able to find minimal length addition chains for small exponents
Luis Guillermo Osorio-Hernández,Efrén Mezura-Montes,Nareli Cruz-Cortés,Francisco Rodríguez-Henríquez +3 more
- 18 May 2009
TL;DR: An improved Genetic Algorithm that is able to find the shortest addition chains for a given exponent e.g. two new variation operators (special two-point crossover and a local-search-like mutation) are proposed as a means to improve the GA search capabilities.
31
Evolutionary programming for the length minimization of addition chains
TL;DR: The use of evolutionary programming to minimize the length of addition chains is presented, able to find competitive or even better results by requiring a lower number of evaluations with respect to those required by state-of-the-art nature-inspired algorithms.
24
Evolutionary Computation and Cryptology
Stjepan Picek
- 20 Jul 2016
TL;DR: This tutorial will concentrate on several topics from cryptography that are successfully tackled up to now with EC and discuss why those topics are suitable to apply EC, and give a special emphasis on the perspective that cryptography presents a source of benchmark problems for the EC community.
12
Evolutionary Algorithms for Finding Short Addition Chains: Going the Distance
Stjepan Picek,Carlos A. Coello Coello,Domagoj Jakobovic,Nele Mentens +3 more
- 30 Mar 2016
TL;DR: A genetic algorithm with a novel representation of solutions and new crossover and mutation operators to minimize the length of the addition chains corresponding to a given exponent, and develops a repair strategy that significantly enhances the performance of this approach.
12
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