Evolutionary programming for the length minimization of addition chains

Domínguez-Isidro, Saúl; Mezura-Montes, Efrén; Osorio-Hernández, Luis Guillermo

doi:10.1016/j.engappai.2014.09.003

Cited by 19 publications

(20 citation statements)

References 12 publications

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“…As demonstrated in (Nedjah and Mourelle, 2005;Bos and Coster, 1990;Cruz-Cortés et al, 2008;Dominguez-Isidro and Mezura-Montes, 2011;Dominguez-Isidro et al, 2015), an even better result is expectedly obtainable with a well-formulated AS algorithm.…”

Section: Note On Windows-based Addition Chain Methodsmentioning

confidence: 95%

“…Similarly, c(r) is defined as smallest e for which l(e) = r. Furthermore, Knuth (1998) compiled c(1) to c(27) while Clift (2011) found up to c(38). Dominguez-Isidro et al (2015) reported another set of es termed "hard" because of the relative hardness in finding their optimal AC by deterministic heuristics. The set consists of 3243679; 3493799; 3459835; 3235007; 3230591; 3182555; 3440623; 3704431; 3234263; 3352927; 3926651; 3922763; 2948207, each having l = 27.…”

Section: Theoretical Results and Asymptotic Bounds For Acpmentioning

confidence: 99%

“…ACP is an NPhard, while its generic ASP is proven as NP-complete problem (Downey et al, 1981). Therefore, heuristic (Bos and Coster, 1990;Gelgi and Onus, 2006;Koc, 1995;Lee et al, 2006;Park et al, 1999) and metaheuristic (Cruz-Cortés et al, 2005;2008;Jose-Garcia et al, 2011;León-Javier et al, 2009;Osorio-Hernández et al, 2009;Dominguez-Isidro and Mezura-Montes, 2011;Dominguez-Isidro et al, 2015) have become alternative approaches in searching for near optimal solution for the ACP.…”

Section: Definitionmentioning

confidence: 99%

“…For example, Kunihiro and Yamamoto (2000) show that optimality of AC for e is a function of H(e): Those e with very few and those with almost all non-zero bits in the corresponding binary forms represent the integers for which their optimal ACs are easy to obtain; Whereas those es having on the average n/2 bits density constitute the most difficult set for which to find the optimal ACs. Cruz-Cortés et al (2008) and Dominguez-Isidro et al (2015) opined that metaheuritics, generally, return (nearest) optimal ACs for relatively small to medium integers. In their respective Artificial Immune System (AIS) and AC Evolutionary Programming (ACEP), SWM is integrated to handle large integers.…”

Section: Metaheuristic Algorithms For Acpmentioning

confidence: 99%

See 3 more Smart Citations

A Review on Heuristics for Addition Chain Problem: Towards Efficient Public Key Cryptosystems

Noma¹,

Muhammed²,

Mohamed³

et al. 2017

Journal of Computer Science

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Field exponentiation and scalar multiplication are the pillars of and the most computationally expensive operations in the public key cryptosystems. Optimizing the operation is the key to the efficiency of the systems. Analogous to the optimization is solving addition chain problem. In this study, we survey from the onset of the addition chain problem to the state-of-the-art heuristics for optimizing it, with the view to identifying fundamental issues that when addressed renders the heuristics most optimal mean of minimizing the two operations in various public key cryptosystems. Thus, our emphasis is specifically on the heuristics: Their various constraints and implementations efficiencies. We present possible ways forwards toward the optimal solution for the addition chain problem that can be efficiently applied for optimal implementation of the public key cryptosystems.

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Section: Note On Windows-based Addition Chain Methodsmentioning

confidence: 95%

Section: Theoretical Results and Asymptotic Bounds For Acpmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: Metaheuristic Algorithms For Acpmentioning

confidence: 99%

See 2 more Smart Citations

A Review on Heuristics for Addition Chain Problem: Towards Efficient Public Key Cryptosystems

Noma¹,

Muhammed²,

Mohamed³

et al. 2017

Journal of Computer Science

View full text Add to dashboard Cite

show abstract

“…They proved that binary method is not optimal for integers of hamming weight 3. In [13], Saul Dominguez-Isidro presented Evolutionary Programming (EP) to minimize the length of addition chains. Also, four experiments were designed to test the performance of EP algorithm.…”

Section: Related Workmentioning

confidence: 99%

A New Method of Generating Optimal Addition Chain Based on Graph

Mani¹,

Viswambari²

2017

IJMSC

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In many number theoretic cryptographic algorithms, encryption and decryption is of the form x n mod p, where n and p are integers. Exponentiation normally takes more time than any arithmetic operations. It may be performed by repeated multiplication which will reduce the computational time. To reduce the time further fewer multiplications are performed in computing the same exponentiation operation using addition chain. The problem of determining correct sequence of multiplications requires in performing modular exponentiation can be elegantly formulated using the concept of addition chains. There are several methods available in literature in generating the optimal addition chain. But novel graph based methods have been proposed in this paper to generate the optimal addition chain where the vertices of the graph represent the numbers used in the addition chain and edges represent the move from one number to another number in the addition chain. Method 1 termed as GBAPAC which generates all possible optimum addition chains for the given integer n by considering the edge weight of all possible numbers generated from every number in addition chain. Method 2 termed as GBMAC which generates the minimum number of optimum addition chains by considering mutually exclusive edges starting from every number. Further, the optimal addition chain generated for an integer using the proposed methods are verified with the conjectures which already existed in the literature with respect to addition chains.

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