The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
A survey of fast exponentiation methods
Journal of Algorithms
Efficient Generation of Minimal Length Addition Chains
SIAM Journal on Computing
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Minimal Addition-Subtraction Chains Using Genetic Algorithms
ADVIS '02 Proceedings of the Second International Conference on Advances in Information Systems
Visualization of cryptographic protocols with GRACE
Journal of Visual Languages and Computing
Finding Minimal Addition Chains with a Particle Swarm Optimization Algorithm
MICAI '09 Proceedings of the 8th Mexican International Conference on Artificial Intelligence
Finding optimal addition chains using a genetic algorithm approach
CIS'05 Proceedings of the 2005 international conference on Computational Intelligence and Security - Volume Part I
An Artificial Immune System Heuristic for Generating Short Addition Chains
IEEE Transactions on Evolutionary Computation
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Online visualization systems have come to be heavily used in education, particularly for online learning Most e-learning systems, including interactive learning systems, have been designed to simplify understanding the ideas of some main problems or in general overall course materials This paper presents a novel interactive visualization system for one of the most important operation in public-key cryptosystems This operation is modular exponentiation using addition chains. An addition chain for a natural number e is a sequence 1=a0a1ar=e of numbers such that for each 0i≤r, ai=aj+ak for some 0≤k≤ji Finding an addition chain with minimal length is NP-hard problem The proposed system visualizes how to generate addition chains with minimal length using depth-first branch and bound technique and how to compute the modular exponentiation using addition chains.