Algorithms for very large integer arithmetic
Technique et Science Informatiques
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
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Year after year, there is an incredible increase in computation power delivered by parallelism and new kinds of processors. The length of integers involved in the computations in public-key cryptosystems will probably be more than one thousand bits within the next ten years to warranty a good security level.The proposal is to present a possible software solution for 1024, 1279, 2048 and 4096-bit numbers which combine software and general purpose hardware. The algorithm implemented makes use of the Discrete fast Fourier Transform (DFT) and produces first results which can be compared to others like Karatsuba's or modular arithmetic.Considering these results for cryptographic applications using asymmetric systems for key exchanges one can take advantage of such a method for modular exponentiation but only for operations involving more than 1024 bits. While using a quad-transputer board equipped with the new T9000 family transputers, software modular exponentiation involving 1024-bit numbers should take under 0.5 s, 2048 bits under 2 s and 4096 bits less than 8 s which takes more time but insures the future of the just necessary security ! Expected results for this method show possible software applications for highly secure key exchange protocols using general purpose new chips (INMOST9000 transputer or Intel 1860 when used in piped mode).