VLSI Architectures for Computing Multiplications and Inverses in GF(2m)
IEEE Transactions on Computers
Algorithms for very large integer arithmetic
Technique et Science Informatiques
VLSI implementation of public-key encryption algorithms
Proceedings on Advances in cryptology---CRYPTO '86
Hardwired polynomial evaluation
Journal of Parallel and Distributed Computing - Parallelism in Computer Arithmetic
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
On the Time Required to Perform Multiplication
Journal of the ACM (JACM)
Fast Multiple-Precision Evaluation of Elementary Functions
Journal of the ACM (JACM)
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
Cryptography and data security
Cryptography and data security
A Single Chip Parallel Multiplier by MOS Technology
IEEE Transactions on Computers
The computational complexity of algebraic numbers
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
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Computing the long multiplication in fixed-radix representation is described first which suggests the use of two mixed solutions: first the sequentialisation of Karatsuba's algorithm by its extension to hexa and octo-mul then their judicious combination plus Implementation in Occam 2 language.Computing the long multiplication in modular representation. Including the principles of modular arithmetic and the Chinese remainder theorem, with efficient methods, is given in detail, together with their implementation for transformation from integer fixed-radix to modular and back again. Choice for modulil are made to compute the Inverse modulo efficiently without a need for the Euclid's algorithm.Montgomery's method for "Computing the long multiplication without trial division" for avoiding time consuming integer-modular-integer conversions at each multiplication-square step is described. It is possible to apply such a method to the exponentiation process currently used in Cryptography because It is interesting to use it only in the case where several multiplications are done in modulo N.Cryptographic applications for computing asymmetric keys may use the present implementations to advantage.