Spigot algorithm

Abstract: Modular exponentiation is fundamental operation in the many cryptosystem such as RSA. This operation is implemented by repeating modular multiplication which is time consuming for large operands. This paper presents a new modified Montgomery modular multiplication algorithm based on multiple bit scan-multiple bit shift technique, sliding window method and signed-digit representation. This new algorithm skips from zero digit partial multiplication and the following required addition. Then it shifts the partial results by using Barrel shifter in only one cycle instead of several cycles. In addition, we proposed new modular exponentiation algorithm based on this new modular multiplication algorithm and common-multiplicand-multiplication method. In this new algorithm, the common part of modular multiplication is computed once rather than several times. So the security of the cryptosystem which used this new algorithm increased considerably. The analysis results show that the number of multiplication steps in the proposed exponentiation algorithm is reduced on average at about 72%–87%, 66%–84%, 15%–61% and 54%–79% in compare with Dusse-Kaliski's algorithm, Ha-Moon's algorithm, Wu et al.'s algorithm and Wu's algorithm respectively for d=3–8.

Abstract: Modular exponentiation is a fundamental and most time-consuming operation in several public-key cryptosystems such as the RSA cryptosystem. In this paper, we propose two new parallel algorithms. The first one is a fast parallel algorithm to multiply n numbers of a large number of bits. Then we use it to design a fast parallel algorithm for the modular exponentiation. We implement the parallel modular exponentiation algorithm on Google cloud system using a machine with 32 processors. We measured the performance of the proposed algorithm on data size from $$2^{12}$$ to $$2^{20}$$ bits. The results show that our work has a fast running time and more scalable than previous works.

Abstract: Paper 13: Stanley Rabinowitz and Stan Wagon, “A spigot algorithm for the digits of π,” American Mathematical Monthly, vol. 102 (March 1995), p. 195–203. Copyright 1995 Mathematical Association of America. All Rights Reserved.

Abstract: We present a method for computing some numbers bit by bit using only a ruler and compass, and illustrate it by applying it to arctan(X)/π. The method is a spigot algorithm and can be applied to numbers that are constructible over the unit circle and the ellipse. The method is precise enough to produce about 20 bits of a number, that is, 6 decimal digits in a matter of minutes. This is surprising, since we do no actual calculations.