Another look at some fast modular arithmetic methods

One of the important parts for computing ME mod N is the modular exponentiation, where M is a plaintext, N is a modulus, and E is a large exponent. The performance of the modular exponentiation algorithm depends on the numbers of modular square and modular multiplication for the exponent. The computational complexity for the strategy which is provided to speed up the triple modular exponentiation can be reduced to 1.875j multiplications, where j is the bit length of the exponent.

Electronic Colloquium on Computational Complexity, 2008

We give anO(N logN 2O(log N) ) algorithm for multiplying twoN-bit integers that improves the O(N logN log logN) algorithm by Schonhage-Strassen (SS71). Both these algorithms use modular arithmetic. Recently, Furer (Fur07) gave an O(N logN 2O(log N) ) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication

Radio Electronics, Computer Science, Control, 2021

Context. Providing the problem of fast calculation of the modular exponentiation requires the development of effective algorithmic methods using the latest information technologies. Fast computations of the modular exponentiation are extremely necessary for efficient computations in theoretical-numerical transforms, for provide high crypto capability of information data and in many other applications. Objective-the runtime analysis of software functions for computation of modular exponentiation of the developed programs based on parallel organization of computation with using multithreading. Method. Modular exponentiation is implemented using a 2 k-ary sliding window algorithm, where k is chosen according to the size of the exponent. Parallelization of computation consists in using the calculation of the remainders of numbers raised to the power of 2 i modulo, and their further parallel multiplications modulo. Results. Comparison of the runtimes of three variants of functions for computing the modular exponentiation is performed. In the algorithm of parallel organization of computation with using multithreading provide faster computation of modular exponentiation for exponent values larger than 1K binary digits compared to the function of modular exponentiation of the MPIR library. The MPIR library with an integer data type with the number of binary digits from 256 to 2048 bits is used to develop an algorithm for computing the modular exponentiation with using multithreading. Conclusions. In the work has been considered and analysed the developed software implementation of the computation of modular exponentiation on universal computer systems. One of the ways to implement the speedup of computing modular exponentiation is developing algorithms that can use multithreading technology on multi-cores microprocessors. The multithreading software implementation of modular exponentiation with increasing from 1024 the number of binary digit of exponent shows an improvement of computation time with comparison with the function of modular exponentiation of the MPIR library.

Arabian Journal for Science and Engineering, 2017

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.

TJPRC, 2014

Security is an important technique for many applications including private networks, e-commerce and secure internet access. Public key cryptosystems like the RSA cryptosystem and the El -Gamal cryptosystem are popular security techniques. These cryptosystems have to perform modular exponentiation with large exponent and modulus for security considerations. Modular exponentiation is performed by repeated modular multiplicat ions. This paper proposes an improvised algorithm for modulo n mult iplication, where n is odd.  The remainder with modulus n is derived from the remainders with modulus (2n+2) and (2n+6)  (2n+2) and (2n+6) can be decomposed into products of relatively prime factors even if n is prime or difficult to be factorized into prime factors. The efficiency of the proposed algorith is estimated. On comparing the computational complexit iy with the conventional method, the new improvised algorithm is more efficient.

— In the paper, we propose a new method of modular multiplication computation, based on Residue Number System. We use an approximate method to find the approximate method a residue from division of a multiplication on the given module. We substitute expensive modular operations, by fast bit right shift operations and taking low bits. The carried-out simulation on Kintex7 XC7K70T board showed that the offered method allows to win in time on average for 75%, and in the area-on average for 80% relatively to modified method from work [1] that makes it more applicable for the hardware implementation of the cryptography primitives constructed over a simple finite field.

IEE Proceedings - Computers and Digital Techniques, 1998

We present an algorithm for computing the residue R = X mod M. The algorithm is based on a sign estimation technique that estimates the sign of a number represented by a carry-sum pair produced by a carry save adder. Given the (n + k)-bit X and the n-bit M , the modular reduction algorithm computes the n-bit residue R in O(k + log n) time, and is particularly useful when the operand size is large. We also present a variant of the algorithm that performs modular multiplication by interleaving the shift-and-add and the modular reduction steps. The modular multiplication algorithm can be used to obtain efficient VLSI implementations of exponentiation cryptosystems.

18th IEEE Symposium on Computer Arithmetic (ARITH '07), 2007

It is widely acknowledged that efficient modular multiplication is a key to high-performance implementation of public-key cryptography, be it classical RSA, Diffie-Hellman, or (hyper-) elliptic curve algorithms. In the recent decade, practitioners have relied mainly on two popular methods: Montgomery Multiplication and regular long-integer multiplication in combination with Barrett's modular reduction technique. In this paper, we propose a modification to Barrett's algorithm that leads to a significant reduction (25% to 75%) in multiplications and additions.

Information Processing Letters, 2009

In this paper, some issues concerning the Chinese remaindering representation are discussed. Some new converting methods, including an efficient probabilistic algorithm based on a recent result of von zur Gathen and Shparlinski [5], are described. An efficient refinement of the NC 1 division algorithm of Chiu, Davida and Litow [2] is given, where the number of moduli is reduced by a factor of log n.

JOURNAL OF ADVANCES IN MATHEMATICS, 2019

This paper introduces a computational scheme for calculating the exponential bw where b and w are positive integers. This two-step method is based on elementary number theory that is used routinely in this and similar contexts, especially the Chinese remainder theorem (CRT), Lagrange’s theorem, and a variation on Garner’s algorithm for inverting the CRT isomorphism. We compare the performance of the new method to the standard fast algorithm and show that for a certain class of exponents it is significantly more efficient as measured by the number of required extended multiplications.