An Efficient Modular Exponentiation Proof Scheme (original) (raw)

Fast verification, testing, and generation of large primes

Theoretical Computer Science, 1979

We present a prime certification method which permits shorter certificates of primality than the method analyzed by Pratt. We analyze the expected time required by a stochastic method for showing that n is prime, given a factorization of n-1. We use this method, together with Rabin's stochastic m.c;hOd for verifying COmpOSiteneSs, to obtain an algorithm for generating arbitrarily large pssnes and short ce&kates Of their primahty. We give plausibility arguments that this method can g(znerate primes larger than n in expected time polynomial in log n. We analyze several such prime generation algorithms*

Formal and Efficient Primality Proofs by Use of Computer Algebra Oracles

Journal of Symbolic Computation, 2001

This paper focuses on how to use Pocklington's criterion to produce efficient formal proof-objects for showing primality of large positive numbers. First, we describe a formal development of Pocklington's criterion, done using the proof assistant Coq. Then we present an algorithm in which computer algebra software is employed as oracle to the proof assistant to generate the necessary witnesses for applying the criterion. Finally, we discuss the implementation of this approach and tackle the proof of primality for some of the largest numbers expressible in Coq.

Taxonomy and Practical Evaluation of Primality Testing Algorithms

ArXiv, 2020

Modern cryptography algorithms are commonly used to ensure information security. Prime numbers are needed in many asymmetric cryptography algorithms. For example, RSA algorithm selects two large prime numbers and multiplies to each other to obtain a large composite number whose factorization is very difficult. Producing a prime number is not an easy task as they are not distributed regularly through integers. Primality testing algorithms are used to determine whether a particular number is prime or composite. In this paper, an intensive survey is thoroughly conducted among the several primality testing algorithms showing the pros and cons, the time complexity, and a brief summary of each algorithm. Besides, an implementation of these algorithms is accomplished using Java and Python as programming languages to evaluate the efficiency of both the algorithms and the programming languages.

A Fast Parallel Modular Exponentiation Algorithm

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.

PPT: New Low Complexity Deterministic Primality Tests Leveraging Explicit and Implicit Non-Residues. A Set of Three Companion Manuscripts

ArXiv, 2019

In this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as lemmas/corollaries/claims whenever we have complete analytic proof(s); otherwise the results are introduced as conjectures. In Part/Article 1, we start with the Baseline Primality Conjecture~(PBPC) which enables deterministic primality detection with a low complexity = O((log N)^2) ; when an explicit value of a Quadratic Non Residue (QNR) modulo-N is available (which happens to be the case for an overwhelming majority = 11/12 = 91.67% of all odd integers). We then demonstrate Primality Lemma PL-1, which reveals close connections between the state-of-the-art Miller-Rabin method and the renowned Euler-Criterion. This Lemma, together with the Baseline Primality Conjecture enables a synergistic fusion of Miller-Rabin iterations and our method(s), resulting in hybrid ...

Primality testing revisited

Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92, 1992

Rabin's algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom* computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm.

Another look at some fast modular arithmetic methods

Journal of Mathematical Cryptology, 2009

In this work we reexamine a modular multiplication and a modular exponentiation method. The multiplication method, proposed by Hayashi in 1998, uses knowledge of the factorization of both N + 1 and N + 2 to compute a multiplication modulo N. If both N + 1 and N + 2 can be factored into k equally sized relatively prime factors then the computations are done modulo each of the factors and then combined using the Chinese Remainder Theorem. It was suggested that the (asymptotic) computational costs of the method is 1/k of simply multiplying and reducing modulo N. We show, however, that the computational costs of the method is (asymptotically) at least as costly as simply multiplying and reducing modulo N for both squarings and general multiplications when efficient arithmetic is used. The exponentiation method, proposed by Hwang, Su, Yeh and Chen in 2005, is based on Hayashi's method and uses knowledge of the factorization of P + 1 and P − 1 to compute an exponentiation modulo an odd prime P. We begin by showing that the method cannot be used as a general purpose exponentiation method and then modify the method so that it can work as a general purpose modular multiplication method. Like Hayashi's method, however, this method is at best (asymptotically) only as efficient as simply multiplying and reducing modulo P .

From theory to practice: horizontal attacks on protected implementations of modular exponentiations

Journal of Cryptographic Engineering, 2018

Nowadays, horizontal or single-shot side-channel attacks against protected implementations of RSA and similar algorithms constitute a theoretic threat against secure devices. Nevertheless, in practice their application remains very difficult not only because of their complexity, but also because of environmental countermeasures integrated by designers that render their application even more difficult. Horizontal side-channel attacks take place in multiple steps. Among them, the most important are the acquisition of a complete trace with a sufficiently high sampling rate, its cutting into regular patterns, the realignment of the obtained patterns, the reduction as far as possible of noise in the acquired trace, the identification of the points of interest and the application of an effective distinguisher. Each of these steps is crucial and leads, if performed without enough attention, to an unsuccessful attack. In this context, this paper introduces effective solutions to efficiently perform all these steps, i.e., practicable means for implementing efficient horizontal attacks. Keywords Side-channel analysis • Horizontal attacks • Collision attacks 1 Introduction Modular exponentiation plays a fundamental role in public key cryptosystems, such as RSA [31], DSA encryption of El Gamal [16], etc. Unfortunately, most physical implementations of modular exponentiation algorithms are vulnerable to side-channel attacks (SCA) exploiting the power consumption [25], the electromagnetic radiations [17] or the computation times [24] if the designers do not implement hardware and software countermeasures. At the beginning of the century, irregular implementations of modular exponentiation, like the binary square-andmultiply algorithms, in left-to-right or in the right-to-left version, were used. Unfortunately, these energy-efficient implementations can easily be broken by a simple power B Ibrahima Diop

Spectacular Exponents: A semi modular Approach to Fast Exponentiation

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.

A Simple Algorithm for Prime Factorization and Primality Testing

Journal of Mathematics

We propose a new simple and faster algorithm to factor numbers based on the nature of the prime numbers contained in such composite numbers. It is well known that every composite number has a unique representation as a product of prime numbers. In this study, we focus mainly on composite numbers that contain a product of prime numbers that are greater than or equal to 5 which are of the form 6 k + 1 or 6 k + 5 . Therefore, we use the condition that every prime or composite P of primes greater than or equal to 5 satisfies P 2 ≡ 1 mod 24 . This algorithm is very fast especially when the difference in the prime components of a composite number (prime gap) is not so large. When the difference between the factors (prime gap) is not so large, it often requires just a single iteration to obtain the factors.