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Journal of Discrete Mathematical Sciences and Cryptography

A quantum algorithm for computing multiplicative order of integers modulo a prime

Maheswara Rao Valluri

Published Jan 1, 2023• 0 citations
Publisher

Abstract

One of the challenging problems in number theory is to compute efficiently multiplicative order of integers modulo a prime. This problem was listed under open problems in the number theoretic complexity II proposed by Leonard M. Adleman and Kevin S.McCurley in 1994. This paper offers a polynomial-time quantum algorithm for computing the order k = min {x|x∈, ax ≡ 1(modp)}, if p∈ Primes and gcd(a,p) = 1, where a, p∈, and it requires 2 (1) 2 ((log p) +O )  quantum operations.

DOI
10.47974/jdmsc-1557

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One of the challenging problems in number theory is to compute efficiently multiplicative order of integers modulo a prime. This problem was listed under open problems in the number theoretic complexity II proposed by Leonard M. Adleman and Kevin S.McCurley in 1994. This paper offers a polynomial-time quantum algorithm for computing the order k = min {x|x∈, ax ≡ 1(modp)}, if p∈ Primes and gcd(a,p) = 1, where a, p∈, and it requires 2 (1) 2 ((log p) +O )  quantum operations.
Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer N, the test tries to find an element of the multiplicative group of integers modulo N with order N-1. If one is found, the number is known to be prime. During the test, we can also show most of the times N is composite with certainty (and a witness) or, after \log\log N unsuccessful attempts to find an element of order N-1, declare it composite with high probability. The algorithm requires O((\log n)^2 n^3) operations for a number N with n bits, which can be reduced to O(\log\log n (\log n)^3 n^2) operations in the asymptotic limit if we use fast multiplication.
Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer NNN with a probability as close to 1 as desired. The algorithm requires O((log⁡n)3n3)O((\log n )^3n^3)O((logn)3n3) quantum operations, or O(log⁡log⁡n(log⁡n)4n2)O(\log\log n (\log n)^4 n^2)O(loglogn(logn)4n2) operations using fast multiplication. Verification, quantum optimizations and applications to RSA and primality tests are also discussed.
It is assumed that P is the product of two prime numbers q1and q2. If there is an integer 02= C (mod P), i.e., the congruence has a solution, then C is said to be a quadratic congruence (mod P). Quadratic congruence (mod P) is a NP-complete problem. If the value of C is equal to one, then four integer solutions for M2= 1 (mod P) are, respectively, b, P - b, 1 and P - 1, where 12= 1 (mod P) can be found by means of the proposed quantum algorithms with polynomial quantum gates, polynomial quantum bits and the successful probability that is the same as that of Shor's order-finding algorithm.
One of the most well known mathematically hard problems in number theory is the integer factorization problem, roughly stated that decomposition of a composite number into its prime factors. In modern cryptography, RSA encryption algorithm whose security is based on integer factorization problem is highly practical, widespread and up to date no classical algorithm having polynomial running time for the factorization of large numbers is known. In 1994, Peter Shor proposed an efficient algorithm on quantum computer. In this paper, we  mention about the fundamentals of  Shor's quantum algorithm illustrating a concrete example.
We present a quantum version of the classical probabilistic algorithms as studied by Rabin. The quantum algorithm is based on the essential use of Grover's operator for the quantum search of a database and of Shor's Fourier transform for extracting the periodicity of a function, and their combined use in the counting algorithm originally introduced by Brassard et al. One of the main features of our quantum probabilistic algorithm is its full unitarity and reversibility, which would make its use possible as part of larger and more complicated networks in quantum computers. As an example of this we describe polynomial time algorithms for studying some important problems in number theory, such as the test of the primality of an integer, the so called prime number theorem, and Hardy and Littlewood's conjecture about the asymptotic number of representations of an even integer as a sum of two primes.