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arXiv:2408.11606 · Mathematical Problems of Computer Science

A Quantum Diophantine Equation Solution Finder

Lara Tatli, P. D. Stevenson

Published Jun 1, 2025• 0 citations
arXivHTMLPDF

Abstract

Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer values for the unknown variables and check for equality. Grover’s algorithm is a quantum search algorithm which can find marked indices in a list very efficiently. By treating the indices as the integer variables in the Diophantine equation, Grover’s algorithm can be used to find solutions in a brute force way more efficiently than classical methods. We present a hand-coded example for the simplest possible Diophantine equation, and results for a more complicated, but still simulable, equation encoded with a high-level quantum language.

DOI
10.51408/1963-0132

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Diophantine equations are multivariate equations, usually polynomial, in which only integer solutions are admitted. A brute force method for finding solutions would be to systematically substitute possible integer values for the unknown variables and check for equality. Grover’s algorithm is a quantum search algorithm which can find marked indices in a list very efficiently. By treating the indices as the integer variables in the Diophantine equation, Grover’s algorithm can be used to find solutions in a brute force way more efficiently than classical methods. We present a hand-coded example for the simplest possible Diophantine equation, and results for a more complicated, but still simulable, equation encoded with a high-level quantum language.
In a fundamental paper [Phys. Rev. Lett. 78, 325 (1997)] Grover showed how a quantum computer can find a single marked object in a database of size NNN by using only O(N)O(\sqrt{N})O(N​) queries of the oracle that identifies the object. His result was generalized to the case of finding one object in a subset of marked elements. We consider the following computational problem: A subset of marked elements is given whose number of elements is either MMM or KKK, our task is to determine which is the case. We show how to solve this problem with a high probability of success using iterations of Grover's basic step only, and no other algorithm. Let mmm be the required number of iterations; we prove that under certain restrictions on the sizes of MMM and KKK the estimation $m<2\sqrt{N}∕(\sqrt{K}\ensuremath{-}\sqrt{M})$ obtains. This bound reproduces previous results based on more elaborate algorithms, and is known to be optimal up to a constant factor. Our method involves simultaneous Diophantine approximations, so that Grover's algorithm is conceptualized as an orbit of an ergodic automorphism of the torus. We comment on situations where the algorithm may be slow, and note the similarity between these cases and the problem of small divisors in classical mechanics.
Grover's algorithm is normally presented as a method of searching a database, however it would be more accurately described as a method of identifying elements of an interval of the integers which satisfy some logical clause - an example might be identifying binary strings which correspond to the solutions of a Sudoku problem. In this paper we present the first method of performing a true database search using Grover's search algorithm, by first creating a mapping from a set of indices in the range 0:2^n-1 to a set of database elements, then applying the clause to these elements. We then demonstrate the feasibility of an attack against the Diffie-Hellman cryptosystem based on a Grover's search of a database of candidate solutions generated via the number field sieve algorithm.
Grover’s search algorithm is one of the most famous quantum algorithms, which enables search in unstructured dataspaces with a quadratic speedup over any known classical algorithm. This is one of the first algorithms to show quantum advantage over classical methods of computation. Besides its most anticipated application of brute force attack on symmetric key encryptions like AES, it is today being used in many modern applications as a subroutine. In this chapter, we describe the mathematical model of Grover’s quantum search algorithm, its behavior, and the theory of its circuit implementation. This chapter also discusses an exact variant of this algorithm and recent applications of Grover’s quantum search.
Several mathematical problems can be modeled as a search in a database. An example is the problem of finding the minimum of a function. Quantum algorithms for solving this problem have been proposed and all of them use the quantum search algorithm as a subroutine and several intermediate measurements are realized. In this work, it is proposed a new quantum algorithm for finding the minimum of a function in which quantum search is not used as a subroutine and only one measurement is needed. This is also named asymptotic quantum search. As an example, we propose a quantum algorithm based on asymptotic quantum search and quantum counting able to calculate a lower bound of the probability of finding a Diophantine equation with integer solution.
We consider Grover's algorithm of quantum search for one or several integers out of N = 2n, where n is a number of quantum bits in the memory register. There is a black-box or subroutine containing information about hidden integers and it can easily recognize these integers but we do not know which ones out of N they are. To find the hidden items we can do no better with a classical computer than to apply the subroutine repeatedly to all possible integers until we hit on the special one and in the worst case we have to repeat this procedure N times. We have analyzed the Grover algorithm carefully and showed that it enables to speed up this search quadratically, although its realization requires to know a number of hidden items. The lower bound for the probability of successful solving the search problem has been obtained. The validity of the results was demonstrated by simulation of the Grover search algorithm using the package QuantumCircuit written in the Wolfram Mathematica language.
In this paper, we consider three types of polynomial equations in quantum computer: linear divisibility equation, which belongs to a special type of binary-quadratic Diophantine equation; quadratic congruence equation with restriction in the solution and exponential congruence equation in finite field. Quantum algorithms based on Grover's algorithm and Shor's algorithm to these problems are given. As for the exponential congruence equation, which has been considered by Dam and Shparlinski \cite{dam} at 2008, a relatively simple quantum algorithm is given here. And some other results and generalizations are discovered.