Quantum Algorithm Efficiently Computes, Unit Groups, Enabling Polynomial Time Solutions for Ideal Class Problems

The efficient calculation of fundamental algebraic structures within number theory remains a significant challenge, with implications for cryptography and our understanding of mathematical foundations. Jean-Francois Biasse from the University of South Florida and Fang Song from Portland State University present a new quantum algorithm that achieves a breakthrough in computing -units, essential components in the study of number fields. This algorithm not only provides a faster method for calculating these units, but also unlocks polynomial time solutions to a range of related problems, including the calculation of class groups and the resolution of the principal ideal problem. By accelerating these core calculations, the research paves the way for advancements in areas such as lattice cryptography and the decomposition of ideal classes within cyclotomic fields.

Lattice Reduction and Number Theoretic Algorithms

This extensive compilation details research in computational number theory and cryptography, covering topics from lattice reduction and factoring to quantum algorithms and solving Diophantine equations. The work encompasses core areas of computational number theory and algebraic number theory, including foundational texts by Neukirch and Cohen. A cornerstone of many algorithms is the LLL algorithm, developed by Lenstra, Lenstra, and Lovász, which efficiently reduces lattices. Further advancements, such as those by Cramer, Ducas, Peikert, and Regev, focus on recovering short generators of principal ideals, crucial for lattice-based cryptography.

Fieker’s work provides algorithmic improvements for solving norm equations, while Simons demonstrates methods for solving these equations using S-units. Research in lattice reduction extends to Lovász’ algorithm by Babai, and methods for computing lattice bases developed by Buchmann and Pohst, and Buchmann and Kessler. The field also explores quantum algorithms, most notably Shor’s algorithm, which enables polynomial-time factorization and discrete logarithms, driving the development of post-quantum cryptography. Hallgren’s work extends this to fast quantum algorithms for computing unit and class groups, and evaluating Pell’s equation, while Childs, Jao, and Soukharev developed a subexponential quantum algorithm for isogenies.

Eisenträger, Hallgren, Kitaev, and Song further advanced quantum algorithms for computing unit groups. Significant progress has been made in understanding class and unit groups, with Cohen and Lenstra providing heuristics on class groups, and Hallgren developing quantum algorithms for their computation. Solving Diophantine equations and norm equations remains a central focus, with contributions from Fieker and Simons. Research into isogenies and elliptic curves, led by Jao and Soukharev, has yielded subexponential algorithms for evaluating isogenies. The work also addresses cryptographic applications and security, including fully homomorphic encryption by Smart and Vercauteren, and candidate multilinear maps from Garg, Gentry, and Halevi.

Bröker, Charles, and Lauter have advanced isogeny-based cryptography, while Campbel, Groves, and Shepherd highlight potential side-channel attacks. Littlewood and Lenstra and Lenstra provide foundational work on class numbers and integer factoring. The collective research underscores the urgent need for post-quantum cryptography, driven by the threat posed by Shor’s algorithm. Algorithmic improvements continue to reduce the computational complexity of key problems, and the work demonstrates strong connections between algebraic number theory, cryptography, and quantum computing.

Polynomial Time Computation of Number Field S-Units

Scientists have developed a quantum algorithm to compute the S-unit group of a number field in polynomial time, representing a significant advance in computational number theory. Building upon prior work resolving the Hidden Subgroup Problem (HSP), particularly the continuous HSP definition on Rm developed by Eisenträger, Hallgren, Kitaev and Song, the research overcomes limitations of earlier approaches by enforcing stringent continuity properties crucial for efficient quantum Fourier sampling. This breakthrough directly addresses a long-standing challenge, as no previously known algorithms could compute these groups in polynomial time. The study pioneers a method that leverages the resolution of the HSP within a bounded and discretized approximation of Rm, a technique refined to accommodate number fields of arbitrary degree.

Scientists engineered a continuous HSP definition, enabling a more amenable function for quantum Fourier sampling. This algorithm not only computes the S-unit group but also has direct implications for calculating class groups, S-class groups, relative class groups, and the unit group itself. Furthermore, the research extends to solving the principal ideal problem, a critical task with applications in computing relative class groups and unit groups, and is relevant to lattice-based cryptography. By efficiently resolving the principal ideal problem, the algorithm allows for the identification of short generators of a principal ideal, directly impacting the security of cryptographic schemes reliant on the hardness of this problem. The team demonstrated that solving the principal ideal problem in polynomial time induces a polynomial time attack on schemes relying on the hardness of finding the short generator of a principal ideal. This work builds upon existing quantum polynomial time algorithms for the Hidden Subgroup Problem, extending their applicability to number fields of arbitrary degree and opening new avenues for research in both number theory and cryptography.

Quantum Algorithm Solves Number Field S-Unit Problem

This research presents a quantum algorithm for computing the S-unit group of a number field, achieving a significant breakthrough in computational number theory. The team demonstrated a method that operates in polynomial time with respect to key parameters including the degree of the number field, n, the logarithm of the discriminant, log|∆|, the size of the prime set, |S|, and the maximum norm of primes within that set, maxp∈S{log(N(p))}. This advancement directly implies polynomial time solutions for a range of related problems, including the computation of ideal class groups, S-class groups, relative class groups, and the resolution of the principal ideal problem. The researchers established a quantum reduction from the S-unit group problem to the Hidden Subgroup Problem (HSP) on Rm, leveraging a previously developed quantum HSP algorithm for efficient solution.

Crucially, the method delivers exact, compact representations of field elements, unlike prior approaches that provided only rational approximations. These compact representations are essential for further algebraic processing, enhancing the practicality of the algorithm. Furthermore, the team’s results induce a quantum polynomial-time attack on cryptosystems reliant on the difficulty of finding short generators of principal ideals, representing a substantial impact on cryptographic security. The method utilizes E-ideals, which are lattices in complex space, and defines their properties in relation to the ring of integers of the number field. The achievement directly implies efficient methods for calculating several related algebraic structures, including class groups, S-class groups, and ray class groups, and for solving key problems such as the principal ideal problem and decomposing ideal classes. These algorithmic advances have implications for various areas of number theory and cryptography, potentially enabling more efficient solutions to problems involving ideal lattices in cyclotomic fields. The team’s work, combined with a result from Cramer, Ducas, Peikert and Regev, allows for the identification of short generators of principal ideals, a crucial step in certain cryptographic applications. Furthermore, the resolution of the principal ideal problem and the decomposition of ideal classes, as developed by Cramer, Ducas and Wesolowski, facilitate the discovery of.