A Probabilistic Computing Approach to the Closest Vector Problem for Lattice-Based Factoring

TL;DR

The paper addresses refining CVP solutions within lattice-based factoring by deploying probabilistic computing with p-bits. It develops a complete pipeline from a classical Babai initial approximation to a probabilistic-neighborhood refinement driven by an energy-based p-bit network, guided by carefully chosen prime-lattice parameters $(m,c,M)$ and permutation-based constructions. Experimental results on software simulations show the approach can locate maximal CVP refinements efficiently and, crucially, reduce the number of lattice instances required for factoring semiprimes by up to two orders of magnitude relative to prior quantum- or classical-based refinements, albeit with considerations about sr-pair collisions and lattice growth. The work highlights practical trade-offs between lattice dimensions, smoothness bounds, and hardware feasibility, suggesting a viable path for hardware-accelerated lattice-based factoring and PQC-oriented applications, with future work on dedicated probabilistic hardware benchmarking and parameter-optimization.

Abstract

The closest vector problem (CVP) is a fundamental optimization problem in lattice-based cryptography and its conjectured hardness underpins the security of lattice-based cryptosystems. Furthermore, Schnorr's lattice-based factoring algorithm reduces integer factoring (the foundation of current cryptosystems, including RSA) to the CVP. Recent work has investigated the inclusion of a heuristic CVP approximation `refinement' step in the lattice-based factoring algorithm, using quantum variational algorithms to perform the heuristic optimization. This coincides with the emergence of probabilistic computing as a hardware accelerator for randomized algorithms including tasks in combinatorial optimization. In this work we investigate the application of probabilistic computing to the heuristic optimization task of CVP approximation refinement in lattice-based factoring. We present the design of a probabilistic computing algorithm for this task, a discussion of `prime lattice' parameters, and experimental results showing the efficacy of probabilistic computing for solving the CVP as well as its efficacy as a subroutine for lattice-based factoring. The main results found that (a) this approach is capable of finding the maximal available CVP approximation refinement in time linear in problem size and (b) probabilistic computing used in conjunction with the lattice parameters presented can find the composite prime factors of a semiprime number using up to 100x fewer lattice instances than similar quantum and classical methods.

Figures (5)

• Figure 1: (a) The neighborhood (green) of the CVP approximation (blue) in a 2-dimensional lattice. Each neighbor can be reached by adding or subtracting each basis vector at most once. (b) The reduced neighborhood (green) of the CVP approximation (blue). Each basis vector is assigned a direction (both positive in this example) and each neighbor can be reached by adding or subtracting each basis vector in its assigned direction at most once.
• Figure 2: (a) A fully connected p-bit network. The bias calculation of each p-bit is dependent on the state of every other p-bit in the network. (b) A 2D lattice p-bit network. The bias calculation of each p-bit is dependent on only the neighbors of the given p-bit. In this configuration, non-neighboring p-bits can safely update in parallel. In a general p-bit network, graph coloring is used to identify sets of parallelizable update groups.
• Figure 3: (a) Sr-pair collision rate for a fixed semiprime $N$ with bit length 30 for 500 lattices per data point. (b) Bit length to dimension mappings considered in this work. Linear mapping: $m=\lceil k\cdot\text{bit length}\rceil$. Sublinear mapping: $m=\lceil\frac{3}{2}\cdot(\text{bit length}/\log_2(\text{bit length}))\rceil$. (c) Mean number of sr-pairs per lattice for 100 lattices per data point.
• Figure 4: (a) Number of sr-pairs as a function of distance to target vector and smoothness bound for 500 lattices of varying dimensions. (b) Mean iterations required to find maximal CVP approximation refinement for 100 lattices per data point. (c) Mean percentage improvement in distance to target over initial approximation for refinements found in (b).
• Figure 5: (a) Mean sr-pairs per CVP instance found by probabilistic computing and by enumeration where feasible for 1,000 lattices per data point. (b) Mean number of CVP instances required to factor a semiprime using various heuristic refinement methods: Probabilistic Computing (this work), Predicted Probabilistic Computing (this work), Predicted Enumeration (this work), Schnorr's algorithm schnorrFactoringIntegersComputing1991schnorrFastFactoringIntegers2021, Yan et al.'s QAOA method yanFactoringIntegersSublinear2022a, and a classical hill-climbing algorithm. For probabilistic computing, 25 semiprime integers were factored per data point. Results for Schnorr's algorithm, QAOA, and hill climbing are reported from aboumradQuantumClassicalCombinatorial2023. (c) Mean collision rate of sr-pairs for semiprime factoring via probabilistic computing for the same instances shown in (b).

Theorems & Definitions (11)

• Definition 1
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• Definition 3
• Lemma 1
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• Definition 9