🤖 AI Summary
This work resolves the unconditional correctness verification problem of Regev’s (2023) low-gate-count multidimensional Shor algorithm for integer factorization and discrete logarithms. The algorithm hinges on a key number-theoretic conjecture: that almost all elements of the multiplicative group modulo $N$, $(mathbb{Z}/Nmathbb{Z})^ imes$, can be expressed as short products of very small primes. Prior to this work, the conjecture remained unproven, impeding rigorous security analysis. We bridge analytic number theory and fine-grained structural analysis of $(mathbb{Z}/Nmathbb{Z})^ imes$, combining zero-density estimates for Dirichlet $L$-functions with precise character-sum bounds to establish the conjecture unconditionally for all sufficiently large $N$. As a result, we prove the unconditional correctness of Regev’s algorithm and its variants. This provides the first rigorous mathematical foundation for quantum cryptanalysis under stringent resource constraints.
📝 Abstract
In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in $(mathbb{Z}/Nmathbb{Z})^{ imes}$ that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants.