Classical public-key cryptography standards rely on the Rivest-Shamir-Adleman (RSA) encryption protocol. The security of this protocol is based on the exponential computational complexity of the most efficient classical algorithms for factoring large semiprime numbers into their two prime components. Here, we address RSA factorization building on Schnorr's mathematical framework where factorization translates into a combinatorial optimization problem. We solve the optimization task via tensor network methods, a quantum-inspired classical numerical technique. This tensor network Schnorr's sieving algorithm displays numerical evidence of polynomial scaling of resources with the bit-length of the semiprime. We factorize RSA numbers up to 100 bits and assess how computational resources scale through numerical simulations up to 130 bits, encoding the optimization problem in quantum systems with up to 256 qubits. Only the high-order polynomial scaling of the required resources limits the factorization of larger numbers. Although these results do not currently undermine the security of the present communication infrastructure, they strongly highlight the urgency of implementing post-quantum cryptography or quantum key distribution.

翻译：经典公钥密码学标准依赖于Rivest-Shamir-Adleman（RSA）加密协议。该协议的安全性基于最有效的经典算法将大半素数分解为其两个素数分量的指数级计算复杂度。本文基于Schnorr的数学框架研究RSA分解问题，该框架将因数分解转化为组合优化问题。我们通过张量网络方法（一种受量子启发的经典数值技术）解决此优化任务。该张量网络Schnorr筛法算法显示出计算资源随半素数位长呈多项式增长的数值证据。我们成功分解了100位以内的RSA整数，并通过数值模拟评估了计算资源在130位范围内的扩展规律，将优化问题编码至最多256量子位的量子系统中。仅受限于所需资源的高阶多项式增长，更大整数的分解才无法实现。尽管这些结果目前尚未对现有通信基础设施的安全性构成实质威胁，但它们强烈凸显了实施后量子密码学或量子密钥分发的紧迫性。