Tensor-Network Schnorr’s Sieve Factoring

• The paper introduces the Tensor-Network Schnorr’s Sieve, recasting the closest vector problem as a spin-glass Hamiltonian to extract smooth-relation pairs for integer factorization.
• It employs variational tree tensor networks to approximate low-energy states, demonstrating empirical polynomial scaling in factoring 100-bit RSA numbers.
• Scalability challenges persist due to high-degree polynomial cost, prompting research into optimized TTN geometries and hybrid quantum-classical approaches for cryptanalysis.

Tensor-network Schnorr’s Sieve Factoring is a quantum-inspired classical algorithmic framework for integer factorization which recasts Schnorr’s lattice-based method as a combinatorial optimization problem, then solves it via tensor network techniques. This approach, referred to as the Tensor-Network Schnorr’s Sieving (TNSS) algorithm, encodes the closest vector problem (CVP) arising in integer factorization into a spin-glass Hamiltonian. The low-energy states of this Hamiltonian are optimized variationally using tree tensor networks (TTNs), and sampled to yield smooth-relation (sr) pairs necessary for the construction of congruences of squares in the factorization process. Demonstrations up to 100-bit RSA numbers, with empirical polynomial scaling in the bit-length $\ell$ of semiprimes, attest to its potential as a high-dimensional, quantum-inspired attack vector against classical cryptographic standards (Tesoro et al., 2024).

1. Schnorr’s Lattice-Based Factoring Framework

Let $N = p \cdot q$ be the semiprime to be factored, with bit-length $\ell = \lfloor \log_2 N \rfloor + 1$. The approach employs two prime bases: $P_1 = \{p_1, \dots, p_{\pi_1}\}$ of size $\pi_1$, and $P_2 = \{p_1, \dots, p_{\pi_2}\}$ of size $\pi_2$, with smoothness bound $B_2 = p_{\pi_2}$. The key step seeks sr-pairs $(u, u - vN)$ such that $u \equiv u-vN \pmod N$ and both $N = p \cdot q$0 and $N = p \cdot q$1 factor completely over $N = p \cdot q$2.

Schnorr’s sieve reframes the search for sr-pairs as a closest vector problem in an $N = p \cdot q$3-rank lattice $N = p \cdot q$4 generated by column vectors $N = p \cdot q$5. Explicitly,

$N = p \cdot q$6

and the CVP seeks $N = p \cdot q$7 for a target $N = p \cdot q$8.

Schnorr’s selection for $N = p \cdot q$9 and $\ell = \lfloor \log_2 N \rfloor + 1$0 uses a random permutation $\ell = \lfloor \log_2 N \rfloor + 1$1 and precision parameter $\ell = \lfloor \log_2 N \rfloor + 1$2, yielding

$\ell = \lfloor \log_2 N \rfloor + 1$3

$\ell = \lfloor \log_2 N \rfloor + 1$4

The lattice point $\ell = \lfloor \log_2 N \rfloor + 1$5 encodes exponents $\ell = \lfloor \log_2 N \rfloor + 1$6 corresponding to $\ell = \lfloor \log_2 N \rfloor + 1$7 (for $\ell = \lfloor \log_2 N \rfloor + 1$8) and $\ell = \lfloor \log_2 N \rfloor + 1$9 (for $P_1 = \{p_1, \dots, p_{\pi_1}\}$0). If $P_1 = \{p_1, \dots, p_{\pi_1}\}$1 is $P_1 = \{p_1, \dots, p_{\pi_1}\}$2-smooth, a valid sr-pair is constructed. LLL reduction and Babai’s nearest-plane algorithm approximate the CVP solution with complexity $P_1 = \{p_1, \dots, p_{\pi_1}\}$3.

2. Tensor Network Mapping and Spin-Glass Hamiltonian

The lattice CVP is recast as finding the ground-state of a spin-glass Hamiltonian acting on $P_1 = \{p_1, \dots, p_{\pi_1}\}$4 qubits. Using $P_1 = \{p_1, \dots, p_{\pi_1}\}$5, the cost function is

$P_1 = \{p_1, \dots, p_{\pi_1}\}$6

where $P_1 = \{p_1, \dots, p_{\pi_1}\}$7 is the Babai-classical nearest lattice vector, $P_1 = \{p_1, \dots, p_{\pi_1}\}$8 are Gram–Schmidt orthogonal columns obtained via LLL, and $P_1 = \{p_1, \dots, p_{\pi_1}\}$9 are associated Gram–Schmidt coefficients and integer offsets.

Eigenstates $\pi_1$0 encode lattice vectors $\pi_1$1, whose energies $\pi_1$2 favor low-energy configurations that are likely to yield valid sr-pairs.

The wavefunction $\pi_1$3 is represented as a binary TTN, with physical indices $\pi_1$4 and virtual indices of dimension $\pi_1$5. Operators such as $\pi_1$6 are diagonal in the computational basis and can be efficiently represented as matrix product operators (MPOs) with bond dimension $\pi_1$7. Integer-smoothness constraints are imposed during post-processing on sampled bit-strings.

3. Algorithmic Implementation and Complexity

The TNSS proceeds via a variational search for the TTN ground-state:

• Initialization of random TTN with bond dimension $\pi_1$8.
• Repeated sweeps update each tensor $\pi_1$9 via diagonalization or SVD of a local effective Hamiltonian ($P_2 = \{p_1, \dots, p_{\pi_2}\}$0), with per-sweep cost $P_2 = \{p_1, \dots, p_{\pi_2}\}$1.
• The OPES (Optimal Projected-Entangled Sampling) algorithm samples $P_2 = \{p_1, \dots, p_{\pi_2}\}$2 bit-strings from the low-energy tail in $P_2 = \{p_1, \dots, p_{\pi_2}\}$3 time, after perturbation with small random transverse fields.

For each sampled bit-string, the exponent vector is reconstructed ($P_2 = \{p_1, \dots, p_{\pi_2}\}$4), $P_2 = \{p_1, \dots, p_{\pi_2}\}$5 is formed, and $P_2 = \{p_1, \dots, p_{\pi_2}\}$6-smoothness is tested via trial division.

The total complexity per CVP is:

$P_2 = \{p_1, \dots, p_{\pi_2}\}$7

where empirically $P_2 = \{p_1, \dots, p_{\pi_2}\}$8 for $P_2 = \{p_1, \dots, p_{\pi_2}\}$9, $\pi_2$0, and $\pi_2$1.

4. Empirical Performance and Scaling

Key empirical findings from benchmark studies include:

• The average number of sr-pairs per CVP, $\pi_2$2, obeys

$\pi_2$3

with $\pi_2$4, $\pi_2$5, $\pi_2$6, $\pi_2$7.

• Achieving $\pi_2$8 for given $\pi_2$9 requires

$B_2 = p_{\pi_2}$0

implying polynomial scaling of $B_2 = p_{\pi_2}$1 with $B_2 = p_{\pi_2}$2.

• Bond dimension follows $B_2 = p_{\pi_2}$3.
• For $B_2 = p_{\pi_2}$4, $B_2 = p_{\pi_2}$5, $B_2 = p_{\pi_2}$6, $B_2 = p_{\pi_2}$7, yielding $B_2 = p_{\pi_2}$8 sr-pairs per CVP.
• A 100-bit RSA challenge ($B_2 = p_{\pi_2}$9, $(u, u - vN)$0) was factored with $(u, u - vN)$1, $(u, u - vN)$2, $(u, u - vN)$3 CVPs, and $(u, u - vN)$4 ($(u, u - vN)$5), extracting 11,278 sr-pairs in a few days on a single Xeon CPU. The result: $(u, u - vN)$6, $(u, u - vN)$7.

These results demonstrate that TNSS achieves empirical polynomial cost scaling in the bit-length $(u, u - vN)$8 for up to 100-bit factorizations.

5. Scalability and Practical Limitations

Although TNSS exhibits polynomial scaling, the dominant complexity terms—$(u, u - vN)$9 and $u \equiv u-vN \pmod N$0 (with $u \equiv u-vN \pmod N$1)—translate, via $u \equiv u-vN \pmod N$2 (with $u \equiv u-vN \pmod N$3), into polynomial degrees as high as $u \equiv u-vN \pmod N$4 for $u \equiv u-vN \pmod N$5. This high-order scaling leads to prohibitive constant factors for practically breaking cryptographically relevant moduli such as RSA-2048 ($u \equiv u-vN \pmod N$6).

Prospective optimizations include:

• Improved TTN geometries or introduction of deeper tree structures to further reduce $u \equiv u-vN \pmod N$7.
• Enhanced sampling strategies for lowering $u \equiv u-vN \pmod N$8.
• Parallelization of CVP instances and TTN sweeps on large-scale HPC clusters.
• Hybrid approaches leveraging quantum subroutines to propose tensor updates.

6. Cryptographic Implications

Currently, the General Number Field Sieve (GNFS), with sub-exponential complexity $u \equiv u-vN \pmod N$9, remains the most effective classical factoring approach for standard RSA key sizes. TNSS does not yet constitute a practical threat to large ($N = p \cdot q$00) moduli. However, if the observed polynomial cost scaling in TNSS extends to much larger sizes, classical tensor-network algorithms could undermine RSA security.

This situation accentuates the urgency of adopting post-quantum cryptography and quantum key distribution, as classical “quantum-inspired” algorithms increasingly erode the computational intractability assumptions underlying modern public-key infrastructures.

7. Summary

Tensor-Network Schnorr’s Sieve recasts lattice factorization attacks as Hamiltonian optimization, leveraging variational TTN ground-state search and OPES sampling to efficiently extract smooth relations. The demonstrated factorization of 100-bit RSA numbers evidences the practical viability of this methodology for moderate parameters. Its polynomial complexity with respect to bit-length, albeit with a high polynomial degree, delineates both a conceptual advance in classical cryptanalytic techniques and key challenges inhibiting immediate scalability. As tensor-network and lattice algorithms continue to mature, the boundaries of classical cryptanalysis against public-key standards are expected to be further tested (Tesoro et al., 2024).