---
title: Tensor-Network Schnorr’s Sieve Factoring
url: https://www.emergentmind.com/topics/tensor-network-schnorr-s-sieve-factoring
type: topic
---

# Tensor-Network Schnorr’s Sieve Factoring

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 [2410.16355].

## 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 $u$ and $u-vN$ factor completely over $P_2$.

Schnorr’s sieve reframes the search for sr-pairs as a closest vector problem in an $n = \pi_1$-rank lattice $\Lambda(B) \subset \mathbb{Z}^{n+1}$ generated by column vectors $b_j \in \mathbb{Z}^{n+1}$. Explicitly,

$$
\Lambda(B) = \{\, B\mathbf{e} \mid \mathbf{e} \in \mathbb{Z}^n \,\},
$$

and the CVP seeks $b^{\mathrm{opt}} = \arg\min_{b \in \Lambda(B)} \|b - t\|_2$ for a target $t \in \mathbb{Z}^{n+1}$.

Schnorr’s selection for $B$ and $t$ uses a random permutation $f$ and precision parameter $c$, yielding

$$
B_{f,c} = \begin{pmatrix} \lfloor f(1)\rfloor & 0 & \cdots & 0 \\
0 & \lfloor f(2)\rfloor & \cdots & 0 \\
\vdots & & \ddots & \vdots \\
0 & 0 & \cdots & \lfloor f(n)\rfloor \\
10^c \ln p_1 & 10^c \ln p_2 & \cdots & 10^c \ln p_n
\end{pmatrix},
$$

$$
t_{N,c} = (0, \dots, 0, 10^c \ln N)^{T}.
$$

The lattice point $b = B\mathbf{e}$ encodes exponents $e_j$ corresponding to $u$ (for $e_j > 0$) and $v$ (for $e_j < 0$). If $u - vN$ is $B_2$-smooth, a valid sr-pair is constructed. LLL reduction and Babai’s nearest-plane algorithm approximate the CVP solution with complexity $T_1 = O(n^7 \log_{10}^3 n)$.

## 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 $n$ qubits. Using $Z_j = (1-\sigma^z_j)/2$, the cost function is

$$
H = \sum_{k=1}^{n+1} \left[ t_k - b^{cl}_k - \sum_{j=1}^n \mathrm{sign}(\mu_j - c_j) d_{j,k} Z_j \right]^2,
$$
where $b^{cl}$ is the Babai-classical nearest lattice vector, $d_{j}$ are Gram–Schmidt orthogonal columns obtained via LLL, and $\mu_j, c_j$ are associated Gram–Schmidt coefficients and integer offsets.

Eigenstates $|s_1 \dots s_n\rangle$ encode lattice vectors $b^s = b^{cl} + \sum_j (2s_j-1) d_j$, whose energies $E_s = \|b^s - t\|_2^2$ favor low-energy configurations that are likely to yield valid sr-pairs.

The wavefunction $|\psi\rangle$ is represented as a binary TTN, with physical indices $s_v \in \{0,1\}$ and virtual indices of dimension $m$. Operators such as $H$ are diagonal in the computational basis and can be efficiently represented as matrix product operators (MPOs) with bond dimension $\leq n + 1$. 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 $m$.
- Repeated sweeps update each tensor $A_v$ via diagonalization or SVD of a local effective Hamiltonian ($h_{\mathrm{eff}}$), with per-sweep cost $O(n^3 m^4)$.
- The OPES (Optimal Projected-Entangled Sampling) algorithm samples $Q$ bit-strings from the low-energy tail in $O(nQ m^4)$ time, after perturbation with small random transverse fields.

For each sampled bit-string, the exponent vector is reconstructed ($b^s \mapsto \mathbf{e} = B^{-1}b^s$), $u-vN$ is formed, and $B_2$-smoothness is tested via trial division.

The total complexity per CVP is:

$$
T = T_1 + T_2 + T_3 = O\Bigl(n^7\log^3 n + n^3 m^4 + nQ m^4\Bigr),
$$
where empirically $m(n) \approx 6.6\, n^{0.42}$ for $n \in [32, 256]$, $Q = O(\ell^\gamma)$, and $n = \mathrm{poly}(\ell)$.

## 4. Empirical Performance and Scaling

Key empirical findings from benchmark studies include:

- The average number of sr-pairs per CVP, $\mathrm{sr}(\ell, n, Q)$, obeys

  $$
  \mathrm{sr} \sim C_1 Q^{\gamma} \exp \left\{ - C_2 \left[ \frac{\ell}{n^{1/\omega}} \right]^\mu \right\},
  $$
  with $C_1 \approx 1.0$, $C_2 \approx 0.013$, $\mu \approx 1.61$, $\omega \approx 9.3$.

- Achieving $\mathrm{sr} \ge 1$ for given $\ell, Q$ requires

  $$
  n(\ell, Q) \sim \left[ \frac{C_2 Q^\mu}{ \ln C_1 - \ln(\mathrm{sr}) + Q\ln \ell } \right]^{\omega/\mu},
  $$
  implying polynomial scaling of $n$ with $\ell$.

- Bond dimension follows $m(n) \approx 6.6\, n^{0.42}$.
- For $\ell = 70$, $n=32$, $Q \approx 70^3$, $m=8$, yielding $\approx 432$ sr-pairs per CVP.
- A 100-bit RSA challenge ($N=791{,}339\,171\,587\,617\,359\,026\,543\,582\,309$, $\ell=100$) was factored with $n=64$, $Q=2$, $28\,980$ CVPs, and $\pi_2=12\,801$ ($B_2 \approx 1.37 \times 10^5$), extracting 11,278 sr-pairs in a few days on a single Xeon CPU. The result: $p=428\,949\,705\,601\,033$, $q=1\,844\,829\,734\,709\,373$.

These results demonstrate that TNSS achieves empirical polynomial cost scaling in the bit-length $\ell$ for up to 100-bit factorizations.

## 5. Scalability and Practical Limitations

Although TNSS exhibits polynomial scaling, the dominant complexity terms—$T_1 \sim n^{7}$ and $T_2 \sim n^{3+4\zeta}$ (with $\zeta \approx 0.42$)—translate, via $n = O(\ell^{\omega/\mu})$ (with $\omega/\mu \approx 5.8$), into polynomial degrees as high as $T = O(\ell^{66})$ for $\ell \lesssim 30$. This high-order scaling leads to prohibitive constant factors for practically breaking cryptographically relevant moduli such as RSA-2048 ($\ell=2048$).

Prospective optimizations include:

- Improved TTN geometries or introduction of deeper tree structures to further reduce $\zeta$.
- Enhanced sampling strategies for lowering $Q$.
- 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 $L_N[1/3]$, remains the most effective classical factoring approach for standard RSA key sizes. TNSS does not yet constitute a practical threat to large ($\ell \sim 10^3$) 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 [2410.16355].

Source: https://www.emergentmind.com/topics/tensor-network-schnorr-s-sieve-factoring