Lattice Cryptanalysis and Reduction

Breaking Cryptographic Assumptions with Lattice Algorithms

Lattice reduction algorithms are the primary tool for attacking cryptographic schemes based on hard lattice problems — the same problems that underpin post-quantum standards like Kyber and Dilithium. I build the tools that test these assumptions: a complete lattice sieving and reduction library in Rust (g6k-rs), covering LLL, BKZ, sieving, CVP, and specialized attacks on LWE and ECDSA. The library implements algorithms from Lenstra-Lenstra-Lovasz through modern sieving techniques (BDGL, BGJ1, HK3), with SIMD acceleration, GPU compute, and arbitrary-precision arithmetic for real-world cryptanalytic workloads.

What This Means for Your Business

Lattice reduction is where the security of post-quantum cryptography gets tested. When NIST evaluates whether Kyber-768 provides 128-bit security, the answer depends on how fast BKZ with sieving can find short vectors in lattices of dimension 1483. Every improvement in sieving performance — better LSH bucketing, tighter enumeration bounds, faster inner products — shifts the security boundary. Building these tools from scratch means understanding the algorithms at the level where optimizations live: the SIMD inner product that cuts sieve time by 4x, the incremental GSO update that avoids O(n^3) recomputation, the extended GCD insertion that lets BKZ exploit SVP solutions it previously had to discard.

Beyond post-quantum, lattice techniques crack classical cryptography when implementations leak information. The Hidden Number Problem reduces ECDSA nonce leakage to a shortest vector problem: given a few bits of each nonce across multiple signatures, LLL or BKZ can recover the private key. Coppersmith's method finds small roots of polynomials modulo a composite — breaking RSA with weak padding. These are not theoretical attacks; they have been used against real Bitcoin wallets, TLS implementations, and hardware security modules.

How I Have Used This in Production

  1. g6k-rs: Full Lattice Sieving Library in Rust

    Open Source — github.com/ovasylenko/g6k-rs

    Solo-built a 16K-line Rust library implementing the complete lattice reduction stack. LLL with incremental GSO and deep insertion (Schnorr-Euchner). BKZ 2.0 with Schnorr-Euchner enumeration — extended GCD insertion for non-unimodular SVP solutions, Gaussian heuristic pruning, and sieve-based SVP oracles. Three sieve algorithms: BDGL with FHT-LSH bucketing, BGJ1, and HK3 triple sieve. Multi-threaded and Metal GPU-accelerated variants. Babai Nearest Plane and Rounding CVP solvers. 215 tests across 6 suites, CI on Linux and macOS.

  2. Coppersmith Small-Root Finding

    g6k-rs

    Implemented the Coppersmith/Howgrave-Graham method for finding small integer roots of polynomials modulo a composite N. Builds the shifted polynomial lattice, LLL-reduces it, and recovers roots via exact integer polynomial evaluation. Applicable to RSA attacks with weak padding schemes.

  3. ECDSA Private Key Recovery via HNP

    g6k-rs

    Built an ECDSA attack tool that recovers private keys from nonce bit leakage using lattice reduction. Constructs the Hidden Number Problem lattice from leaked LSBs of per-message secrets across multiple signatures, applies LLL/BKZ reduction on arbitrary-precision lattices, and extracts the key from short vectors in the reduced basis. Tested against SECP256K1 with 4-8 known nonce bits.

  4. LWE Solver via Kannan Embedding

    g6k-rs

    Implemented the primal attack on Learning With Errors using Kannan's embedding technique. Constructs the embedding lattice, applies LLL and BKZ reduction, and extracts the secret and error vectors. Solves small to medium LWE instances (secret dimension up to 8, modulus up to 97) with BKZ block sizes 10-20.

Technologies

RustLattice ReductionLLLBKZBDGL SieveSIMDMetal GPUGMPCoppersmithECDSAHidden Number ProblemLWEPost-Quantum Cryptography

g6k-rs is the Rust-native counterpart to the C++/Python g6k toolkit. See the C++ sieving engine optimization and the 100x spherical code generator.

Post-Quantum Cryptography Tooling — Optimizing the Leading Lattice Sieving Toolkit

Cryptanalytic tools demand correctness — a silent numerical bug produces wrong security estimates. See why Rust for systems where failure is not an option.

Rust for Mission-Critical Systems — Why Rust When Failure Is Not an Option

Lattice cryptography and zero-knowledge proofs share the intersection of hard mathematical problems and practical implementation. See how I build production ZK systems.

ZK-SNARK Development — From Circuits to Production Wallets

Need lattice reduction expertise?

Whether you are estimating the security of a post-quantum scheme, testing an ECDSA implementation for nonce leakage, or building cryptanalytic tools — I have built the algorithms from scratch and understand them at the level where optimizations and correctness bugs live. If your work touches lattice problems, let's talk.

Discuss lattice cryptanalysis