Lattice Cryptanalysis and Reduction

Breaking Cryptographic Assumptions with Lattice Algorithms

Updated

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.

Why Sieving Tooling Is Security-Relevant

Modern lattice sieves spend most of their time deciding which pairs of vectors deserve a real reduction attempt. G6K answers that with a SimHash-style filter: a bank of sparse probe vectors produces a bit fingerprint per lattice vector, and a single XOR-popcount rejects most candidate pairs before any inner product is computed. Across a full sieve, that filter is the difference between hours and weeks of compute.

The filter is only as good as its probe bank. The probes are sparse spherical codes, and a poorly distributed bank either rejects good pairs or lets noise through - both collapse sieving performance. This is why engineering work at the tooling level, from probe bank generation to SIMD inner-product kernels, translates directly into which lattice dimensions are practically reachable, and therefore into the concrete security estimates behind schemes like Kyber and Dilithium.

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

Frequently Asked Questions

What is lattice cryptanalysis and why does it matter?

Lattice cryptanalysis uses lattice reduction algorithms like LLL and BKZ to attack cryptographic schemes built on hard lattice problems — the same problems underpinning post-quantum standards like Kyber and Dilithium. It is where post-quantum security gets tested: whether Kyber-768 provides 128-bit security depends on how fast BKZ with sieving finds short vectors in lattices of dimension 1483. Every sieving improvement — better LSH bucketing, tighter enumeration bounds, faster inner products — shifts that security boundary.

When should a team apply lattice techniques to classical cryptography like ECDSA or RSA?

Lattice techniques apply whenever an implementation leaks partial 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.

What hands-on experience does Oleksii Vasylenko have with lattice cryptanalysis?

Oleksii Vasylenko solo-built g6k-rs, a 16K-line open-source Rust library implementing the complete lattice reduction stack. It covers LLL with incremental GSO and deep insertion, BKZ 2.0 with Schnorr-Euchner enumeration and sieve-based SVP oracles, three sieve algorithms (BDGL with FHT-LSH bucketing, BGJ1, HK3), Babai CVP solvers, Coppersmith small-root finding, and an LWE solver via Kannan embedding, plus an ECDSA key-recovery tool tested against SECP256K1 with 4-8 known nonce bits. The library has 215 tests across 6 suites.

What goes wrong when teams build their own cryptanalytic tools?

Cryptanalytic tools demand correctness, because a silent numerical bug produces wrong security estimates. The hard parts live at the algorithm level: incremental GSO updates that avoid O(n^3) recomputation, extended GCD insertion that lets BKZ exploit non-unimodular SVP solutions it would otherwise discard, and SIMD inner products that cut sieve time by 4x. Teams that implement reduction naively get tools that are too slow for real workloads or quietly wrong — both undermine the security conclusions drawn from them.

How can a company engage Oleksii Vasylenko for lattice reduction expertise?

Oleksii Vasylenko works on engagements involving lattice problems: estimating the security of a post-quantum scheme, testing an ECDSA implementation for nonce leakage, or building cryptanalytic tools. He has built the algorithms from scratch in g6k-rs and understands them at the level where optimizations and correctness bugs live — from SIMD inner products to GPU-accelerated sieving on arbitrary-precision lattices. Teams whose work touches lattice problems can contact him through ovasylenko.com.

Further Reading

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