The LLL algorithm takes as input a basis of a Euclidean lattice, and, within a polynomial number of operations, it outputs another basis of the same lattice but consisting of rather short vectors. We provide a generalization to R-modules contained in $K^n$ for arbitrary number fields K and dimension n, with R denoting the ring of integers of K. Concretely, we introduce an algorithm that efficiently finds short vectors in rank-n modules when given access to an oracle that finds short vectors in rank-2 modules, and an algorithm that efficiently finds short vectors in rank-2 modules given access to a Closest Vector Problem oracle for a lattice that depends only on K. The second algorithm relies on quantum computations and its analysis is heuristic.

An LLL Algorithm for Module Lattices

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An LLL Algorithm for Module Lattices.

We introduce a new lattice basis reduction algorithm with approximation guarantees analogous to the LLL algorithm and practical performance that far exceeds the current state of the art. We achieve these results by iteratively applying precision management techniques within a recursive algorithm structure and show the stability of this approach. We analyze the asymptotic behavior of our algorithm, and show that the heuristic running time is $$O(n^{\omega }(C+n)^{1+\varepsilon })$$ for lattices of dimension n, $$\omega \in (2,3]$$ bounding the cost of size reduction, matrix multiplication, and QR factorization, and C bounding the log of the condition number of the input basis B. This yields a running time of $$O\left( n^\omega (p + n)^{1 + \varepsilon }\right) $$ for precision $$p = O(\log \Vert B\Vert _{max})$$ in common applications. Our algorithm is fully practical, and we have published our implementation. We experimentally validate our heuristic, give extensive benchmarks against numerous classes of cryptographic lattices, and show that our algorithm significantly outperforms existing implementations. 

Fast Practical Lattice Reduction Through Iterated Compression

We introduce a new lattice basis reduction algorithm with approximation guarantees analogous to the LLL algorithm and practical performance that far exceeds the current state of the art. We achieve these results by iteratively applying precision management techniques within a recursive algorithm structure and show the stability of this approach. We analyze the asymptotic behavior of our algorithm, and show that the heuristic running time is O(n(C+n)) for lattices of dimension n, ω ∈ (2, 3] bounding the cost of size reduction, matrix multiplication, and QR factorization, and C bounding the log of the condition number of the input basis B. This yields a running time of O ( n(p+ n) ) for precision p = O(log ∥B∥max) in common applications. Our algorithm is fully practical, and we have published our implementation. We experimentally validate our heuristic, give extensive benchmarks against numerous classes of cryptographic lattices, and show that our algorithm signi cantly outperforms existing implementations.

Fast Practical Lattice Reduction through Iterated Compression

Key Exchange is an essential primitive in cryptography that originated from the breakthrough idea of Diffie Hellman (DH) protocol presented in 1976. Following this work, there are many successful variants of DH using Elliptic curves, Isogeny and RLWE (Ring Learning with Errors). The Ring-Leaning with Errors (RLWE), Search and Decision version are hard problems in mathematics and it has been shown that solving worst case hard problems in ideal lattices can be reduced to solving RLWE in the average case. The reduction forms the basis of provable security for RLWE based key exchange against passive adversaries. The hardness of the decision version of RLWE, which means indistinguishability of RLWE samples from Uniform, makes it challenging to validate a RLWE form public key. This proves to be an advantage for malicious users to create public keys that deviate from a protocol to recover the secret of reused keys in Key Exchange (KE) protocols. In this thesis, we provide a cryptanalysis of RLWE key exchange, presenting two polynomial time strategies to exploit key reuse. Then we propose a defense against such exploits by presenting a Zero Knowledge authentication protocol to verify the prover’s knowledge of a secret corresponding to his public key. The protocol is Statistical Zero Knowledge with negligible soundness and completeness errors so that the prover does not reveal any information about the secret in the proof. This is to ensure that a malicious verifier cannot impersonate the prover after the proof. The authentication protocol uses the signal function defined in RLWE key exchange to perform the verification.

Zero-Knowledge Proof for Knowledge of RLWE (Ring-Learning with Errors) Secret Keys

Let B be a basis of a Euclidean lattice, and B an approximation thereof. We give a sufficient condition on the closeness between B and B so that an LLL-reducing transformation U for B remains valid for B. Further, we analyse an efficient reduction algorithm when B is itself a small deformation of an LLL-reduced basis. Applications include speeding-up reduction by keeping only the most significant bits of B, reducing a basis that is only approximately known, and efficiently batching LLL reductions for closely related inputs.

/pdf/lll-reducing-with-the-most-significant-bits-dl3nfzagbk.pdf

LLL reducing with the most significant bits

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LLL reducing with the most significant bits