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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 ...
We introduce stochastic sandpile models which imitate numerous aspects of the practical behavior of the LLL algorithm with compelling accuracy. In addition, we argue that the physics and mathematics o...
Quantum LLL with an Application to Mersenne Number Cryptosystems
quantum attack lattice reduction Grover's algorithm
2019/9/16
In this work we analyze the impact of translating the well-known LLL algorithm for lattice reduction into the quantum setting. We present the first (to the best of our knowledge) quantum circuit repre...
The LLL algorithm (from Lenstra, Lenstra and Lovász) and its generalization BKZ (from Schnorr and Euchner) are widely used in cryptanalysis, especially for lattice-based cryptography. Precisely unders...
We describe an asymptotically fast variant of the LLL lattice reduction algorithm. It takes as input a basis B∈Zn×nB∈Zn×n and returns a (reduced) basis CC of the Euclidean lattice LL spanned by BB, wh...
Adaptive precision LLL and Potential-LLL reductions with Interval arithmetic
Lattice reduction lattice techniques
2016/5/30
Lattice reduction is fundamental in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra–Lenstra–Lovász reduction algorithm (called LLL or L3) has be...
Cryptanalysis of a public key cryptosystem based on Diophantine equations via weighted LLL reduction
Weighted LLL reduction Public-key cryrtosystem Post-quantum cryptosystem
2016/1/26
In this paper, we give an attack against a public key cryptosystem based on Diophantine
equations of degree increasing type (DEC) proposed by the third author ([Oku15]). We show
that the security of...
Rounding and Chaining LLL: Finding Faster Small Roots of Univariate Polynomial Congruences
Coppersmith's Algorithm Small Roots of Polynomial Equations LLL
2016/1/9
In a seminal work at EUROCRYPT '96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysi...
Rounding LLL: Finding Faster Small Roots of Univariate Polynomial Congruences
Coppersmith's Algorithm Small Roots of Polynomial Equations
2014/3/10
In a seminal work at EUROCRYPT '96, Coppersmith showed how to find all small roots of a univariate polynomial congruence in polynomial time: this has found many applications in public-key cryptanalysi...