A hierarchy of polynomial time lattice basis reduction algorithms

Schnorr, Claus-Peter

doi:10.1016/0304-3975(87)90064-8

Cited by 525 publications

(432 citation statements)

References 7 publications

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“…By enumerating all possible short vectors, we can then upper bound the probability that there exists a nonzero lattice point shorter than the target vector for a randomly chosen instance. From a practical point of view, one hopes to solve the problem by using standard lattice reductions algorithms [21,[30][31][32] as lattice shortest vector oracles.…”

Section: Lattice-based Methods For Noisy Polynomial Interpolationmentioning

confidence: 99%

“…For a randomly chosen instance, we built the corresponding sublattice Λ. For lattice reduction, we successively applied three different types of reduction : plain LLL [21], Schnorr's BKZ reduction [30,31] with block size 20, and when necessary, Schnorr-Hörner's pruned BKZ reduction [32] with block size 54 and pruning factor 14. We stopped the reduction as soon as the reduced basis contained the target vector.…”

Section: Methodsmentioning

confidence: 99%

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Noisy Polynomial Interpolation and Noisy Chinese Remaindering

Bleichenbacher¹,

Nguyêñ

2000

Advances in Cryptology — EUROCRYPT 2000

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Abstract. The noisy polynomial interpolation problem is a new intractability assumption introduced last year in oblivious polynomial evaluation. It also appeared independently in password identification schemes, due to its connection with secret sharing schemes based on Lagrange's polynomial interpolation. This paper presents new algorithms to solve the noisy polynomial interpolation problem. In particular, we prove a reduction from noisy polynomial interpolation to the lattice shortest vector problem, when the parameters satisfy a certain condition that we make explicit. Standard lattice reduction techniques appear to solve many instances of the problem. It follows that noisy polynomial interpolation is much easier than expected. We therefore suggest simple modifications to several cryptographic schemes recently proposed, in order to change the intractability assumption. We also discuss analogous methods for the related noisy Chinese remaindering problem arising from the well-known analogy between polynomials and integers.

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Section: Lattice-based Methods For Noisy Polynomial Interpolationmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Noisy Polynomial Interpolation and Noisy Chinese Remaindering

Bleichenbacher¹,

Nguyêñ

2000

Advances in Cryptology — EUROCRYPT 2000

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“…The statement is a combination of the Schnorr modification [13] of the lattice basis reduction algorithm of Lenstra, Lenstra and Lovász [5] with a result of Kannan [6] about reduction of the closest vector problem to the shortest vector problem.…”

Section: Latticesmentioning

confidence: 99%

Security of the most significant bits of the Shamir message passing scheme

Vasco

Shparlinski

2001

Math. Comp.

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Abstract. Boneh and Venkatesan have recently proposed a polynomial time algorithm for recovering a "hidden" element α of a finite field Fp of p elements from rather short strings of the most significant bits of the remainder modulo p of αt for several values of t selected uniformly at random from F * p . Unfortunately the applications to the computational security of most significant bits of private keys of some finite field exponentiation based cryptosystems given by Boneh and Venkatesan are not quite correct. For the Diffie-Hellman cryptosystem the result of Boneh and Venkatesan has been corrected and generalized in our recent paper. Here a similar analysis is given for the Shamir message passing scheme. The results depend on some bounds of exponential sums.

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“…C.P. Schnorr [16] proved that the factor can be replaced by (1 + ) n for any fixed > 0. However Schnorr's algorithm has a running time with 1/ in the exponent.…”

Section: Lattice Problems With Worst-case/average-case Equivalencementioning

confidence: 99%

A Lattice- Based Public-Key Cryptosystem

Cai

Cusick

1999

Selected Areas in Cryptography

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Abstract. Ajtai recently found a random class of lattices of integer points for which he could prove the following worst-case/average-case equivalence result: If there is a probabilistic polynomial time algorithm which finds a short vector in a random lattice from the class, then there is also a probabilistic polynomial time algorithm which solves several problems related to the shortest lattice vector problem (SVP) in any n-dimensional lattice. Ajtai and Dwork then designed a public-key cryptosystem which is provably secure unless the worst case of a version of the SVP can be solved in probabilistic polynomial time. However, their cryptosystem suffers from massive data expansion because it encrypts data bit-by-bit. Here we present a public-key cryptosystem based on similar ideas, but with much less data expansion.

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A hierarchy of polynomial time lattice basis reduction algorithms

Cited by 525 publications

References 7 publications

Noisy Polynomial Interpolation and Noisy Chinese Remaindering

Noisy Polynomial Interpolation and Noisy Chinese Remaindering

Security of the most significant bits of the Shamir message passing scheme

A Lattice- Based Public-Key Cryptosystem

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