Noisy interpolation of sparse polynomials in finite fields

Shparlinski, Igor E.; Winterhof, Arne

doi:10.1007/s00200-005-0180-1

Cited by 11 publications

(21 citation statements)

References 19 publications

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“…Extending the bound of [84] to composite moduli is already an interesting problem. The bound of [46] has been applied to various problems in number theory [10], computer science [124] and cryptography [125]. Several other bounds (stronger, but more restrictive on the class of polynomials to which they apply) on exponential sums with sparse polynomials can be found in [23,27,47,111].…”

Section: Beatty Sequencesmentioning

confidence: 99%

Open Problems on Exponential and Character Sums

Shparlinski

2009

Number Theory

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Section: Beatty Sequencesmentioning

confidence: 99%

Open Problems on Exponential and Character Sums

Shparlinski

2009

Number Theory

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“…It has also found some applications to a problem on the complexity of approximation of the permanent [5] and to noisy polynomial interpolation [26]. On the other hand, it should be remarked that our approach requires an increase in the number of t ∈ G for which MSB η (αt) is requested (which remains polynomial nevertheless, in particular it never exceeds (log p) 4 ).…”

Section: Introductionmentioning

confidence: 98%

A hidden number problem in small subgroups

Shparlinski

Winterhof

2005

Math. Comp.

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Abstract. Boneh and Venkatesan have proposed a polynomial time algorithm for recovering a hidden element α ∈ F p , where p is prime, from rather short strings of the most significant bits of the residue of αt modulo p for several randomly chosen t ∈ F p . González Vasco and the first author have recently extended this result to subgroups of F * p of order at least p 1/3+ε for all p and to subgroups of order at least p ε for almost all p. Here we introduce a new modification in the scheme which amplifies the uniformity of distribution of the multipliers t and thus extend this result to subgroups of order at least (log p)/(log log p) 1−ε for all primes p. As in the above works, we give applications of our result to the bit security of the Diffie-Hellman secret key starting with subgroups of very small size, thus including all cryptographically interesting subgroups.

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“…, c. The MMO problem with known moduli is to recover the polynomials f and g. This is a natural extension of the well known polynomial interpolation problem. We want to mention now that if p is much larger than q, then the MMO problem with known moduli can be easily transformed in a noisy polynomial interpolation problem (see [10]), where the evaluation of the polynomial g modulo q can be seen as random "noise" and the attacker tries to recover f . Rigorous bounds for the noise of the results in [10] depend heavily on the performance of finding a close vector in the lattice and it seems that there is some gap between the theoretic results and the practical experiments.…”

Section: Introductionmentioning

confidence: 99%

The MMO problem

García-Morchón

Gómez-Pérez

Gutiérrez

et al. 2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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We consider a two polynomials analogue of the polynomial interpolation problem. Namely, we consider the Mixing Modular Operations (MMO) problem of recovering two polynomials f ∈ Zp[x] and g ∈ Zq[x] of known degree, where p and q are two (un)known positive integers, from the values of f (t) mod p + g(t) mod q at polynomially many points t ∈ Z. We show that if p and q are known, the MMO problem can be reduced to computing a close vector in a lattice with respect to the infinity norm. Using the Gaussian heuristic we also implemented in the SAGE system a polynomial-time algorithm. If p and q are kept secret, we do not know how to solve this problem. This problem is motivated by several potential cryptographic applications.

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Noisy interpolation of sparse polynomials in finite fields

Cited by 11 publications

References 19 publications

Open Problems on Exponential and Character Sums

Open Problems on Exponential and Character Sums

A hidden number problem in small subgroups

The MMO problem

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