Polynomial evaluation via the division algorithm the fast Fourier transform revisited

Fiduccia, Charles M.

doi:10.1145/800152.804900

Cited by 50 publications

(24 citation statements)

References 2 publications

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“…For the usual Chinese remainder theorem, efficient algorithms are well known for carrying out the corresponding conversions [6,22,47]. These algorithms are based on the technique of so-called remainder trees, for which recent improvements can be found in [4,10,35].…”

Section: Multi-modular Representationsmentioning

confidence: 99%

Accelerated tower arithmetic

Hoeven

Lecerf

2019

Journal of Complexity

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Nowadays, asymptotically fast algorithms are widely used in computer algebra for computations in towers of algebraic field extensions of small height. Yet it is still unknown how to reach softly linear time for products and inversions in towers of arbitrary height. In this paper we design the first algorithm for general ground fields with a complexity exponent that can be made arbitrarily close to one from the asymptotic point of view. We deduce new faster algorithms for changes of tower representations, including the computation of primitive element representations in subquadratic time.

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Section: Multi-modular Representationsmentioning

confidence: 99%

Accelerated tower arithmetic

Hoeven

Lecerf

2019

Journal of Complexity

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“…, α ∆−1 ∈ Z N . Using Horner's evaluation rule, it is easy to propose a solution that uses O(∆ 2 ) addition and multiplication in Z N but it is well-known that there exists an algorithm with quasi-linear complexity O(∆ log 2 ∆) =Õ(∆) operations in Z N using a divide-and-conquer approach [33,34]. The multipoint evaluation of univariate polynomials has found numerous application in cryptanalysis (e.g.…”

Section: Time/memory Tradeoof Attack Using Multi-evaluation Of Polynomentioning

confidence: 99%

Cryptanalysis of Server-Aided RSA Protocols with Private-Key Splitting

Mefenza

Vergnaud

2019

The Computer Journal

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We analyze the security and the efficiency of interactive protocols where a client wants to delegate the computation of an RSA signature given a public key, a public message and the secret signing exponent. We consider several protocols where the secret exponent is splitted using some algebraic decomposition. We first provide an exhaustive analysis of the delegation protocols in which the client outsources a single RSA exponentiation to the server. We then revisit the security of the protocols RSA-S1 and RSA-S2 that were proposed by Matsumoto, Kato and Imai in 1988. We present an improved lattice-based attack on RSA-S1 and we propose a simple variant of this protocol that provides better efficiency for the same security level. Eventually, we present the first attacks on the protocol RSA-S2 that employs the Chinese Remainder Theorem to speed up the client's computation. The efficiency of our (heuristic) attacks has been validated experimentally.

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“…There are classic algorithms for fast polynomial arithmetic, which use the Fast Fourier Transformation (FFT) [17,4,5] and have been used in various areas of cryptography; e.g., for efficiency improvement in protocols [10,35,24,2] and in cryptanalysis [11,8,18]. In this paper, we use two fast polynomial arithmetic algorithms, each denoted by Alg F F T P oly and Alg F F T M P E , as subroutines; the algorithm Alg F F T P oly takes points as inputs and outputs a monic degree-polynomial over Z x 0 having input points as roots.…”

Section: Fast Polynomial Algorithmsmentioning

confidence: 99%

Security Analysis of Multilinear Maps over the Integers

Lee

Seo

2014

Advances in Cryptology – CRYPTO 2014

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Abstract. At Crypto 2013, Coron, Lepoint, and Tibouchi (CLT) proposed a practical Graded Encoding Scheme (GES) over the integers, which has very similar cryptographic features to ideal multilinear maps. In fact, the scheme of Coron et al. is the second proposal of a secure GES, and has advantages over the first scheme of Garg, Gentry, and Halevi (GGH). For example, unlike the GGH construction, the subgroup decision assumption holds in the CLT construction. Immediately following the elegant innovations of the GES, numerous GES-based cryptographic applications were proposed. Although these applications rely on the security of the underlying GES, the security of the GES has not been analyzed in detail, aside from the original papers produced by Garg et al. and Coron et al. We present an attack algorithm against the system parameters of the CLT GES. The proposed algorithm's complexityÕ(2 ρ/2 ) is exponentially smaller thanÕ(2 ρ ) of the previous best attack of Coron et al., where ρ is a function of the security parameter. Furthermore, we identify a flaw in the generation of the zero-testing parameter of the CLT GES, which drastically reduces the running time of the proposed algorithm. The experimental results demonstrate the practicality of our attack.

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Polynomial evaluation via the division algorithm the fast Fourier transform revisited

Cited by 50 publications

References 2 publications

Accelerated tower arithmetic

Accelerated tower arithmetic

Cryptanalysis of Server-Aided RSA Protocols with Private-Key Splitting

Security Analysis of Multilinear Maps over the Integers

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