Computing Frobenius maps and factoring polynomials

Gathen, Joachim von zur; Shoup, Victor

doi:10.1007/bf01272074

Cited by 148 publications

(89 citation statements)

References 31 publications

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“…Currently, the asymptotically fastest (probabilistic) algorithms use O((n 2 + n log q)(log n) 2 loglog n) operations in F q (von zur Gathen and Shoup, 1992) or O(n 1.815 log q) operations in F q (Kaltofen and Shoup, 1995) for factoring a polynomial of degree n. Hence we obtain the following:…”

Section: The Chinese Remainder Theoremmentioning

confidence: 99%

Factoring Modular Polynomials

Gathen

Hartlieb

1998

Journal of Symbolic Computation

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This paper gives an algorithm to factor a polynomial f (in one variable) over rings like Z /rZ for r ∈ Z or F q [y]/rF q [y] for r ∈ F q [y]. The Chinese Remainder Theorem reduces our problem to the case where r is a prime power. Then factorization is not unique, but if r does not divide the discriminant of f , our (probabilistic) algorithm produces a description of all (possibly exponentially many) factorizations into irreducible factors in polynomial time. If r divides the discriminant, we only know how to factor by exhaustive search, in exponential time.

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Section: The Chinese Remainder Theoremmentioning

confidence: 99%

Factoring Modular Polynomials

Gathen

Hartlieb

1998

Journal of Symbolic Computation

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“…This will allow us to make use of von zur Gathen and Shoup (1992) algorithm to compute quickly all conjugates of an element in F over K (see below). Such an element can be computed with log q operations in K by repeated squaring, though for convenience, we consider it pre-computation and do not count this cost in algorithms using this technique.…”

Section: Representing Skew-polynomial Ringsmentioning

confidence: 99%

“…For convenience, we assume throughout the paper that M(µ) = Ω(µ log µ). We can also compute a −1 for any a ∈ F with O(M(µ) log µ) operations in K. Using an algorithm of von zur Gathen and Shoup (1992), for any a ∈ F we can compute all conjugates a, τ (a), τ 2 (a), . .…”

Section: Representing Skew-polynomial Ringsmentioning

confidence: 99%

Factoring in Skew-polynomial Rings over Finite Fields

Giesbrecht

1998

Journal of Symbolic Computation

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“…Remark 2: For a univariate polynomial of degree over , there is a deterministic algorithm to find zeros using field operations (ignoring factors). See [13] and [14]. In order to prove Theorem 1, we use a generalization of the MTF balls-in-bins game first introduced in [10].…”

Section: A Wcr and Unforgeabilitymentioning

confidence: 99%

Multiproperty-Preserving Domain Extension Using Polynomial-Based Modes of Operation

Lee

Steinberger

2012

IEEE Trans. Inform. Theory

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In this paper, we propose a new double-piped mode of operation for multiproperty-preserving domain extension of message authentication codes (MACs), pseudorandom functions (PRFs), and pseudorandom oracles (PROs). Our mode of operation performs twice as fast as the original double-piped mode of operation of Lucks while providing comparable security. Our construction, which uses a class of polynomial-based compression functions proposed by Stam, makes a single call to a 3n-bit to n-bit primitive f 1 at each iteration and uses a finalization function f 2 at the last iteration, producing an n-bit hash function H[f 1 ; f 2 ] 1 We are not claiming that polynomial-based compression functions provide significant practical advantages over the existing rate 1=2 schemes, since field multiplication is not trivial to implement. In this paper, we only focus on the theoretical aspects in terms of "rate." For a software benchmark for a wide range of blockcipher-based hash functions including a blockcipher-based variant of a polynomial-based compression function, we refer to [5].

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Computing Frobenius maps and factoring polynomials

Cited by 148 publications

References 31 publications

Factoring Modular Polynomials

Factoring Modular Polynomials

Factoring in Skew-polynomial Rings over Finite Fields

Multiproperty-Preserving Domain Extension Using Polynomial-Based Modes of Operation

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