Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal

Buchberger, Bruno

doi:10.1016/j.jsc.2005.09.007

Cited by 572 publications

(527 citation statements)

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“…Then, it S S S S S S S S S S S S S S S S solves the corresponding system which turns out to be overdefined and sparse. The methods already proposed for solving polynomial system of equations are Gröbner basis including Buchberger algorithm [4], F4 [15], F5 [16] and algorithms like ElimLin [9], XL [6] and its family [7], and Raddum-Semaev algorithm [26]. Converting these equations to Boolean expressions in Conjunctive Normal Form (CNF) [9] and deploying various SAT-solver programs is another strategy [14].…”

Section: Algebraic Analysismentioning

confidence: 99%

Linear (Hull) and Algebraic Cryptanalysis of the Block Cipher PRESENT

Nakahara

Sepehrdad

Zhang

et al. 2009

Lecture Notes in Computer Science

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Abstract. The contributions of this paper include the first linear hull and a revisit of the algebraic cryptanalysis of reduced-round variants of the block cipher PRESENT, under known-plaintext and ciphertextonly settings. We introduce a pure algebraic cryptanalysis of 5-round PRESENT and in one of our attacks we recover half of the bits of the key in less than three minutes using an ordinary desktop PC. The PRESENT block cipher is a design by Bogdanov et al., announced in CHES 2007 and aimed at RFID tags and sensor networks. For our linear attacks, we can attack 25-round PRESENT with the whole code book, 2 96.68 25-round PRESENT encryptions, 2 40 blocks of memory and 0.61 success rate. Further we can extend the linear attack to 26-round with small success rate. As a further contribution of this paper we computed linear hulls in practice for the original PRESENT cipher, which corroborated and even improved on the predicted bias (and the corresponding attack complexities) of conventional linear relations based on a single linear trail.

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Section: Algebraic Analysismentioning

confidence: 99%

Linear (Hull) and Algebraic Cryptanalysis of the Block Cipher PRESENT

Nakahara

Sepehrdad

Zhang

et al. 2009

Lecture Notes in Computer Science

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“…The effective notion stems from Shirshov's Composition Lemma and his algorithm [16] for Lie algebras and independently from Buchberger's algorithm [8] of computing Gröbner bases for commutative algebras. In [2], Bokut applied Shirshov's method to associative algebras, and Bergman mentioned the diamond lemma for ring theory [1].…”

Section: Introductionmentioning

confidence: 99%

Monomial Enumeration of the Full Icosahedral Group

Dong-Il Lee

2017

JMSS

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“…A Gröbner basis is a special kind of generating set for an ideal which provides a computational framework to determine many properties of the ideal. The notion of Gröbner bases was originally introduced in 1965 by Buchberger in his Ph.D. thesis and he also gave the basic algorithm to compute it [2,3]. Later on, he proposed two criteria for detecting superfluous reductions to improve his algorithm [1].…”

Section: Introductionmentioning

confidence: 99%

Improved Computation of Involutive Bases

Binaei

Hashemi

Seiler

2016

Computer Algebra in Scientific Computing

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Abstract. In this paper, we describe improved algorithms to compute Janet and Pommaret bases. To this end, based on the method proposed by Möller et al. [21], we present a more efficient variant of Gerdt's algorithm (than the algorithm presented in [17]) to compute minimal involutive bases. Further, by using the involutive version of Hilbert driven technique, along with the new variant of Gerdt's algorithm, we modify the algorithm, given in [24], to compute a linear change of coordinates for a given homogeneous ideal so that the new ideal (after performing this change) possesses a finite Pommaret basis. All the proposed algorithms have been implemented in Maple and their efficiency is discussed via a set of benchmark polynomials.

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Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal

Cited by 572 publications

References 1 publication

Linear (Hull) and Algebraic Cryptanalysis of the Block Cipher PRESENT

Linear (Hull) and Algebraic Cryptanalysis of the Block Cipher PRESENT

Monomial Enumeration of the Full Icosahedral Group

Improved Computation of Involutive Bases

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