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Prime (order theory)

About: Prime (order theory) is a(n) research topic. Over the lifetime, 15324 publication(s) have been published within this topic receiving 157801 citation(s).
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Journal ArticleDOI
Johan Håstad1Institutions (1)
01 Jul 2001-Journal of the ACM
TL;DR: It is proved optimal, up to an arbitrary ε > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting.
Abstract: We prove optimal, up to an arbitrary e > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.

1,813 citations


Book ChapterDOI
Victor Shoup1Institutions (1)
11 May 1997-
TL;DR: Lower bounds on the complexity of the discrete logarithm and related problems are proved that match the known upper bounds: any generic algorithm must perform Ω(p1/2) group operations, where p is the largest prime dividing the order of the group.
Abstract: This paper considers the computational complexity of the discrete logarithm and related problems in the context of "generic algorithms"--that is, algorithms which do not exploit any special properties of the encodings of group elements, other than the property that each group element is encoded as a unique binary string. Lower bounds on the complexity of these problems are proved that match the known upper bounds: any generic algorithm must perform Ω(p1/2) group operations, where p is the largest prime dividing the order of the group. Also, a new method for correcting a faulty Diffie-Hellman oracle is presented.

1,232 citations


Book
01 Jan 1991-
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

1,089 citations


Journal ArticleDOI
TL;DR: Two closely-related pseudo-random sequence generators are presented: the ${1 / P} generator, with input P a prime, and the $x^2 \bmod N$generator, which outputs the quotient digits obtained on dividing 1 by P.
Abstract: Two closely-related pseudo-random sequence generators are presented: The ${1 / P}$generator, with input P a prime, outputs the quotient digits obtained on dividing 1 by P. The $x^2 \bmod N$generato...

968 citations


Journal ArticleDOI
Abstract: We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite.

964 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202219
2021837
2020858
2019774
2018659
2017705

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Topic's top 5 most impactful authors

Igor E. Shparlinski

116 papers, 961 citations

Zhi-Wei Sun

61 papers, 1K citations

Basudeb Dhara

19 papers, 45 citations

Florian Luca

18 papers, 125 citations

Pieter Moree

16 papers, 173 citations