FAST ISOMORPHISM TESTING IN ARITHMETICAL VARIETIES

2003 ◽  
Vol 13 (04) ◽  
pp. 499-506
Author(s):  
TOMASZ A. GORAZD
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Boolean Algebras ◽  
Heyting Algebras ◽  
Subdirectly Irreducible ◽  
Finite Algebras ◽  
Element Chain ◽  
Subdirectly Irreducible Algebras

Let [Formula: see text] be a finitely generated, arithmetical variety such that all subdirectly irreducible algebras from [Formula: see text] have linearly ordered congruences. We show that there is a polynomial time algorithm that tests the existing of an isomorphism between any two finite algebras from [Formula: see text]. This includes the following classical structures in algebra: • Boolean algebras. • Varieties of rings generated by finitely many finite fields. • Varieties of Heyting algebras generated by an n–element chain.

scholarly journals Independence of algebras with edge term

2015 ◽  
Vol 25 (07) ◽  
pp. 1145-1157 ◽  
Author(s):  
Erhard Aichinger ◽  
Peter Mayr
Keyword(s):  
Polynomial Time ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Universal Algebras ◽  
Finite Algebras ◽  
Necessary And Sufficient ◽  
Independent Varieties ◽  
Subdirect Factorization ◽  
Independence Of Algebras

In [A. L. Foster, The identities of — and unique subdirect factorization within — classes of universal algebras, Math. Z. 62 (1955) 171–188], two varieties [Formula: see text] of the same type are defined to be independent if there is a binary term [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we give necessary and sufficient conditions for two finite algebras with a Mal’cev term (or, more generally, with an edge term) to generate independent varieties. In particular we show that the independence of finitely generated varieties with edge term can be decided by a polynomial time algorithm.

Computing the residual Galois representations

2011 ◽  
Author(s):  
Bas Edixhoven
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Galois Representations ◽  
Las Vegas ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complex Numbers ◽  
Rigorous Proof ◽  
Main Result Theorem ◽  
Result Theorem

This chapter proves the main result on the computation of Galois representations. It provides a detailed description of the algorithm and a rigorous proof of the complexity. It first combines the results of chapters 11 and 12 in order to work out the strategy of Chapter 3. This gives the main result, Theorem 14.1.1: a deterministic polynomial time algorithm, based on computations with complex numbers. The crucial transition from approximations to exact values is done, and the proof of Theorem 14.1.1 is finished later in the chapter. The chapter then replaces the complex computations with the computations over finite fields from Chapter 13, and gives a probabilistic (Las Vegas type) polynomial time variant of the algorithm in Theorem 14.1.1.

scholarly journals Traps to the BGJT-algorithm for discrete logarithms

2014 ◽  
Vol 17 (A) ◽  
pp. 218-229 ◽  
Author(s):  
Qi Cheng ◽  
Daqing Wan ◽  
Jincheng Zhuang
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Time Complexity ◽  
Complexity Analysis ◽  
Discrete Logarithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
New Approach ◽  
Discrete Logarithms ◽  
Polynomial Time Complexity

AbstractIn the recent breakthrough paper by Barbulescu, Gaudry, Joux and Thomé, a quasi-polynomial time algorithm is proposed for the discrete logarithm problem over finite fields of small characteristic. The time complexity analysis of the algorithm is based on several heuristics presented in their paper. We show that some of the heuristics are problematic in their original forms, in particular when the field is not a Kummer extension. We propose a fix to the algorithm in non-Kummer cases, without altering the heuristic quasi-polynomial time complexity. Further study is required in order to fully understand the effectiveness of the new approach.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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