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On taking square roots without quadratic nonresidues over finite fields

2011
*
Mathematics of Computation
*

*In*some cases, the square

*root*algorithm runs

*in*Õ(^2 q) bit operations over

*finite*

*fields*with q elements. ... We present a novel idea to compute square

*roots*over

*finite*

*fields*, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. ... There are several efficient probabilistic algorithms for

*taking*square

*roots*

*in*

*finite*

*fields*. ...

##
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An Algorithms for Finding the Cube Roots in Finite Fields

2021
*
Procedia Computer Science
*

Let F q be a

doi:10.1016/j.procs.2021.01.072
fatcat:xal745khnbewndnoakffqx5rfi
*finite**field*with q elements. Quadratic residues*in*number theory and*finite**fields*is an important theory that has many applications*in*various aspects. ...*In*this paper we examine the solubility of x 3 = a*in*general*finite**fields*. ... We would like also to thank Rafael Herman Yosef for helping to calculate the complexity of the programme*in*Python. ...##
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Efficient Computation of Roots in Finite Fields

2006
*
Designs, Codes and Cryptography
*

We present an algorithm to compute r-th

doi:10.1007/s10623-005-4017-5
fatcat:b4q2szelurfwhkqzi4sweufmmm
*roots**in*Fq m with complexity O((log m + r log q)m 2 log 2 q) for certain choices of m and q. ...*Taking*r-th*roots**in*a*finite**field*Fq m is most commonly computed by means of the Adleman-Manders-Miller algorithm [1] (see also [2, section 7.3] ), which extends Tonelli's square*root*algorithm. ... Conclusion This contribution described an efficient algorithm to compute r-*roots**in*certain*finite**fields*Fq m . ...##
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Korat: A Tool for Generating Structurally Complex Test Inputs

2007
*
Proceedings / International Conference of Software Engineering
*

Korat

doi:10.1109/icse.2007.48
dblp:conf/icse/MilicevicMMK07
fatcat:us2ca6qybzdk3dioza4uzsuooa
*takes*(1) an imperative predicate that specifies the desired structural integrity constraints and (2) a*finitization*that bounds the desired test input size. ... Acknowledgments We thank Brett Daniel for extensive comments*on*an earlier draft of this paper. We thank Jesus DeLaTorre and ChoongHwan Lee for comments*on*the Korat tool. ... Two*finitizations*for SearchTree Each candidate that Korat generates is an object graph with*one**root*. ...##
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Overconvergent unit-root $F$-isocrystals and isotriviality

2017
*
Mathematical Research Letters
*

We show that a semisimple overconvergent "absolutely unit-

doi:10.4310/mrl.2017.v24.n6.a7
fatcat:gmnqxwpb2zb3dn5mpvpkcgzntm
*root*" F -isocrystal*on*a geometrically connected smooth variety over a*finite**field*becomes constant over a*finite*covering. ... Vladimir Drinfeld kindly pointed out a gap*in*an earlier version and suggested Example 2.2, and I would like to thank him. ... Introduction The main theorem We will study overconvergent unit-*root*F -isocrystals*on*a smooth variety over a*finite**field*. ...##
###
Tamely Ramified Towers and Discriminant Bounds for Number Fields—II

2002
*
Journal of symbolic computation
*

*In*1978, Martinet constructed an infinite unramified tower of totally complex number

*fields*with small constant

*root*discriminant, demonstrating that α 0 < 92.4. ... The

*root*discriminant of a number

*field*of degree n is the nth

*root*of the absolute value of its discriminant. ... Consider k = k 0 ( √ 2),

*in*which 7 splits completely, and

*take*K = k( √ −7).

*In*K/k, ten

*finite*and ten infinite places ramify. ...

##
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Infinite Towers of Galois Defect Extensions of Kaplansky Fields

2018
*
Annales Mathematicae Silesianae
*

We also give a constructive proof of the fact that a henselian Kaplansky

doi:10.1515/amsil-2017-0012
fatcat:fyxgyehc3jcjrmc5hevaq35euy
*field*cannot be defectless-by-*finite*. ... We give conditions for Kaplansky*fields*to admit infinite towers of Galois defect extensions of prime degree. ... By induction*on*n, this holds for every extension*in*the tower.*Take*a natural number n. Since (K, v) is a Kaplansky*field*, the same holds for (K n−1 , v). ...##
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On the Foundation of a Constructive Theory of Discrete Commutative Algebra (Second Paper)

1935
*
Proceedings of the National Academy of Sciences of the United States of America
*

*In*order to build up, based VOL. 21, 1935

*on*this notion, the idea of real

*field*, it is necessary to isolate a

*root*interval (ai,bl), called a simple interval

*in*(a,b) such that we cannot have

*root*intervals ... It is easily shown that any

*finite*ring is a quasi-

*field*, and the ring of square matrices of order n with elements

*in*a

*field*has this property. ...

##
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Constructing Finite Field Extensions with Large Order Elements

2007
*
SIAM Journal on Discrete Mathematics
*

polynomial

doi:10.1137/s0895480104445514
fatcat:3olpn3z2uvgwnb5nrpli2natm4
*on*N . ... We present another algorithm that find an integer n ∈ [N, N + O(N 0.77 )] and an element α ∈ F q n of order at least 5.8 √ n ,*in*time polynomial*on*N . ... Let x m − g, g ∈ F r , be an irreducible polynomial over F r and α be*one*of its*roots**in*the extension*field*F r m . ...##
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MODIFIED ALGORITHM FOR SEARCHING THE ROOTS OF THE ERROR LOCATORS POLYNOMINAL WHILE DECODING BCH CODES

2020
*
Radìoelektronika, Ìnformatika, Upravlìnnâ
*

The classical

doi:10.15588/1607-3274-2020-3-14
fatcat:l527vjmzgnapfctvexhkfmdgsu
*roots*search method based*on*the Chan's algorithm is performed using the arithmetic of the Galois*finite**fields*and the laborious calculation,*in*this case depends*on*the number of addition ... A modified*roots*search method for affine polynomials over the*finite**fields*has been proposed to determine error positions*in*the code word while decoding the cyclic BCH and RS codes. Conclusions. ... The proposed algorithm reduces the complexity of*root*calculations at*one*point of the*finite**field*due to the application of a special arrangement of all elements of the*finite**field*. ...##
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Cutting towers of number fields
[article]

2019
*
arXiv
*
pre-print

*In*the tame setting we achieve new records

*on*Martinet constants (

*root*discriminant bounds)

*in*the totally real and totally complex cases. ... Given a prime p, a number

*field*and a

*finite*set of places S of , let _S be the maximal pro-p extension of unramified outside S. ... . — Let K be a p-rational

*field*and let L{K be a

*finite*extension

*in*KSp {K. Then L is p-rational. ...

##
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A design of public key Cryptosystem in an algebraic extension field over a finite field using the difficulty of solving DLP

2020
*
Malaya Journal of Matematik
*

Through this research paper, authors construct a public key cryptosystem which works

doi:10.26637/mjm0802/0022
fatcat:vlk6qvzcxbewxaubmp7hbg5nsa
*in*the*finite*algebraic extension*field*F p n of the*finite**field*F p . ... The security of this system is based*on*difficulty of solving DLP*in*F p n * . The primitive polynomials are used*in*the construction of algebraic extension*fields*. ... [2]*In*this study, the authors propose a novel cryptosystem, which works*in*the*finite*extension*field*of F p . ...##
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A remark on an arithmetic theorem of Chevalley

1965
*
Proceedings of the American Mathematical Society
*

Then the conditions

doi:10.1090/s0002-9939-1965-0184925-x
fatcat:ktudb3ky4jahnj44q5rhotwr3q
*on*|x| <1 require that/1,2 should be a solution of (1). If det(7 -ia) p^O we can find a solution while if det(7 -ia) =0 there exists no solution. ... The algebraic closure of a*finite**field*is generated, as a*field*, by*roots*of unity of prime order. The same is (therefore) true of the maximal unramified extension of a p-adic*field*. Proof. ... ., such that ker (£->(0/p)*) Cker x-It follows immediately that if U is a subgroup of*finite*index*in*£ then ker(£->(©/a)*) C U for a suitable a, which we may*take*to be square free. ...##
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Deterministic equation solving over finite fields

2005
*
Proceedings of the 2005 international symposium on Symbolic and algebraic computation - ISSAC '05
*

All these algorithms

doi:10.1145/1073884.1073932
dblp:conf/issac/Woestijne05
fatcat:jqr3fgcizrb6jokoc53myvg4se
*take*polynomial time*in*n and*in*the logarithm of the*field*size, and are practical as stated. ... Deterministic algorithms are presented for the efficient solution of diagonal homogeneous equations*in*many variables over*finite**fields*. ... (and higher)*roots*(see [12] , [1] , [4] ) • methods for multivariate equations based*on*the above (see Section 8) • Schoof's algorithm for*taking*square*roots**in*prime*fields*(see [10] ) However, ...##
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Overconvergent unit-root F-isocrystals and isotriviality
[article]

2016
*
arXiv
*
pre-print

We show that a semisimple overconvergent "absolutely unit-

arXiv:1511.02884v4
fatcat:x2kt6ffnrfe2zbb6gzfy2hmfsi
*root*" F-isocrystal*on*a geometrically connected smooth variety over a*finite**field*becomes constant over a*finite*covering. ... Vladimir Drinfeld kindly pointed out a gap*in*an earlier version and suggested Example 2.2, and I would like to thank him. I also thank Koji Shimizu and Hélène Esnault for their comments. Notation. ... Introduction We will study overconvergent unit-*root*F -isocrystals*on*a smooth variety over a*finite**field*. ...
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