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On taking square roots without quadratic nonresidues over finite fields

Tsz-Wo Sze
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.  ... 
doi:10.1090/s0025-5718-2011-02419-1 fatcat:tjmitir2nvakfdzdppt3dp6g7u

An Algorithms for Finding the Cube Roots in Finite Fields

Faisal, Rojali, Mohd Sham Bin Mohamad
2021 Procedia Computer Science  
Let F q be a 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.  ... 
doi:10.1016/j.procs.2021.01.072 fatcat:xal745khnbewndnoakffqx5rfi

Efficient Computation of Roots in Finite Fields

Paulo S. L. M. Barreto, José Felipe Voloch
2006 Designs, Codes and Cryptography  
We present an algorithm to compute r-th 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 .  ... 
doi:10.1007/s10623-005-4017-5 fatcat:b4q2szelurfwhkqzi4sweufmmm

Korat: A Tool for Generating Structurally Complex Test Inputs

Aleksandar Milicevic, Sasa Misailovic, Darko Marinov, Sarfraz Khurshid
2007 Proceedings / International Conference of Software Engineering  
Korat 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.  ... 
doi:10.1109/icse.2007.48 dblp:conf/icse/MilicevicMMK07 fatcat:us2ca6qybzdk3dioza4uzsuooa

Overconvergent unit-root $F$-isocrystals and isotriviality

Teruhisa Koshikawa
2017 Mathematical Research Letters  
We show that a semisimple overconvergent "absolutely unit-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.  ... 
doi:10.4310/mrl.2017.v24.n6.a7 fatcat:gmnqxwpb2zb3dn5mpvpkcgzntm

Tamely Ramified Towers and Discriminant Bounds for Number Fields—II

Farshid Hajir, Christian Maire
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.  ... 
doi:10.1006/jsco.2001.0514 fatcat:65q6tkoxlzdj5ocfhtgou3vjwm

Infinite Towers of Galois Defect Extensions of Kaplansky Fields

Anna Blaszczok
2018 Annales Mathematicae Silesianae  
We also give a constructive proof of the fact that a henselian Kaplansky 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).  ... 
doi:10.1515/amsil-2017-0012 fatcat:fyxgyehc3jcjrmc5hevaq35euy

On the Foundation of a Constructive Theory of Discrete Commutative Algebra (Second Paper)

H. S. Vandiver
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.  ... 
doi:10.1073/pnas.21.3.162 pmid:16587951 pmcid:PMC1076554 fatcat:yjqburdi6bguvew75jjfqekwdm

Constructing Finite Field Extensions with Large Order Elements

Qi Cheng
2007 SIAM Journal on Discrete Mathematics  
polynomial 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 .  ... 
doi:10.1137/s0895480104445514 fatcat:3olpn3z2uvgwnb5nrpli2natm4

MODIFIED ALGORITHM FOR SEARCHING THE ROOTS OF THE ERROR LOCATORS POLYNOMINAL WHILE DECODING BCH CODES

V. А. Krylova, Е. Е. Тverytnykova, O. G. Vasylchenkov, T. P. Kolisnyk
2020 Radìoelektronika, Ìnformatika, Upravlìnnâ  
The classical 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.  ... 
doi:10.15588/1607-3274-2020-3-14 fatcat:l527vjmzgnapfctvexhkfmdgsu

Cutting towers of number fields [article]

Farshid Hajir, Christian Maire, Ravi Ramakrishna
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.  ... 
arXiv:1901.04354v1 fatcat:ojzpo46io5dizmyejpxughacve

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

M. I. Saju, Renjith Varghese, E.F. Antony John
2020 Malaya Journal of Matematik  
Through this research paper, authors construct a public key cryptosystem which works 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 .  ... 
doi:10.26637/mjm0802/0022 fatcat:vlk6qvzcxbewxaubmp7hbg5nsa

A remark on an arithmetic theorem of Chevalley

H. Bass
1965 Proceedings of the American Mathematical Society  
Then the conditions 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.  ... 
doi:10.1090/s0002-9939-1965-0184925-x fatcat:ktudb3ky4jahnj44q5rhotwr3q

Deterministic equation solving over finite fields

Christiaan van de Woestijne
2005 Proceedings of the 2005 international symposium on Symbolic and algebraic computation - ISSAC '05  
All these algorithms 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,  ... 
doi:10.1145/1073884.1073932 dblp:conf/issac/Woestijne05 fatcat:jqr3fgcizrb6jokoc53myvg4se

Overconvergent unit-root F-isocrystals and isotriviality [article]

Teruhisa Koshikawa
2016 arXiv   pre-print
We show that a semisimple overconvergent "absolutely unit-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.  ... 
arXiv:1511.02884v4 fatcat:x2kt6ffnrfe2zbb6gzfy2hmfsi
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