Decomposition of algebras over finite fields and number fields

1991 ◽  
Vol 1 (2) ◽  
pp. 183-210 ◽  
Author(s):  
Wayne Eberly
Keyword(s):  
Finite Fields ◽  
Number Fields

scholarly journals Public-key cryptosystem based on invariants of diagonalizable groups

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
František Marko ◽  
Alexandr N. Zubkov ◽  
Martin Juráš
Keyword(s):  
Linear Algebra ◽  
Finite Fields ◽  
Number Fields ◽  
Euclidean Algorithm ◽  
Public Key ◽  
Public Key Cryptosystem ◽  
Finite Rings

AbstractWe develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).

scholarly journals Using number fields to compute logarithms in finite fields

1999 ◽  
Vol 69 (231) ◽  
pp. 1267-1284 ◽  
Author(s):  
Oliver Schirokauer
Keyword(s):  
Finite Fields ◽  
Number Fields

scholarly journals Hessian matrices, automorphisms of p-groups, and torsion points of elliptic curves

2021 ◽  
Author(s):  
Mima Stanojkovski ◽  
Christopher Voll
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Automorphism Groups ◽  
Number Fields ◽  
Explicit Formulas ◽  
Torsion Points ◽  
Curves Over Finite Fields

AbstractWe describe the automorphism groups of finite p-groups arising naturally via Hessian determinantal representations of elliptic curves defined over number fields. Moreover, we derive explicit formulas for the orders of these automorphism groups for elliptic curves of j-invariant 1728 given in Weierstrass form. We interpret these orders in terms of the numbers of 3-torsion points (or flex points) of the relevant curves over finite fields. Our work greatly generalizes and conceptualizes previous examples given by du Sautoy and Vaughan-Lee. It explains, in particular, why the orders arising in these examples are polynomial on Frobenius sets and vary with the primes in a nonquasipolynomial manner.

scholarly journals Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

2020 ◽  
Vol 156 (4) ◽  
pp. 733-743
Author(s):  
John R. Doyle ◽  
Bjorn Poonen
Keyword(s):  
Finite Fields ◽  
Function Field ◽  
Uniform Boundedness ◽  
Number Fields ◽  
Periodic Points ◽  
Irreducible Components

Fix $d\geqslant 2$ and a field $k$ such that $\operatorname{char}k\nmid d$. Assume that $k$ contains the $d$th roots of $1$. Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^{d}+c$ are geometrically irreducible and have gonality tending to $\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of $z^{d}+c$. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.

scholarly journals The multiple number field sieve for medium- and high-characteristic finite fields

2014 ◽  
Vol 17 (A) ◽  
pp. 230-246 ◽  
Author(s):  
Razvan Barbulescu ◽  
Cécile Pierrot
Keyword(s):  
Number Field ◽  
Finite Fields ◽  
Discrete Logarithm ◽  
Number Fields ◽  
Discrete Logarithm Problem ◽  
Discrete Logarithms ◽  
Number Field Sieve ◽  
Characteristic Case ◽  
Asymptotic Complexity ◽  
Improved Algorithm

AbstractIn this paper we study the discrete logarithm problem in medium- and high-characteristic finite fields. We propose a variant of the number field sieve (NFS) based on numerous number fields. Our improved algorithm computes discrete logarithms in $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_{p^n}$ for the whole range of applicability of the NFS and lowers the asymptotic complexity from $L_{p^n}({1/3},({128/9})^{1/3})$ to $L_{p^n}({1/3},(2^{13}/3^6)^{1/3})$ in the medium-characteristic case, and from $L_{p^n}({1/3},({64/9})^{1/3})$ to $L_{p^n}({1/3},((92 + 26 \sqrt{13})/27)^{1/3})$ in the high-characteristic case.

Finite fields and algebraic number fields

2003 ◽  
pp. 191-209
Author(s):  
Graham Everest ◽  
Alf van der Poorten ◽  
Igor Shparlinski ◽  
Thomas Ward
Keyword(s):  
Finite Fields ◽  
Algebraic Number ◽  
Number Fields ◽  
Algebraic Number Fields

scholarly journals -theory of one-dimensional rings via pro-excision

2013 ◽  
Vol 13 (2) ◽  
pp. 225-272 ◽  
Author(s):  
Matthew Morrow
Keyword(s):  
Finite Fields ◽  
Number Fields ◽  
Local Rings ◽  
Perfect Field ◽  
Finite Characteristic ◽  
Mixed Characteristic ◽  
One Dimensional ◽  
Curves Over Finite Fields

AbstractThis paper studies ‘pro-excision’ for the $K$-theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller’s conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller’s conjecture in mixed characteristic, and we establish various finiteness results for the $K$-groups of singularities, covering both orders in number fields and singular curves over finite fields.

scholarly journals Computing abelian varieties over finite fields isogenous to a power

2019 ◽  
Vol 5 (4) ◽  
Author(s):  
Stefano Marseglia
Keyword(s):  
Characteristic Polynomial ◽  
Finite Fields ◽  
Abelian Varieties ◽  
Number Fields ◽  
Explicit Description ◽  
Finite Product ◽  
Real Roots ◽  
Isomorphism Classes ◽  
Ordinary Case ◽  
Theoretic Description

Abstract In this paper we give a module-theoretic description of the isomorphism classes of abelian varieties A isogenous to $$B^r$$Br, where the characteristic polynomial g of Frobenius of B is an ordinary square-free q-Weil polynomial, for a power q of a prime p, or a square-free p-Weil polynomial with no real roots. Under some extra assumptions on the polynomial g we give an explicit description of all the isomorphism classes which can be computed in terms of fractional ideals of an order in a finite product of number fields. In the ordinary case, we also give a module-theoretic description of the polarizations of A.

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