A short proof for explicit formulas for discrete logarithms in finite fields

1990 ◽  
Vol 1 (1) ◽  
pp. 55-57 ◽  
Author(s):  
Harald Niederreiter
Keyword(s):  
Finite Fields ◽  
Short Proof ◽  
Explicit Formulas ◽  
Discrete Logarithms

scholarly journals Function Field Sieve Method for Discrete Logarithms over Finite Fields

1999 ◽  
Vol 151 (1-2) ◽  
pp. 5-16 ◽  
Author(s):  
Leonard M. Adleman ◽  
Ming-Deh A. Huang
Keyword(s):  
Finite Fields ◽  
Function Field ◽  
Sieve Method ◽  
Discrete Logarithms

QUADRATIC FORMULAS FOR GENERALIZED QUATERNIONS

2009 ◽  
Vol 08 (03) ◽  
pp. 289-306 ◽  
Author(s):  
MARCO ABRATE
Keyword(s):  
Finite Fields ◽  
Quaternion Algebra ◽  
Quadratic Polynomial ◽  
Explicit Formulas ◽  
Quaternion Algebras ◽  
Solving Equations ◽  
Generalized Quaternion

In this paper we derive explicit formulas for computing the roots of a quadratic polynomial with coefficients in a generalized quaternion algebra over any field 𝔽 with characteristic not 2. We also give some example of applications for the derived formulas, solving equations in the algebra of Hamilton's quaternions ℍ, in the ring M2(ℝ) of 2 × 2 square matrices over ℝ and in quaternion algebras over finite 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 Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields

2018 ◽  
Vol 12 (4) ◽  
pp. 741-759
Author(s):  
Gora Adj ◽  
◽  
Isaac Canales-Martínez ◽  
Nareli Cruz-Cortés ◽  
Alfred Menezes ◽  
...  
Keyword(s):  
Finite Fields ◽  
Discrete Logarithms
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