Riemann hypothesis and finding roots over finite fields
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Elliptic Curves: Number Theory and Cryptography
Elliptic Curves: Number Theory and Cryptography
Computing the eigenvalue in the schoof-elkies-atkin algorithm using abelian lifts
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
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We define a class of algebras over finite fields, called polynomially cyclic algebras, which extend the class of abelian field extensions. We study the structure of these algebras; furthermore, we define and investigate properties of Lagrange resolvents and Gauss and Jacobi sums. Natural examples of polynomially cyclic algebras are for instance algebras of the form F"p[X]/(F"q(X)) where p,q are distinct odd primes and F"q is the q-th cyclotomic polynomial. Further examples occur similarly on replacing the cyclotomic polynomials with factors of division polynomials of elliptic curves. Finally, Gauss and Jacobi sums over polynomially cyclic algebras are applied for improving current algorithms for counting the number of points of elliptic curves over finite fields.