scholarly journals Constructing nonresidues in finite fields and the extended Riemann hypothesis

1991 ◽  
Author(s):  
Johannes Buchmann ◽  
Victor Shoup
Keyword(s):  
Finite Fields ◽  
Riemann Hypothesis ◽  
Extended Riemann Hypothesis

Short Proofs for Nondivisibility of Sparse Polynomials under the Extended Riemann Hypothesis

1996 ◽  
Vol 28 (3,4) ◽  
pp. 297-301 ◽  
Author(s):  
Dima Grigoriev ◽  
Marek Karpinski ◽  
Andrew M. Odlyzko
Keyword(s):  
Riemann Hypothesis ◽  
Sparse Polynomials ◽  
Extended Riemann Hypothesis

scholarly journals Probabilistic and deterministic analogues of the Miller – Rabin algorithm for ideals of rings of integer algebraic elements of finite extensions of the field 

2020 ◽  
Vol 56 (2) ◽  
pp. 144-156
Author(s):  
N. P. Prochorov
Keyword(s):  
Riemann Hypothesis ◽  
Primality Testing ◽  
Deterministic Algorithms ◽  
Rings Of Integers ◽  
Extended Riemann Hypothesis ◽  
Algebraic Elements

In this paper, we obtained the primality criteria for ideals of rings of integer algebraic elements of finite extensions of the field Q, which are analogues of Miller and Euler’s primality criteria for rings of integers. Also advanced analogues of these criteria were obtained, assuming the extended Riemann hypothesis. Arithmetic and modular operations for ideals of rings of integer algebraic elements of finite extensions of the field Q were elaborated. Using these criteria, the polynomial probabilistic and deterministic algorithms for the primality testing in rings of integer algebraic elements of finite extensions of the field Q were offered.

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