A primality test using cyclotomic extensions

1989 ◽  
pp. 310-323
Author(s):  
Preda Mihailescu
Keyword(s):  
Primality Test ◽  
Cyclotomic Extensions

scholarly journals A primality test for $Kp^n+1$ numbers

2014 ◽  
Vol 84 (291) ◽  
pp. 505-512 ◽  
Author(s):  
José María Grau ◽  
Antonio M. Oller-Marcén ◽  
Daniel Sadornil
Keyword(s):  
Primality Test

scholarly journals COMPARISON STUDY OF FERMAT, SOLOVAY-STRASSEN AND MILLER-RABIN PRIMALITY TEST USING MATHEMATICA 6.0

2012 ◽  
Vol 3 (1) ◽  
pp. 1-10
Author(s):  
Ega Gradini
Keyword(s):  
Quantitative Comparison ◽  
Comparison Study ◽  
Source Codes ◽  
Powerful Test ◽  
Euler’S Theorem ◽  
Primality Test ◽  
Miller Test ◽  
Mathematica Software ◽  
The Given

This paper presents three primality tests; Fermat test, Solovay-Strassen test, and Rabin-Miller test. Mathematica software is used to carry out the primality tests. The application of Fermat’s Litle Theorem as well as Euler’s Theorem on the tests was also discussed and this leads to the concept of pseudoprime. This paper is also discussed some results on pseudoprimes with certain range and do quantitative comparison. Those primality tests need to be evaluated in terms of its ability to compute as well as correctness in determining primality of given numbers. The answer to this is to create a source codes for those tests and evaluate them by using Mathematica 6.0. Those are Miller-Rabin test, Solovay-Strassen test, Fermat test and Lucas-Lehmer test. Each test was coded using an algorithm derived from number theoretic theorems and coded using the Mathematica version 6.0. Miller-Rabin test, SolovayStrassen test, and Fermat test are probabilistic tests since they cannot certainly identify the given number is prime, sometimes they fail. Using Mathematica 6.0, comparison study of primality test has been made and given the Miller- Rabin test as the most powerful test than other.

scholarly journals Cyclotomic extensions of number fields

2003 ◽  
Vol 14 (2) ◽  
pp. 183-196 ◽  
Author(s):  
Henri Cohen ◽  
Francisco Diaz y Diaz ◽  
Michel Olivier
Keyword(s):  
Number Fields ◽  
Cyclotomic Extensions

scholarly journals On the Number of Witnesses in the Miller–Rabin Primality Test

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 890
Author(s):  
Shamil Talgatovich Ishmukhametov ◽  
Bulat Gazinurovich Mubarakov ◽  
Ramilya Gakilevna Rubtsova
Keyword(s):  
Positive Integer ◽  
Euler’S Totient Function ◽  
Number Of Factors ◽  
Primality Test ◽  
Average Probability ◽  
Test Errors ◽  
The Difference

In this paper, we investigate the popular Miller–Rabin primality test and study its effectiveness. The ability of the test to determine prime integers is based on the difference of the number of primality witnesses for composite and prime integers. Let W ( n ) denote the set of all primality witnesses for odd n. By Rabin’s theorem, if n is prime, then each positive integer a < n is a primality witness for n. For composite n, the power of W ( n ) is less than or equal to φ ( n ) / 4 where φ ( n ) is Euler’s Totient function. We derive new exact formulas for the power of W ( n ) depending on the number of factors of tested integers. In addition, we study the average probability of errors in the Miller–Rabin test and show that it decreases when the length of tested integers increases. This allows us to reduce estimations for the probability of the Miller–Rabin test errors and increase its efficiency.

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