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On the least primitive root modulo 2p for odd primes p

Paszkiewicz A.

Electronics and Telecommunications Quarterly

2009

Vol. 55, No 4

649-658

In this paper we present results of numerical investigations concerning primitive roots modulo m = 2p, where p is prime greater than 2. This is one of these cases of integers m for which the multiplicative group of primitive rests modulo m can be generated by one element. Our computations comprise all primes below 10 10 .

On the average of the least quadratic non-residue modulo a prime number

Paszkiewicz A.

Electronics and Telecommunications Quarterly

2009

Vol. 55, No 4

639-647

In this paper we focus on a problem of existence the mean value of the least quadratic non-residue modulo a prime number. We prove that the answer to that question is positive and calculate the exact value of that constant with high accuracy. We also prove that the density of all primes having its least quadratic non-residue equal to k-th prime is 1/2k. Some computational results are included to provide numerical arguments that the convergence is very fast.

A Contribution to the Discrete Logarithm Problem

Paszkiewicz A.

Electronics and Telecommunications Quarterly

2009

Vol. 55, No 3

493-499

In this paper we consider a common, fundamental problem of all cryptographic schemes based on the discrete logarithm modulo a prime. This is the choice a base of the modular exponent function or other words a generator of a prime order multiplicative group. We construct a class of primes for which its least generators can be extremely large.

An experimental investigation of least primitive roots of prime numbers p less than one bilion

Paszkiewicz A.

Biuletyn Wojskowej Akademii Technicznej

1998

Vol. 47, nr 3

41-63

This work presents results of the research of least primitive roots modulo prime number for prime numbers p ≤ 10 9. Also some hypotheses about least primitive roots were discussed and new hypotheses were supposed.

Przedstawiono wyniki badań nad najmniejszymi pierwiastkami modulo liczba pierwsza dla liczb pierwszych p ≤ 109. Przedyskutowane zostały niektóre hipotezy dotyczące najmniejszych pierwiastków oraz wysunięte nowe przypuszczenia.