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1
Content available remote Pairing-Friendly Primes for Abelian Varieties
Grześkowiak M.
Fundamenta Informaticae
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2016
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Vol. 149, nr 4
385--400
EN
We present a method of generating primes r ≡ 1 (mod n), q and a Weil q-number π such that r divides Φn(q) and r divides |A(Fq)|, where A/Fq is an ordinary abelian variety defined over a finite Fq corresponding to π. Such primes can be used for implementing pairing-based cryptographic systems.
2
Content available remote An Algorithmic Construction of Finite Elliptic Curves of Order Divisible by a Large Prime
Grześkowiak M.
Fundamenta Informaticae
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2015
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Vol. 136, nr 4
331--343
EN
Given a square-free integer Δ < 0, we present an algorithm constructing a pair of primes p and q such that q|p + 1 − t and 4p − t2, = Δf2, where |t| ≤ 2√p for some integers f, t. Together with a CM method presented in the paper, such primes p and q are used for a construction of an elliptic curve E over a finite field Fp such that the order of E is divisible by a large prime. It is shown that our algorithm works in polynomial time.
3
Content available remote Prime Numbers and Cryptosystems Based on Discrete Logarithms
Grześkowiak M.
Studia Bezpieczeństwa Narodowego
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2014
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R. 4, Nr 6
163--176
EN
In this paper, we give a short overview of algorithms of generating primes to a DL systems. The algorithms are probabilistic and works in a polynomial time.
PL
W pracy przedstawiamy algorytmy, które generują liczby pierwsze do kryptosystemów opartych na logarytmach dyskretnych. Zaprezentowane algorytmy są probabilistyczne i działają w wielomianowym czasie.
4
Content available remote Algorithms for Relatively Cyclotomic Primes
Grześkowiak M.
Fundamenta Informaticae
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2013
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Vol. 125, nr 2
161--181
EN
We present a general method of generating primes p and q such that q divides Φn(p), where n > 2 is a fixed number. In particular, we present the deterministic method of finding a primitive nth roots of unity modulo q. We estimate the computational complexity of our methods.
5
Content available remote Algorithm for Generating Primes p and q Such that q Divides p^4+p^2+p+1
Grześkowiak M.
Fundamenta Informaticae
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2012
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Vol. 114, nr 3/4
287-299
EN
In the paper we propose an algorithm for generating large primes p and q such that q divides p4+p3+p2+p+1 or p4-p3+p2-p+1, and p, q are key parameters for Giuliani-Gong’s Public Key System. We analyze the computational complexity of considered methods.
6
Content available Generating elements of orders dividing p6 ± p5 + p4 ± p3 + p2 + p ± 1
Grześkowiak M.
Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica
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2011
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Vol. 11, no. 2
113--125
EN
In this paper we propose an algorithm for computing large primes p and q such that q divides p6 + p5 + p4 + p3 + p2 + p + 1 or p6 - p5 + p4 - p3 + p2 - p + 1. Such primes are the key parameters for the cryptosystem based on the 7th order characteristic sequences.
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