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The Identity Transform of a Permutation and its Applications

Frosini A., Battaglino D., Rinaldi S., Socci S.

Fundamenta Informaticae

2015

Vol. 141, nr 2/3

191--205

Starting from a Theorem by Hall, we define the identity transform of a permutation π as C(π) = (0 + π(0), 1 + π(1), ..., (n - 1) + π(n - 1)), and we define the set Cn = {(C(π) : π ∈ Sn}, where Sn is the set of permutations of the elements of the cyclic group Zn. In the first part of this paper we study the set Cn: we show some closure properties of this set, and then provide some of its combinatorial and algebraic characterizations and connections with other combinatorial structures. In the second part of the paper, we use some of the combinatorial properties we have determined to provide a different algorithm for the proof of Hall's Theorem.

The average time complexity of probabilistic algorithms for finding generators in finite cyclic groups

Adamski T., Nowakowski W.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2015

Vol. 63, nr 4

989--996

Generators of finite cyclic groups play important role in many cryptographic algorithms like public key ciphers, digital signatures, entity identification and key agreement algorithms. The above kinds of cryptographic algorithms are crucial for all secure communication in computer networks and secure information processing (in particular in mobile services, banking and electronic administration). In the paper, proofs of correctness of two probabilistic algorithms (for finding generators of finite cyclic groups and primitive roots) are given along with assessment of their average time computational complexity.

Średnia złożoność obliczeniowa probabilistycznego algorytmu wyszukiwania pierwiastków pierwotnych modulo n

Adamski T.

Studia Bezpieczeństwa Narodowego

2014

R. 4, Nr 6

247--257

W pracy oszacowano średnią złożoność obliczeniową probabilistycznego algorytmu wyszukiwania pierwiastków pierwotnych modulo n. Uzyskany wynik może być w naturalny sposób uogólniony na przypadek algorytmu wyszukiwania generatorów dowolnej skończonej grupy cyklicznej jeśli znamy rozkład na czynniki pierwsze rzędu tej grupy.

Primitive roots from a natural number n (i.e. generators of the multiplicative group Z* n) play an important role in many cryptographic algorithms like public key ciphers, digital signatures and key agreement algorithms. In the paper, proof of correctness of the probabilistic algorithm for ﬁnding primitive roots is given along with assessment of its average computational complexity. Results obtained for the multiplicative group Z* n can be in natural easy way generalized on the case of arbitrary ﬁnite cyclic groups.

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 .