On some dynamical systems in finite fields and residue rings
Igor E. Shparlinski
Discrete & Continuous Dynamical Systems - A 2007, 17(4): 901-917 doi: 10.3934/dcds.2007.17.901
We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning.
keywords: finite fields and rings uniformity of distribution orbit length. Character sums
On finite fields for pairing based cryptography
Florian Luca Igor E. Shparlinski
Advances in Mathematics of Communications 2007, 1(3): 281-286 doi: 10.3934/amc.2007.1.281
Here, we improve our previous bound on the number of finite fields over which elliptic curves of cryptographic interest with a given embedding degree and small complex multiplication discriminant may exist. We also give some heuristic arguments which lead to a lower bound which in some cases is close to our upper bound.
keywords: embedding degree. Elliptic curves pairing based cryptography
On the generalized joint linear complexity profile of a class of nonlinear pseudorandom multisequences
Alina Ostafe Igor E. Shparlinski Arne Winterhof
Advances in Mathematics of Communications 2010, 4(3): 369-379 doi: 10.3934/amc.2010.4.369
Recently, multisequences have gained increasing interest for applications in cryptography and quasi-Monte Carlo methods. We study the (generalized) joint linear complexity of a class of nonlinear pseudorandom multisequences introduced by the first two authors as well as the linear complexity of its coordinate sequences. We prove lower bounds which are much stronger than in the case of single sequences since the multidimensional case brings in new and favourable effects.
keywords: nonlinear pseudorandom number generators. Linear complexity

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