On the Distribution of the Power Generator over a Residue Ring for Parts of the Period

This paper studies the distribution of the power generator of pseudorandom numbers over a residue ring for parts of the period. These results compliment some recently obtained distribution bounds of the power generator modulo an arbitrary number for the entire period. Also, the arbitrary modulus case may have some cryptography related applications and could be of interest in other settings which require quality pseudorandom numbers.This paper studies the distribution of the power generator of pseudorandom numbers over a residue ring for parts of the period. These results compliment some recently obtained distribution bounds of the power generator modulo an arbitrary number for the entire period. Also, the arbitrary modulus case may have some cryptography related applications and could be of interest in other settings which require quality pseudorandom numbers

On Some Dynamical Systems in Finite Fields and Residue Rings

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

On the distribution of the power generator modulo a prime power

We study the multidimensional distribution of the power generator of pseudorandom numbers modulo a high power of a fixed prime number. These results complement some recently obtained results about the power generator modulo a product of two distinct primes in which case the generator is of great value for many cryptographic applications. The case of a prime power modulus, although it does not have any immediate cryptography related applications, may still be of interest for other applications which require quality pseudorandom numbers. Moreover, in this case new effects arise which allow us to apply some recent bounds for exponential sums with sparse polynomials to study the multidimensional distribution. In the case of moduli which are the product of two primes such results are known only for power generators with small exponents.9 page(s