A Hybrid Inversive Congruential Pseudorandom Number Generator with High Period
31 Jan 2021-European Journal of Pure and Applied Mathematics (New York Business Global LLC)-Vol. 14, Iss: 1, pp 1-18
TL;DR: In this paper, a hybrid inversive congruential generator (HICG) based on a second order recurrence using the inversive modulo M, a power of 2, was proposed.
Abstract: Though generating a sequence of pseudorandom numbers by linear methods (Lehmer generator) displays acceptable behavior under some conditions of the parameters, it also has undesirable features, which makes the sequence unusable for various stochastic simulations. An extension which showed promise for such applications is a generator obtained by using a first-order recurrence based upon the inversive modulo a prime or a prime power, called inversive congruential generator (ICG). A lot of work has been dedicated to investigate the periods (under some conditions of the parameters), the lattice test passing, discrepancy and other statistical properties of such a generator. Here, we propose a new method, which we call hybrid inversive congruential generator (HICG), based upon a second order recurrence using the inversive modulo M, a power of 2. We investigate the period of this pseudorandom numbers generator (PRNG) and give necessary and sufficient conditions for our PRNG to have periods M (thereby doubling the period of the classical ICG) and M/2 (matching the one of the ICG). Moreover, we show that the lattice test complexity for a binary sequence associated to (a full period) HICG is precisely M/2.
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References
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TL;DR: A nonlinear congruential pseudorandom number generator with modulus M = 2 w is proposed, which may be viewed to comprise both linear as well as inversive Congruential generators.
Abstract: A nonlinear congruential pseudorandom number generator with modulus M = 2 w is proposed, which may be viewed to comprise both linear as well as inversive congruential generators. The condition for it to generate sequences of maximal period length is obtained. It is akin to the inversive one and bears a remarkable resemblance to the latter.
32 citations
"A Hybrid Inversive Congruential Pse..." refers background in this paper
...Later in [14], the authors modified (3) and proposed another nonlinear method, which we will describe below....
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...The main result obtained in [14] is the following theorem....
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TL;DR: It is shown that if c and 9/14 of the most significant bits of two consecutive values un,u of the Pollard generator are given, one can recover in polynomial time the initial value u0 with a probabilistic algorithm.
Abstract: Let p be a prime and let c be an integer modulo p. The Pollard generator is a sequence (un) of pseudorandom numbers defined by the relation un+1equivun 2+c mod p. It is shown that if c and 9/14 of the most significant bits of two consecutive values un,un+1 of the Pollard generator are given, one can recover in polynomial time the initial value u0 with a probabilistic algorithm. This result is an improvement of a theorem in a recent paper which requires that 2/3 of the most significant bits be known
32 citations
"A Hybrid Inversive Congruential Pse..." refers background in this paper
...However, it has been found that the linearity of the recursion leads to some drawbacks and as a result, recently, several nonlinear congruential generators [3–10, 12, 15, 19] have been proposed and also analyzed [11]....
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TL;DR: This work investigates a general method for constructing families of pseudorandom sequences with low correlation and large linear complexity from elliptic curves over finite fields in this correspondence with the help of the tool of exponential sums on elliptic curve.
Abstract: Following the idea of Xing et al., we investigate a general method for constructing families of pseudorandom sequences with low correlation and large linear complexity from elliptic curves over finite fields in this correspondence. With the help of the tool of exponential sums on elliptic curves, we study their periods, linear complexities, linear complexity profiles, distributions of r-patterns, periodic correlation, partial period distributions, and aperiodic correlation in detail. The results show that they have nice randomness.
31 citations
TL;DR: In this article, it was shown that for a pos- itive proportion of the inversive congruential generators with maximal period length, the discrepancy £>iJ2 is at least of the order of magnitude m~xl2 for all k > 2.
Abstract: The inversive congruential method with modulus m = 2W for the generation of uniform pseudorandom numbers has recently been introduced. The discrepancy Z)' L of A:-tuples of consecutive pseudorandom numbers generated by such a generator with maximal period length m/2 is the cru- cial quantity for the analysis of the statistical independence properties of these pseudorandom numbers by means of the serial test. It is proved that for a pos- itive proportion of the inversive congruential generators with maximal period length, the discrepancy £>iJ2 is at least of the order of magnitude m~xl2 for all k > 2 . This shows that the bound D{2)/2 = 0(m_1/2(logw)2) established by the second author is essentially best possible.
27 citations
TL;DR: It is shown that the set of points formed by consecutive pseudorandom numbers has a more complicated lattice structure: it forms a superposition of shifted lattices.
24 citations