Nonlinear Congruential Pseudorandom Number Generators Research Articles

DWT based coding DNA watermarking for DNA copyright protection

DNA watermarking is a technique for copyright protection and ownership authentication of DNA sequences and ensures the security of private genetic information. This paper addresses issues regarding watermarking DNA coding sequences in the frequency domain that confer mutation resistance, amino acid conservation, and security. Multimedia watermarking is designed for robustness and invisibility mainly based on frequency domain representations. However, frequency domain watermarking for a coding DNA sequence is significantly constrained because the transformation and inverse transformation must be performed while completely conserving the amino acid sequence. In this paper, we present a coding DNA watermarking method in a lifting-based discrete wavelet transform (DWT) domain that focuses on the feasibility of frequency domain watermarking for DNA sequences. Our method divides a coding DNA sequence into a number of subsequences and allocates all codons in subsequences to a numerical code using the histogram ranks of the amino acids. Our method then calculates a set of DWT coefficients for subsequences of synonymous codons and finds a subsequence among them with DWT coefficients that are optimal for embedding watermark bits. Finally, our method substitutes this sequence for a subsequence of codons. To secure the watermark, our method generates the binary watermark based on nonlinear congruential – pseudorandom number generator (NC-PRNG) and randomly selects the embeddable position in the DWT domain of the subsequence. We experimentally verified that our method ensures not only amino acid conservation and security but also resists a point mutation rate of approximately 18.5% point mutations.

A new hybrid nonlinear congruential number generator based on higher functional power of logistic maps

We propose a nonlinear congruential pseudorandom number generator consisting of summation of higher order composition of random logistic maps under certain congruential mappings. We change both bifurcation parameters of logistic maps in the interval of U = [ 3.5599 , 4 ) and coefficients of the polynomials in each higher order composition of terms up to degree d. This helped us to obtain a perfect random decorrelated generator which is infinite and aperiodic. It is observed from the simulation results that our new PRNG has good uniformity and power spectrum properties with very flat white noise characteristics. The results are interesting, new and may have applications in cryptography and in Monte Carlo simulations.

ON THE DISTRIBUTION OF COUNTER-DEPENDENT NONLINEAR CONGRUENTIAL PSEUDORANDOM NUMBER GENERATORS IN RESIDUE RINGS

Nonlinear congruential pseudorandom number generators can have unexpectedly short periods. Shamir and Tsaban introduced the class of counter-dependent generators which admit much longer periods. In this paper, using a technique developed by Niederreiter and Shparlinski, we present discrepancy bounds for sequences of s-tuples of successive pseudorandom numbers generated by counter-dependent generators modulo a composite M.

Exponential sums of nonlinear congruential pseudorandom number generators with Rédei functions

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. We give new bounds of exponential sums with sequences of iterations of Rédei functions over prime finite fields, which are much stronger than bounds known for general nonlinear congruential pseudorandom number generators.

On the Linear Complexity Profile of Nonlinear Congruential Pseudorandom Number Generators with Dickson Polynomials

Linear complexity and linear complexity profile are important characteristics of a sequence for applications in cryptography and Monte-Carlo methods. The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. Recently, a weak lower bound on the linear complexity profile of a general nonlinear congruential pseudorandom number generator was proven by Gutierrez, Shparlinski and the first author. For most nonlinear generators a much stronger lower bound is expected. Here, we obtain a much stronger lower bound on the linear complexity profile of nonlinear congruential pseudorandom number generators with Dickson polynomials.

On the linear complexity profile of nonlinear congruential pseudorandom number generators with Rédei functions

Linear complexity and linear complexity profile are important characteristics of a sequence for applications in cryptography and quasi-Monte Carlo methods. The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. We prove lower bounds on the linear complexity profile of nonlinear congruential pseudorandom number generators with Rédei functions which are much stronger than bounds known for general nonlinear congruential pseudorandom number generators.

On the linear complexity profile of nonlinear congruential pseudorandom number generators of higher orders

Nonlinear congruential methods are attractive alternatives to the classical linear congruential method for pseudorandom number generation. Generators of higher orders are of interest since they admit longer periods. We obtain lower bounds on the linear complexity profile of nonlinear pseudorandom number generators of higher orders. The results have applications in cryptography and in quasi-Monte Carlo methods.

The combination of LPτ vectors and shift register RNGs as a way to generate independent random sequences for parallel computations

DNA watermarking is a technique for copyright protection and ownership authentication of DNA sequences and ensures the security of private genetic information. This paper addresses issues regarding watermarking DNA coding sequences in the frequency domain that confer mutation resistance, amino acid conservation, and security. Multimedia watermarking is designed for robustness and invisibility mainly based on frequency domain representations. However, frequency domain watermarking for a coding DNA sequence is significantly constrained because the transformation and inverse transformation must be performed while completely conserving the amino acid sequence. In this paper, we present a coding DNA watermarking method in a lifting-based discrete wavelet transform (DWT) domain that focuses on the feasibility of frequency domain watermarking for DNA sequences. Our method divides a coding DNA sequence into a number of subsequences and allocates all codons in subsequences to a numerical code using the histogram ranks of the amino acids. Our method then calculates a set of DWT coefficients for subsequences of synonymous codons and finds a subsequence among them with DWT coefficients that are optimal for embedding watermark bits. Finally, our method substitutes this sequence for a subsequence of codons. To secure the watermark, our method generates the binary watermark based on nonlinear congruential – pseudorandom number generator (NC-PRNG) and randomly selects the embeddable position in the DWT domain of the subsequence. We experimentally verified that our method ensures not only amino acid conservation and security but also resists a point mutation rate of approximately 18.5% point mutations.

On a nonlinear congruential pseudorandom number generator

A nonlinear congruential pseudorandom number generator with modulus M = 2 w 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.

On the statistical independence of nonlinear congruential pseudorandom numbers

Recently, several nonlinear congruential methods for generating uniform pseudorandom numbers have been proposed and analysed. In the present note, further statistical independence properties of a general class of nonlinear congruential pseudorandom number generators are established. The results that are obtained are essentially best possible in an asymptotic sense and show that the generated pseudorandom numbers model truly random numbers very closely in terms of asymptotic discrepancy.

The lattice structure of nonlinear congruential pseudorandom numbers

Several known deficiencies of the classical linear congruential method for generating uniform pseudorandom numbers led to the development of nonlinear congruential pseudorandom number generators. In the present paper a general class of nonlinear congruential methods with prime power modulus is considered. It is proved that these generators show certain undesirable linear structures, too, which stem from the composite nature of the modulus.

Inversive congruential pseudorandom numbers avoid the planes

Nonlinear congruential pseudorandom number generators based on inversions have recently been introduced and analyzed. These generators do not show the lattice structure of the widely used linear congruential method. In the present paper it is proved that the points formed by d consecutive pseudorandom numbers of an inversive congruential generator with prime modulus possess an even stronger property: Any hyperplane in d-space contains at most d of these points, that is to say, the hyperplane spanned by d arbitrary points of an inversive congruential generator contains no further points. This feature makes the inversive congruential method particularly attractive for simulation problems where linear structures within the generated points should be avoided.

Inversive Congruential Pseudorandom Numbers Avoid the Planes

Nonlinear congruential pseudorandom number generators based on inversions have recently been introduced and analyzed. These generators do not show the lattice structure of the widely used linear congruential method. In the present paper it is proved that the points formed by d consecutive pseudorandom numbers of an inversive congruential generator with prime modulus possess an even stronger property: Any hyperplane in d-space contains at most d of these points, that is to say, the hyperplane spanned by d arbitrary points of an inversive congruential generator contains no further points. This feature makes the inversive congruential method particularly attractive for simulation problems where linear structures within the generated points should be avoided.

On the lattice structure of a nonlinear generator with modulus 2 α

Nonlinear congruential pseudorandom number generators based on inversions have been introduced and analysed recently. These generators do not show the simple lattice structure of the widely used linear congruential generators which are too regular for certain simulation purposes. In the present paper a nonlinear congruential generator based on inversions with respect to a power of two modulus is considered. 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. The corresponding lattice bases are explicitly determined and analysed.

A Nonlinear Congruential Pseudorandom Number Generator with Power of Two Modulus

A nonlinear congruential pseudorandom number generator is studied where the modulus is a power of two. Investigation of this generator was suggested by Knuth [7]. A simple necessary and sufficient condition is given for this generator to have the maximal period length.

Statistical independence of nonlinear congruential pseudorandom numbers

Statistical independence properties of recently proposed nonlinear congruential pseudorandom number generators are analyzed by means of the serial test. The results that are established are essentially best possible. The method relies heavily on bounds for exponential sums.

A nonlinear congruential pseudorandom number generator with power of two modulus

A nonlinear congruential pseudorandom number generator is studied where the modulus is a power of two. Investigation of this generator was suggested by Knuth [7]. A simple necessary and sufficient condition is given for this generator to have the maximal period length.