A pseudo-random number generation device calculates a value st[i] of b[i] bits by using a function F[i] taking a value st[i1] as input for each integer value i with i=1, . . . , n in ascending order. The pseudo-random number generation device calculates a value x[i] of r[i] bits by using a function g[i] taking as input at least a part of bits of a value st[j] and at least a part of bits of the value st[i] for at least a part of an integer value i with i=1, . . . , n, where a value j is an integer value smaller than the integer value i. The pseudo-random number generation device combines the values x[i] calculated by using the function g[i] to obtain a pseudo random number.

Described embodiments relate to systems and method for conditioning, de-biasing and/or whitening raw entropy data or for hashing data. The method comprises receiving data; determining at least a first algebraic number from the data; calculating at least one solution to one or more transcendental equations using the at least the first algebraic number as an input parameter value, wherein the one or more transcendental equations comprise a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs; determining one or more sequences of pseudo random numbers based on the at least one solution; and determining an output based on the one or more sequences of pseudo random numbers. For example, the data may be received from a raw entropy source and comprise raw entropy to be transformed. Alternatively, the data may be data to be hashed and the output may comprise a hash of the data.

A method and apparatus is provided for generating pseudorandom numbers in a way that is deterministic (i.e., repeatable), that passes statistical tests, can have multiple instances of objects generating pseudorandom numbers at the same time. Also, it is desirable that the collection of pseudorandom numbers generated by multiple instances have the same statistical properties as numbers generated by a single instance (i.e., randomness). Embodiments described herein generate pseudorandom values by using a plurality of subsidiary linear congruential generators and combining their outputs nonlinearly. According to embodiments, after their outputs have been combined, a mixing function is applied. Embodiments include an on-demand split method in the style of the SplitMix algorithm.

Various embodiments are provided for fully digital chaotic differential equation-based systems and methods. In one embodiment, among others, a digital circuit includes digital state registers and one or more digital logic modules configured to obtain a first value from two or more of the digital state registers; determine a second value based upon the obtained first values and a chaotic differential equation; and provide the second value to set a state of one of the plurality of digital state registers. In another embodiment, a digital circuit includes digital state registers, digital logic modules configured to obtain outputs from a subset of the digital shift registers and to provide the input based upon a chaotic differential equation for setting a state of at least one of the subset of digital shift registers, and a digital clock configured to provide a clock signal for operating the digital shift registers.

A random number generating device of the present disclosure includes: an arithmetic random number generator that generates an arithmetic random number sequence; an arithmetic random number converter that sequentially reads at least one arithmetic random number from the arithmetic random number sequence and converts a value of the read arithmetic random number into a voltage or current value of at least two predetermined levels of gray scale having an identical polarity; a hysteresis unit that outputs values depending on a presently-input voltage or current value and a previously-input voltage or current value with respect to the sequentially-input voltage or current value; and a threshold processor that binarizes the output of the hysteresis unit.

A new mathematical technique called the T-sequence is a primality testing method. A similar approach can be applied to perform fast factoring for numerous special cases, a method that can, in all likelihood, be extended to the general case, making possible a general and fast factoring algorithm. The same T-sequence can be used to construct a prime number formula and a good random number generator.

A methods comprises: receiving, by a pseudo random number generator module, an instruction to generate pseudo random numbers from a security application; determining, by the pseudo random number generator module, at least one algebraic input parameter value for a transcendental equation from a randomness library in memory of the device, wherein the transcendental equation comprises a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs; calculating, by the pseudo random number generator module, a solution to the transcendental equation based on the at least one algebraic input parameter value; determining, by the pseudo random number generator module, pseudo random number(s) based on the solution; and storing, by the pseudo random number generator module, the pseudo random number(s) in a randomness library for use as seeds for keys by the security application and as subsequent input parameter values for the pseudo random number generator module.