The generation of random numbers
is too important to be left to chance.

Robert R. Coveyou (1915-1996)

Stochastic Technology and Cryptography.
Random Number Generators (Random Generator)

Random Number Generation is a problem that has more than 50 years of history, which even today remains one of the most urgent and not fully solved problems, despite the efforts of scientists, the international cryptographic community and the creative teams. For obvious reasons, even if not to distinguish between pseudo-random and true-random numbers, the essence of the problem does not change.
During this time, only two methods for generating random numbers, based on a rigorous algebraic basis, have been generally recognized and developed, i.e., a Linear Recurrence Method and a Linear Congruential Method.
Studies show that a linear congruential method and the algebraic basis represented by this method are far from being fully explored and contain many unsolved problems. Indicative of this is the existence of nonlinear extensions of the linear (polynomial, quadratic, cubic and higher-order) Congruential Method in a binary modulus формула, and the possibility of using variable coefficients in it.
One can also single out additionally the algebraic ‘Feedback With Carry Shift Register’ (FCSR) method proposed in 1993, which is built on 2-additive numbers; however, the method unfortunately has many system and technical drawbacks that seriously limit its application.
Meanwhile, Random Number Generators (RNG, PRNG, TRNG), constructed on the basis of the mentioned linear methods, are not cryptographically strong. To overcome this drawback, it is usually needed to use more complex architectures and to introduce into generators the complexity function that is close to block ciphers in terms of cost, which makes these methods unsuitable for practical applications.

Discovery of such algebraic systems as Incomplete Arithmetic, Pre-Arithmetic (Prearithmetic), its varieties and Dichotomic sequences generated on their basis, as well as creation of Stochastic Technologies (Random Method, Stochastic Method) and the Random Method represented by them (Randomization Method, Stochastic Method), which have significantly pronounced nonlinear properties, and if necessary, an insurmountable functional complexity, helped to solve the problems mentioned above.
Random Generators (Randomization Generator, RAGN), constructed on the basis of Stochastic Cryptography (Minimalistic Cryptography, Light-Weight Cryptography), have an enormous potential and overwhelming superiority in all respects over the present well-known analogues, as indicated by the studies and calculations.

Random Generators have a maximum repetition period equal to формула, and within the period they may be of nonrepetitive or equally repetitive character. In particular, the nonrepetitive generators can be used to generate cryptographic strength Passwords, IDs and unique Keys, with a capacity of billions of keys per second, whereas equally repetitive generators – to generate gamma random numbers in Stream Cipher Systems, with the performance of hundreds of gigabits per second. In this case, the expenses necessary for the hardware implementation of such generators are very smal.
One of the remarkable properties of randomization generator is the ability of multiplicative complexation of Linear Feedback Shift Registers (LFSR), with a period equal to формула(формула – 1).


With the creation of effective Entropy Sources (hardware random numbers source), Random Generators undergo transition into high-quality, minimalistic and low-cost True Random Number Generators (TRNG), whose performance is far ahead of all known today analogs, including the promising ones produced by Intel (Security Driver).

Igor Kulakov, Игорь Кулаков