The God is sophisticated, but not ill-intentioned.
Albert Einstein (1879 -1955)

Pre-Arithmetic (PreArithmetic). Number Theory

Launched in 1988, studies of Random Number Generators (RNG, PRNG, TRNG) constructed using mixed operations, i.e., usual arithmetic operations and bitwise addition without carry (modulo-2 addition), revealed the properties that do not fit into the known framework of mathematics and number theory. Only by 2002, it had become possible to come close to solving this problem.
In fact, it turned out that the solution lies close at hand. The first results were presented in the Application for International Patent PCT/RU03/00141 dated 7 April 2003 «A Method of the Randomiza- tion Properties Imparting to a Real Object and a Randomization System». Subsequent development of the ideas presented in the patent, which were tested at scientific conferences in Slovakia (2003) and Russia (2004, 2005), led, in August 2005, to the discovery of Pre-Arithmetic and Incomplete Arithmetic, preceding the conventional classical Arithmetic.
The term «Pre-Arithmetic» in a phenomenological algebraic, rather than in the general educational mundane sense was voiced for the first time at the conference “RusCrypto 2006” and at a symposium in China in September 2006.

Discovery of Pre-arithmetic (Prearithmetic) marked the beginning of a new direction in mathematics. Later, during the period from 2007 to 2010, its other varieties were discovered.

«Pre-Arithmetics» generate new algebra and have a fundamental cha- racter, which is reflected in the article of the same title. Pre-Arithmetics along with Incomplete Arithmetic, Arithmetic and Number Theory show that it is now time to breathe new life into the undeservedly forgotten Higher Arithmetic.

In Pre-Arithmetic, the operation of addition and subtraction is chara- cterized by the result of the operation – n-bit base variable G and its non-linear complement P.

In contrast to arithmetic, when a number of elements that differ by unity are formed, there may be the presence of a transient non-linear segment (attractor) of lengthформула, dependent on the initial conditions. In this case, strict behaviour is observed, with the passage through the transient segment, due to the phenomenal self-regulation of the base variable G and its complement P, the base variable G reaches a maximum period of формула, and then everywhere, within the boundaries of every subsequent period, behaves in a stationary and repetition-free manner.
In Incomplete Arithmetic and Pre-Arithmetic, non-linear complement P is an integral part of operations, has no independent function, does not degenerate into a constant and does not vanish, as is the case in Arithmetic.

In general, based on the results of years of research, the introduction of Pre-Arithmetic and its varieties, may enrich the theory of numbers, physics and mathematics, and, using them, achieve a quality leap in the development of the theory of Dynamic Systems, Nonlinear Dynamics (Non-Linear Dynamics), Stochastic Systems and Chaos Theory as well as in the development of methods and technologies for digital processing.
First of all, we deal here with the Generation of Random Numbers (Random Number Generation, PRNG, TRNG), with the construction (based on the Dichotomic Properties inherent in natural phenomena and processes) of Binary Nonlinear Functions, One-Sided Functions, One-Way Functions, Hash Functions, Ciphers, i.e., Stream Ciphers and Block Ciphers.
In other words, we speak here of Cryptography as a whole, which in its essence and in the achievable technical characteristics is Minimalistic and Light-Weight.


The first applied and theoretical results, laying a foundation and framework of Stochastic Technology (Random Method, Stochastic Method), as well as Stochastic Cryptography (Minimalistic Cryptography, Light-Weight Cryptography) following from it, are considered at pages Hypothesis About the Nature of Arithmetic and Random Method (Randomization Method, Stochastic Method).

Igor Kulakov, Игорь Кулаков