nontrivial divisor for a large number. In 2009, the number RSA-768 was factorized in 2009 within the RSA-competition that was already closed. Specified 768-bit number (232 decimal digits) is currently the largest number, known from public sources, which was factorized. To factorize it, the researchers from 6 countries united; more than 10 20 operations 6

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A survey of the main properties of three classes of curves in the generalized Edwards form is given: complete, quadratic and twisted Edwards curves. The analysis of the Montgomery algorithm for differential addition of points for the Montgomery curve is carried out. An estimation of the record low cost of computing the scalar product kP of a point P is given, which is equal to 5M+4S+1U on one step of the iterative cycle (M is the cost of finite field multiplication, S is the cost of squaring, U is the cost of field multiplication by a known constant). A detailed derivation of the formulas for addition-subtraction and doubling points for the curve in the generalized Edwards form in projective coordinates of Farashahi-Hosseini is carried out. Moving from three-dimensional projective coordinates (X: Y: Z) to two-dimensional coordinates (W: Z) allows achieving the same minimum computational cost for the Edwards curves as for the Montgomery curve. Aspects of the choice of an Edwards-form curve acceptable for cryptography and its parameters optimization in the problem of differential addition of points are discussed. Twisted Edwards curves with the order of NE=4n (n is prime) at p≡5mod8 are recommended, minimizing the parameters a and d allows achieving the minimum cost estimation 5M+4S for one step of computing the point product. It is shown that the transition from the Weierstrass curves (the form used in modern cryptographic standards) to the Edwards curves makes it possible to obtain a potential gain in the speed of computing the scalar product of the point by a factor of 3.09.

УДК 004 (056.5+421.5) sequence, but also perform such task effectively. Unfortunately, a compromise is achieved quite hard in this case and modern suites also have statistical dependencies, which unreasonably increase such suites operating time. Thus, one of the main question of tests suite construction and the statistical tests using is statistical independence of tests from this suite. To create an effective suite, tests without statistical dependencies should be used and, at the same time, the tests set should remain sufficiently complete. However, modern questions of forming texts suite, its number determination, type I error value, etc. are solved intuitively and empirically. This article provides the existing evaluation methods overview of statistical randomness tests independence verification and proposes a new, mathematically grounded method, which can be applied to arbitrary tests number and arbitrary random sequences number. The proposed method has advantages in speed and implementation. The paper also presents the experimental research results of the new method application to the statistical randomness test suite.