Chevardin V. Methods of deterministic random bit generators building based on isomorphic transformations of elliptic curves

The analysis of modern situation of the cryptographic security of information in a world and in the country is received. The impossibility of future developing of cryptographic security of information systems without safe deterministic random bit generators (DRBG) with increasing a security and a rapid is defined. It becomes more actuality in conditions of future increasing power and number of quantum computers. In the work much results of researches in the cryptographic security field were analyzed. Using the results of analysis there was defined the conditions for search of new solutions in cryptography field and methods of building safe and resistance of DRBG based on theoretical problems. The research results showed much more possibilities for improvement of modern DRBG based on elliptic curves with measures: security resistance, prediction resistance, backtracking resistance and computational complexity (performance). However except of known shortcomings of DRBG on elliptic curves in research process the pseudorandom sequences with anomaly small periods were detected. The results were received with famous approach to building DRBG based on elliptic curves arithmetic. The results of the analysis of different approaches to building RBG, the results of estimate cryptographically resistance, statistical properties of sequences and performances of famous DRBGs with considering requires ISO/IEC 18031, ANSI X.9.82, AIS 20 were showed in the work. In the chapter the limitations and advantages of RGB based on different cryptographic primitives: block ciphers, hash functions, theoretical problems are detected. The pseudorandom sequences with anomaly small periods and preperiods from DRBG on elliptic curve are showed. There was received practical estimates of anomaly small numbers before first cycling DRBG on elliptic curves that justify the necessity of their theoretical estimation. The theoretical estimates give more justify values of numbers before first DRBG cycle. The received theoretical estimates of number of generator iterations before first DRBG cycle are smaller than idealized simplified model nearly in √N. There were enhanced DRBGs based on the scalar multiplication of elliptic curve points by using transformation in group of Edwards curve points over Galois field like a generator of DRBG internal states. It allowed to reduce of the DRBG calculate complexity in 2 – 3 times in the pseudorandom sequences generation in comparison with the standard Dual_EC_DRBG as a result the elliptic curve DRBG speed was increased. The number of elliptic curve transforms from canonical form to canonical form under Galois field shows linear dependence of isomorphic transformation highest bound from characteristic field p. For transformation from canonical to elliptic curve normal form that boundary increases proportionally р4. The method of generation of pseudorandom sequences based on double scalar elliptic curve point multiplication over Galois field with characteristic p≠2,3 was developed, which defers from existing by using full elliptic curve isomorphic transformation set. It allowed to increase number of internal states of DRBG proportionally field characteristic р and it increases the period of pseudorandom sequence and increase the DRBG resistance proportionally р2 in comparison with existing standard.