Algorithm for creating a digital signature with error detection and correction
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ALGORITHM FOR CREATING A DIGITAL SIGNATURE WITH ERROR DETECTION AND CORRECTION R. G. Biyasheva† and S. E. Nyssanbayeva a‡
UDC 004.056.5
Abstract. This paper proposes an algorithm that creates a digital signature in a nonpositional polynomial notation and forms three surplus residues modulo one extension polynomial base number. Such a signature allows for single-error detection and correction (which is shown to be a unique procedure) and multiple error detection. A formula of cryptographic security is obtained for such a digital signature. Keywords: digital signature, nonpositional polynomial notation, complete residue system, cryptostrength.
Digital (or electronic) signature systems (DS systems or DSSs) include the following two algorithms: the formation of a digital signature and its verification. In developing DS schemes, fundamentally different approaches are used that can be divided into three groups. These schemes are based on · public key cryptographic systems; · symmetric enciphering systems; · specially developed signature computation and verification algorithms. The digital signature scheme proposed below belongs to the latter group. Well-known DSSs are based on the algorithms RSA, ElGamal, and also DSA proposed in 1991 as the Digital Signature Standard (DSS) in the USA. The Russian standard GOST R 34.10–94 has taken effect in 1995. These algorithms and standards are developed on the basis of public-key cryptosystems. In the Republic of Kazakhstan, the state standard ST RK 1073-2007 specifies four security levels of DSSs [1]. According to its requirements on means of cryptographic protection of information of the first, second, third, and fourth levels, a DS key length must be no less than 60, 100, 150, and 200 bits, respectively. The proposed algorithm is developed using nonpositional polynomial notations (NPNs) whose synonyms are modular arithmetic and residue notations. In NPNs, base numbers are not prime numbers [2] but irreducible polynomials over the field GF(2) [3]. Algorithms and methods created on the basis of these systems are called nonconventional. The use of NPNs in constructing nonconventional algorithms of cryptographic protection of stored and transmitted information makes it possible to considerably increase their efficiency and cryptostrength [4–7]. Based on NPNs, two electronic signature systems are created. The first DS scheme is presented in [4, 6]. The results of constructing the second DS system are presented below. Let us consider a procedure for constructing an NPN. Assume that irreducible polynomials p1 ( x ), p 2 ( x ), K , p S ( x ) of degrees m1 , m2 , K , mS , respectively, over the field GF ( 2) are selected as NPN base numbers. With allowance for their arrangement, these polynomials form systems of base numbers and are called working (informational polynomials or symbols). According to the Chinese remainder theorem, all the base numbers must be different including the case when their S
degrees are equal. The main working range in NPNs is a polynomial PS ( x ) = p1 ( x )
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