Research On Some Problems Of Public-key Cryptography

Public-key cryptography is a method for secret communication between two parties without requiring an initial exchange of secret keys. It can also be used to create digital signatures. Public key cryptography is a fundamental and widely used technology around the world, and enables secure transmission of information on the Internet.The security of RSA depends on the difficulty in factoring a large integer. Given an integer, there are many methods to factor it into its prime factors, such as Pollard's rho-method, elliptic curves method, quadratic sieve and number fields sieve. Each method has its own advantages and disadvantages. If there is t = uv such that ,where , the third part of this paper propose an efficient algorithm to factor N .There are some special requirements in the selection of N = pq for the security of RSA: 1. p ,q is large prime numbers. 2. p and q are almost of the same length. 3. p -1 and q - 1 both have at least one large prime factor. 4. gcd( p - 1, q- 1) is very small. To avoid a factorization by our method, a new suggestion is proposed for the selection of the RSA number according to continue fractions:In the continued fraction expansions of p|q , if there is a convergent qpn n such that then t = pn qn must be large enough.Hyperelliptic curve cryptosystem (HECC) is based on the discrete logarithm problem on the Jacobian of hyperelliptic curves over finite fields. NUCOMP algorithm can compute the reduced sum of two divisors of the Jacobian. In the fourth part and the fifth part, we propose the explicit formulae derived by NUCOMP algorithm for the hyperelliptic curves of genus 2 and 3 respectively. In particular, for the most common case in HECC: 1. deg(u 1 ) = deg(u 2)= g, gcd(u 1 , u 2) = 1, compute [u 1 , v1 ] + [u 2 , v2], 2. gcd(u , h + 2 v) = 1, compute 2[u , v ], we use the following technique to improve the NUCOMP algorithm: 1. avoiding the computation of the polynomial's inverse by resultant, 2. Montgomery's trick of simultaneous inversions, 3. reordering of normalization steps to save some field operations. Authentication code is an essential technology in the modern communication. In the sixth part of this paper, A construction method of Cartesian authentication codes from BN pair decomposition of GLn(Fn) is proposed and the parameters are computed.: Moreover, assuming that the encoding rules are chosen according to a uniform probability distribution, the successful impersonation attack of these codes and the successful substitution attack are also computed:A Coxeter system is a pair (W, S), where W is a Coxeter group and S is a fundamental set of generators for W. In the seventh part, the formulae for the length of the element of maximal length in the Coxeter system of type An, Bn/ Cn, and Dn are proposed as following: 1. the length of the element of maximal length in the Coxeter system of type An is C n2+ 1 . 2. the length of the element of maximal length in the Coxeter system of type Bn/ Cn is n 2. 3. the length of the element of maximal length in the Coxeter system of type Dn is 2C n2.