Study On Elliptic Curve Cryptography Fast Algorithm In Finite Field

With the rapid development of information technology, information security has become an important issue. The cryptographic technique is the effective way to realize information hiding, integrity verification and identity authentication, and thus becomes one of the key technologies to ensure the information system security. Elliptic curve cryptosystem has the same functionality as RSA cryptosystem, and at the same time, elliptic curve cryptography has the prominent advantages of faster computations, smaller keys and certificates, etc, which makes it especially suitable for environments where processing power, storage, bandwidth, or power consumption are constrained. After more than20years of research, elliptic curve cryptography has gradually been moving from academic theory study phase to the application stage and now it attracts wide attention as the most promising public key cryptosystem.In recent years, how to realize efficient elliptic curve cryptography is a research focus of information security field. On the basis of the rapid implementation of elliptic curve cryptography, this paper delves into the finite field operations, scalar representations and rapid scalar multiplication algorithms.The elements multiplication algorithm of GF(2n) improved by sliding window technology greatly reduces the number of exclusive-or operation, compared with the comb multiplication algorithm based on window technology. The theoretical analysis and experimental results show that the efficiency increases obviously.Considering the problem that the comb multiplication algorithms involve more shift operation and exclusive-or operation, we improve precomputation and elements multipli-cation algorithm of GF(2n), completely avoiding shift operation, and greatly reduces the number of exclusive-or operation, effectively improve the computational efficiency and reduce the space consumption.This thesis analyzes some typical representations of scalar and improves the counting algorithms of binary signed digit representation and optimal binary signed digit representa-tion of scalar, and designs a scalar representation algorithm with given scalar and hamming weight.The thesis researches the influence of the width of the window on the efficiency of scalar multiplication based on window technology and obtains the corresponding best window width values.This thesis puts forward two scalar representation methods. Based on direct computing technology and improving the scalar multiplication algorithm, we obtain a fast scalar multiplication algorithm without precomputation and a fast scalar Multiplication algorithm with low memory requirement. The analysis results indicate that the efficiency of the algorithms is improved obviously and can enhance the ECC’s efficiency.This thesis discusses the timing attacks, simple power analysis attacks, differential power analysis attacks, and the corresponding preventive strategy. And the thesis researches the influence of the width of the window on the efficiency of guarding against side-channel attacks scalar multiplication and obtains the best window width value.