Improvement And Application Of Elliptic Curve Scalar Multiplication Algorithm

With the development of technology and the arrival of the information age, both shopping and science are inseparable from the computer. Now with more and more computers on the network information, the protection of data is not so seriously, so the information security must become an important problem in the whole society. In order to make people use the Internet better, guarantee data security, guarantee network information security, cryptography is developed rapidly.Cryptography ensures that data is safe on the Internet, not compromised,and better secured.For the early cryptography, there were only simple substitutions.With the development of computer and mathematical theory, the system of symmetrical ciphers gradually emerged. The symmetric cryptography to be encrypted and decrypted, so the keys are exchanged in advance, and the key leaks are sometimes occured during the exchange of keys. In order to solve this problem, the asymmetric cipher is produced. The main non-symmetric cryptographic codes now include RSA algorithm encryption, backpacking algorithm encryption, elliptic curve encryption,etc. In the asymmetric encryption algorithm, elliptic curve encryption algorithm (ECC) and other kinds of asymmetric encryption algorithm, has advantages of low storage, high efficiency, high security, and great attention in recent years.Both the decryption and the digital signature require scalar multiplication (kP), while scalar multiplication is the most time-consuming and important part of the ECC calculation. The work of this paper is mainly to analyze scalar multiplication, center point by analyzing the scalar multiplication operations and times point operations,reduces the time complexity to find a new scalar multiplication algorithm,this method is to use the hybrid coordinate under coordinate shift scalar multiplication algorithm. After the validation algorithm is correct, the improved scalar multiplication algorithm is applied in the ECC digital signature.The main work is as follows: first, the innovation of scalar multiplication algorithms. The scalar multiplication algorithm is improved by coordinate displacement of the mixed coordinate system.Through the improvement of scalar multiplication algorithm, the improvement of scalar multiplication rate is realized. Compared with the original scalar multiplication algorithm, the time was reduced by 16.4%.Second, the application of scalar multiplication algorithm in digital signature. The scalar multiplication algorithm applied in this paper is applied to traditional ESCDA,T-ECDSA, and improved ESCDA. The time space complexity of the three algorithms. Finally, the conclusion is that the improved scalar multiplication in this paper has some improvement in time. Third, the programming implementation of ECC encryption algorithm in different coordinate systems. The implementation of the encryption programming is performed on the coordinates of affine coordinates, projection coordinates, and jacobian coordinates. Fourth, the three algorithms of ESCDA, T-ECDSA and improved ESCDA are compared to verify the correctness of the three algorithms. The feasibility of the improved scalar multiplication algorithm in the three signature algorithms is also verified by programming. And the complexity of the time space before and after the application is improved.