Elliptic Curve Cryptography Scalar Multiplication Study

With the high speed development of economy, science and technology, especially along with the computer network and communication technology development, the electronic commerce, electronic government affairs, internet banking, digital city and cloud computing have been developing full of swing, and the informationization wave is affecting every aspect of our lives. At the same time, viruses, hackers, electronic eavesdropping and electronic fraud issues are increasingly serious. How to make people get information effectively and conveniently and how to effectively guarantee people's information safe has become the focus of attention.Cryptography is a study to research how to realize the security and confidentiality of the information of the discipline, which provide an important support for people to protect the data and resources not leak out, and to ensure the authenticity of the data and information.In1985, elliptic curve cryptography system (ECC) was put forward by Koblitz and Miller. After that, elliptic curve cryptography has now become a public key cryptography of a research focus, because of its unique features, it has a wide range of applications in the field on information encryption and digital signature, news authentication and so on. Use of elliptic curve cryptography for encryption and decryption, digital signatures and verification to digital signature, require kP or kP+1Q operation, this operation is called the elliptic curve scalar multiplication, which is the most time consuming operation in the process of implementing, its operation speed decided efficiency of the elliptic curve cryptography system itself.The inversion operation in scalar multiplication is the most time-consuming operation. In this paper, On the basis of the existing calculation of the3P and5P algorithm,a fast direct computing3kP and5kP algorithm was deduced, which only need one inversion.Through comparison and analysis on some of the elliptic curve of existing scalar multiplication, this paper puts forward two methods of scalar k to be coding, and on the basis of the this method it is proposed an improved algorithm of scalar multiplication.(1) Use of symmetrical ternary to code the scalar k, then use this kind of coding method to put forward a new algorithm of elliptic curve scalar multiplication, by comparing this algorithm and the existing algorithm, efficiency was obviously ascension. Using pretreatment, symmetrical ternary algorithm is expected to be18%of ascension than binary algorithm efficiency. Relative to the binary algorithm, symmetrical ternary algorithm is expected to be the less number of points due to short string, so in storage limited Spaces, bandwidth limited application field, it has the corresponding application value and advantage.(2) Through using the ternary symmetrical coding method in scalar multiplication, the paper gave the scalar algorithm and the analysis on the time complexity. By comparing with other scalar algorithm, the operation efficiency also had some improvement.(3) Based on the characteristics of elliptic curve on a scalar halving operation to save more time than double operation, the application of halving algorithm in double-base chain,put forward a new double-base chain algorithm based on halving algorithm. It can improve the scalar multiplication efficiency by using this kind of coding method. Through comparative analysis of performance with Dimitiov algorithm and Wong algorithm, we can choose different domain in long data, relative to Dimitiov algorithm and Wong algorithm, average efficiency of scalar multiplication algorithm re-encoded, respectively84.03%and15.4%improvement, the improved algorithm operation has the obvious efficiency of ascension, the improvement effect is obvious.