With the rapid development of communication networks, especially the Internet, the phenomenon of information exchange and information processing using the network is becoming more and more common, and the traditional affairs and business operation mode of society are under unprecedented impact. At present, both the national government and enterprises are integrating into this network revolution, evolving from the original traditional business model to the network model. The future e-government, e-commerce and e-commerce will become an irreversible development trend. In the increasing network activities, people pay more and more attention to information security. This is reflected in:
(1) Network Authentication-Confirm the true identity of network customers.
(2) Confidentiality of information and data-personal or system confidential information and data protection.
(3) Information and data integrity-prevent illegal data modification.
(4) Non-repudiation-the non-repudiation of ex post facto behavior (digital signature) in the network environment.
The core technology of information security is cryptography, which can be basically divided into three types: sequential cryptography, symmetric cryptography (also known as block cryptography) and asymmetric cryptography (also known as public key cryptography).
Asymmetric cryptographic algorithm is the core to support and solve the above four key problems. At present, solutions based on PKI system model are becoming more and more popular. In the PKI system model, the client needs a better personal information security carrier, and smart card or smart password key will be an ideal way, both of which must support public key algorithm, and ECC is the most suitable client product with limited resources.
2 elliptic curve cryptosystem ECC
Since the advent of public key cryptography, scholars have proposed many public key encryption methods, whose security is based on complex mathematical problems. According to the classification based on mathematical problems, there are currently three types of systems that are considered to be safe and effective:
(1) large integer factorization system (represented by RSA),
(2) Finite field (an algebraic structure in mathematics) discrete logarithm system (represented by DSA),
(3) Elliptic Curve Discrete Logarithmic System (ECC).
At present, RSA, the most famous and widely used public key system, was put forward by Rivet, Shamir and Adelman (RSA system for short). Its security is based on the difficulty of prime number decomposition of large integers, and the decomposition of large integers is a famous mathematical problem, and there is no effective method to solve it, so the security of RSA algorithm can be guaranteed. RSA system is the most typical method in public key system. Most products and standards that use public key cryptography for encryption and digital signature use RSA algorithm. The advantages of RSA method mainly lie in its simple principle and convenient use. However, with the progress and perfection of large integer decomposition method, the improvement of computer speed and the development of computer network (thousands of machines can be used to decompose large integers at the same time), the requirements of large integers as RSA encryption and decryption security are getting higher and higher. In order to ensure the security of RSA, the number of its keys has been increasing. For example, it is generally believed that RSA needs a word length greater than 1024 bits to ensure security.
However, the increase of key length greatly reduces the speed of encryption and decryption, and the hardware implementation becomes more and more unbearable, which brings a heavy burden to applications using RSA, especially e-commerce that conducts a large number of secure transactions, thus limiting its application scope more and more. DSA(Data Signature Algorithm) is a digital signature standard based on discrete logarithm problem in finite field, which only provides digital signature, but does not provide data encryption function. ECC(Elliptic Curve Cryptography) is a public key system encryption algorithm with higher security and better algorithm performance, but it is difficult to calculate the discrete logarithm of elliptic curves over finite fields. Human beings have studied elliptic curves for more than one hundred years, but the application of elliptic curves to cryptography was put forward by Koblitz (University of Washington, USA) and Miller(IBM) in 1985. The point (x, y) defined on the elliptic curve (y2=x3+ax+b) of a finite field (Fp or F(2m)), plus the point o at infinity, will form a group (an algebraic structure in mathematics) if it is operated according to certain rules (estimated as multiplication). Elliptic curve multiplication groups over finite fields also have corresponding difficulties in calculating discrete logarithms. Therefore, many public cryptosystems are developed based on this problem, such as ECES and ECDSA, similar to elgamal and DSA.
Advantages of elliptic curve encryption algorithm ECC
Compared with RSA method, elliptic curve encryption algorithm ECC has many technical advantages:
● Higher safety performance.
The security performance of encryption algorithm is generally reflected by the anti-attack strength of the algorithm. Compared with other public key systems, elliptic curve cryptosystem has absolute advantages in resisting attacks. At present, the computational complexity of elliptic curve discrete logarithm (ECDLP) is completely exponential, while RSA is sub-exponential. This shows that ECC has higher security performance per bit than RSA.
Low computational complexity and fast processing speed.
Under certain conditions of the same computing resources, although choosing a smaller public key (as small as 3) in RSA can improve the speed of public key processing, that is, the speed of encryption and signature verification can be compared with ECC, but the speed of private key processing (decryption and signature) in ECC is much faster than RSA and DSA. So the overall speed of ECC is much faster than RSA and DSA. At the same time, the key generation speed of ECC system is faster than RSA 100 times. Therefore, under the same conditions, ECC has higher encryption performance.
● Small storage space occupation
The key size and system parameters of ECC are much smaller than RSA and DSA. 160 bit ECC has the same security strength as 1024 bit RSA and DSA. While 2 10 bit ECC has the same security strength as 2048 bit RSA and DSA. This means it takes up less storage space. This is of great significance to the application of encryption algorithms in resource-constrained environments (such as smart cards).
Low bandwidth requirement
When encrypting and decrypting long messages, the three types of cryptosystems have the same bandwidth requirements, but when applied to short messages, the ECC bandwidth requirements are much lower. However, public key encryption system is mainly used for short messages, such as digital signature and session key transmission in symmetric systems. The low bandwidth requirement makes ECC have a broad application prospect in the field of wireless networks.
4 elliptic curve encryption algorithm ECC related standards
These characteristics of ECC make it replace RSA in some fields (such as PDA, mobile phone and smart card) and become a general public key encryption algorithm. Many international standardization organizations (government, industry, finance, commerce, etc. ) Various elliptic curve cryptosystems have been published all over the world as their standardized documents. ECC standards can be roughly divided into two forms: one is a technical standard, which describes the ECC system mainly supported by technology, including IEEE/KOOC-0/363, ANSI X9.62, ANSI X9.63, SEC/KOOC-0/,SEC2, FIP/KOOC-0/86-2 and ISO/IEC/KOOC-0. The selection of ECC parameters is standardized, and a set of ECC parameters with different security strength is given. The other is the application standard, that is, it is suggested to use ECC technology in a specific application environment, mainly including ISO/IEC 15946, IETF PKIX, IETF TLS, WAP WTLS, etc. At the same time of standardization, some software and hardware of elliptic curve encryption, signature and key exchange based on standards (or drafts) have come out one after another. RSA Data Security Company of the United States published a cryptographic engine toolkit containing ECC at 1997. The security company headed by Canada Certicom also jointly developed and produced cryptographic products with elliptic curve cryptography algorithm as the core, and also proposed a reward challenge against elliptic curve discrete logarithm attacks under various security conditions. It is believed that ECC technology will be more and more widely used in the field of information security.