Research On Public Key Cryptography Based On Identity

In the traditional public key cryptography, the issuation, inquirement, transmission, validation and storage of public key certificate, all need to spend a lot of time and cost. In order to simplify certificate management and improve the efficiency of public key cryptography, in1985, Shamir put forward the concept of Identity-Based Cryptosystem(IBC inshort). In IBC, the user’s identity is bound together to public key in the most natural way. The user’s identity as a user’s public Key, while the user’s private key is generated by the Private Key Generator (PKG in short). IBC has so many advantages compared with the traditional public key cryptography. First, it makes any two users can communicate directly; Second, it does not need to exchange public key certificate; Third, it doesn’t have to keep public key certificate list, last but not least, it also don’t have to use online third party. Therefore, IBC simplifies the management of public key certificate process and reduces the computing and storage costs. Because of this, IBC can be used as a good alternative of the traditional public key cryptography, especially in the storage and computing limited circumstances.In this paper, we introduce the IBC related theory systematically and completely from three aspects, such as IBC mathematics foundation safety, IBC mechanism as well as the IBC mechanism safty. And then we answer the four questions for urgent attention:1. Why do we have to study IBC? Compared with the traditional public key cryptography, what advantages IBC has?2. What other problems are worth studying about IBC?3. Why do we believe that IBC is safe?4. What are the key technologies involved in IBC?In this paper, the studies about IBC can be divided into three parts:In the first part, we introduce the theoretical knowledge related with elliptic curve and bilinear pairings, maily from IBC mathematics foundation safety. Especially, we focus on the arithmetic properties of elliptic curve and Dirichlet characteristics sums. The main research results are as follows:1. We study IBC mathematics foundation safety systematically. From elliptic curve theory to begin, first, we briefly introduce the arithmetic theory of elliptic curve, and then, we introduce the concept of Weil pairings through elliptic curve divisor, maily discuss about the properties of Weil and divisor and give some proofs. Then, we briefly introduced how to transform Weil parings into effective bilinear paings through the deformation mapping. Because the bilinear-based mathematical problems and assumptions are the foundation of constructing the secure IBC, and the level of safety directly determines the security strength of a scheme, So several popular mathematics problems based on bilinear pairings are analyzed and compared. The research results indicate that IBC mathematics foundation safety can be summed up to the difficult solution of elliptic curve discrete logarithm problem (ECDLP in short). For all kinds of existing ECDLP solutions, how to build a safe and effective elliptic curve are discussed at finally.2. We analyze the relation between the point computing problem of elliptic curve and Dirichlet characteristic. And we focus on the distribution about the fourth mixed mean of Dirichlet characteristics and exponential sums. In additional, we obtain some pretty accurate formulas. Because the core of elliptic curve security selection problem is looking for the suitable elliptic curve which has a large prime order or the elliptic curve order contains a large prime factor. That can be summed up to the computation of the elliptic curve order, namely the point computing problem of elliptic curve. Because Dirichlet characteristic is closely related to the point computing problem of elliptic curve, we try to solve the point computing problem of elliptic curve by studying the arithmetic propertie of Dirichlet characteristic. For different positive integer n, Dirichlet characteristic’s distribution is irregular. However, its mixed mean value’s distribution is relatively stable after weighted with some arithmetic function. So we tried to study the distribution of its mixed mean and then to reveal the mystery of the mean distribution of Dirichlet characteristic sums.In the second part, we research IBC key escrow problem, mainly from IBC mechanism and its safety. The so-called IBC key escrow problem is refers to that as follows:the system must unconditional trust PKG, which knows the private keys of all users. Thus, once the PKG is not honest, it can achieve to forge valide signature of a legitimate user. So, how to solve the IBC key escrow problems becomes a problem to be solved urgently. First, the existing key escrow solutions are analyzed and compared. And then, according to the existing key escrow problems of Cha-Cheon scheme, we put forward a new identity-based certificateless signature scheme. The user’s private is produced by user and PKG, so the PKG can’t forge valid signature. Even though it’s able to forge, that will also be found. Thus, that could solve the IBC key escrow problem. In the random oracle model, the new scheme has been proved secure against existential forgery on adaptively chosen message and identity under the CDHP assumption. The new scheme not only solves the key escrow problem, but also its efficiency is higer than the others.In the third part, we study the "untraceability’’analysis method about ID-based blind signature, mainly from the IBC security. As we know, the analysis method is being in a state of debate. In order to solving the problem, we proposed a new "untraceability" analysis method, which can analyze the "untraceability" of all the signatures related with blind signature. It is worth mentioning that, in order to keep consistent with the topic of this thesis, we take Zhang, Hu, Cai and Nong et al.’s blind signature schemes for example. By reductio, this paper shows that the analysis method proposed by Zhang Xuejun is wrong. On that basis, this paper pinpoints a new analysis method of "Untraceability". In order to explain our proposed method is correct, we use the method to analyze the Nong, Cai and Hu et al.’s schemes. The analysis results show that Nong and Cai et al.’s schemes indeed have the "untraceability" but Hu et al.’s not. In additional, the method also successfully analyzes Carmenisch’s scheme.