On the Design of Cryptographic Primitives

Several approaches to the design of cryptographic protocols have been carried out from different angles. Some of them have had the aim of developing a set of standards that can be applied to cryptographic protocols in general, whereas others have proposed new specific protocols. The simplest approach to analyze cryptographic protocols consists in considering them in an abstract environment where absolute physical and cryptographic security is assumed. The main disadvantage of this formal approach is that it does not address potential flaws in actual implementations of concrete algorithms. On the other hand, the traditional approach has consisted in guaranteeing the security of specific protocols based on Finite Mathematics such as the Quadratic Residuosity Problem and the Discrete Logarithm Problem. In general such an approach does not allow the composition of protocols in order to design more complex protocols because it requires re-modelling the entire system and re-proving its security. In this paper we propose a mixed approach where security conditions are guaranteed for certain types of actual protocols. These algorithms may be used as modules in order to build complex protocols while maintaining security conditions. This work is organized as follows. Firstly, basic concepts and necessary tools are introduced in section 2. Afterwards, specific notation used throughout the work and general properties of two-party cryptographic protocols are described in section 3. In section 4 special attention is paid to the general-purpose protocol of Oblivious Transfer and its different versions and applications. Section 5 is devoted to the other primitive of Bit Commitment and its main application, the so-called Zero-Knowledge Proof. Finally, several conclusions and possible future works are mentioned in section 6.

This work addresses the topic of secure distributed computing through the proposal of general and specific schemes for some two-party cryptographic protocols. In such a context, two parties who are mutually unreliable have to cooperate in order to reach a common goal in an insecure distributed environment.

The design of cryptographic protocols typically includes two basic phases corresponding to specification and verification. Up to now, most works have concentrated on this latter step while a systematic specification of protocols is almost an undiscovered area yet. The best known formal methods to analyze cryptographic protocols that have been published may be classified into three types. The modal logic based approach is represented by the BAN logic model for analyzing cryptographic protocols first published in [7]. On the other hand, one of the earliest works that used the idea of developing expert systems to generate and investigate various scenarios in protocol design was [9]. A different approach to protocol verification was based on algebraic systems [12]. Regarding research in the emerging area of formal and systematic specification of cryptographic protocols, a modular approach was proposed in [21]. Finally, a methodology for both specification and verification of protocols was presented in [20], and several basic informal design principles were proposed in [1].

Note that the design of protocols is not difficult if a Third Trusted Party (TTP) is available. In such a case, all input information may be given by both parties to it, and then the TTP can distribute corresponding outputs to each party. However, the enormous costs of extra communications, establishment and maintenance of TTP justify the search for secure non-arbitrated protocols. In fact, the importance of cryptographic protocol design lies in the fact that TTP becomes non necessary. Typical solutions to avoid TTP in cryptographic protocol design include the use of two powerful tools: computational complexity assumptions and random choices. In most specific cryptographic protocol designs the computing power of one or both parties is supposed bounded. Also usually, some unproven assumption on the intractability of some finite mathematical problems, and some sort of interaction between both parties are required. Most two-party cryptographic protocols include the use of two general techniques, the so-called Cut-and-Choose and Challenge-Response methods. Cut-and-Choose technique consists in two stages. First, a party cuts a secret piece of information in several parts and

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