On the Design of Cryptographic Primitives
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On the Design of Cryptographic Primitives Pino Caballero-Gil · Amparo Fúster-Sabater
Published online: 4 August 2006 © Springer Science + Business Media B.V. 2006
Abstract The main objective of this work is twofold. On the one hand, it gives a brief overview of the area of two-party cryptographic protocols. On the other hand, it proposes new schemes and guidelines for improving the practice of robust protocol design. In order to achieve such a double goal, a tour through the descriptions of the two main cryptographic primitives is carried out. Within this survey, some of the most representative algorithms based on the Theory of Finite Fields are provided and new general schemes and specific algorithms based on Graph Theory are proposed. Mathematics Subject Classifications (2000) 94A60 · 11T99 · 14G50 · 11T71. Key words cryptography · secure communications · finite fields · discrete mathematics.
1. Introduction A two-party cryptographic protocol may be defined as the specification of a sequence of computations and communications performed by two entities in order to accomplish some common goal. For instance, several algorithms may be described in the form of two-party protocols, which allow to perform in the telecommunication world some usual actions such as flipping a coin, putting a message in an envelope, signing a contract or sending a certified mail. This work surveys known protocols based on finite fields, and proposes new general and specific solutions based on graphs.
P. Caballero-Gil (B) Departamento de Estadistica, Investigacion Operativa y Computacion, University of La Laguna, 38271 La Laguna, Tenerife, Spain e-mail: [email protected] A. Fúster-Sabater Institute of Applied Physics, C.S.I.C. Serrano 144, 28006 Madrid, Spain e-mail: [email protected]
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Acta Appl Math (2006) 93: 279–297
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 remodelling 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 condition
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