Cryptographic Applications of Generalized Lucas-Sequences

The increasing importance of electronic information and communication systems in various fields of our society has raised a large number of new security problems, questions and needs. Especially distributed computer networks, distributed data bases and new applications in telecommunications have generated the risk of exposing confidential information to unauthorized personnel or the illegal alteration of data. Many of the identified problems can be solved by the application of cryptologic procedures and mechanisms. Nearly all these systems require the provision of large random prime numbers with special additional properties as parameters. Most of current solutions to generate large primes are based on generalizations of Fermat`s Little Theorem. Unfortunately, many of those probabilistic primality testing algorithms involve the danger not to identify so-called pseudoprimes. These are composite numbers which are not disclosed by certain tests. But if composite numbers instead of primes, too small primes or primes which do not satisfy the known security requirements are used as parameters in a cryptographic system the procedure might be broken easily by an attacker or even not work properly. The principal tasks of the proposed project are: Determination of number-theoretic, algebraic and arithmetic properties of Lucas-related sequences and their associated polynomials, analysis and further development of existing primality tests used in public-key cryptography, classification and investigation of properties of different types of pseudoprimes, cryptanalysis und further development of cryptographic algorithms based on generalized Lucas- sequences, studies of the structure, properties and magnitude of parameters for the generation of secure Lucas- based cryptographic procedures.