The talk will give an introduction to elliptic curve cryptography and explain some recent developments regarding attacks on the discrete logarithm problem on elliptic curves. In practice, elliptic curve cryptography can for example be employed in the TLS/SSL and IPsec protocols. It is well suited for use in constraint devices such as smart cards. Two main tasks of cryptography are encryption and digital signatures. In order to solve these tasks one employs in general suitable one way functions as fundamental building blocks in public key encryption and digital signature schemes. These are functions for which images are very easy but preimages very hard to compute. The security of encryption and digital signatures is then directly linked to the hardness of computing preimages under such one way functions. Strictly speaking, the existence of one way functions is not known. There are some candidates however, and under current knowledge particularly efficient candidates can be obtained from exponentiation in elliptic curves over finite fields. The reverse operation is called discrete logarithm problem. The investigation of the hardness of the discrete logarithm problem is mathematically challenging. The talk will explain the relevant mathematical objects and discuss these and further issues in simple terms. Towards the end of the talk I intend to report on some advanced attacks on the discrete logarithm problem.
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