Cryptography based on error correcting codes
Abstract
The idea to use error-correcting codes in order to construct public key cryptosystems was published in 1978 by McEliece [ME1978]. In his original construction, McEliece used Goppa codes, but various later publications suggested the use of different families of error-correcting codes. The choice of the code has a crucial impact on the security of this type of cryptosystem. Some codes have a structure that can be recovered in polynomial time, thus breaking the cryptosystem completely, while other codes have resisted attempts to cryptanalyze them for a very long time now. In this thesis, we examine different derivatives of the McEliece cryptosystem and study their structural weaknesses. The main results are the following: In chapter 3 we devise an effective structural attack against the McEliece cryptosystem based on algebraic geometry codes defined over elliptic curves. This attack is inspired by an algorithm due to Sidelnikov and Shestakov [SS1992] which solves the corresponding problem for Reed-Solomon codes. The presented algorithm is heuristic polynomial time and thus inverts trapdoors even for very large codes. In chapter 4, we show that the Sidelnikov cryptosystem [S1994], which is based on binary Reed-Muller codes, is insecure. The basic idea of our attack is to use the fact that minimum weight words in a Reed-Muller code have very particular properties. This attack relies on the ability to find minimum weight words in the code, a problem that is, in this specific…
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