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The Mathematics Department Algebra Seminar features Francisco Javier Lobillo Borrero, Universidad de Granada, Spain, discussing "Skew differential Goppa codes and their application to McEliece cryptosystem" on Tuesday, Sept. 27, from 4-5 p.m. via Zoom. 

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Abstract: Code-based cryptography  proposals  still alive after the Round 4 for the NIST Post-Quantum Cryptography competition.  The strength of these technologies rests upon the hardness of the decoding problem for a general linear code. Of course, an efficient decoding algorithm is required in practice. So, what is already needed is a family of codes with some conveniently masked  properties that allow their efficient decoding. The original McEliece criptosystem took advantage of such  features enjoined by classic Goppa binary codes.

 

One way to introduce Goppa codes is the following. Let \(F \subseteq L\) be a field extension and let \(g \in L[x]\) be a polynomial. A subset of group of units in \(L[x]/\langle g \rangle\) represented by linear polynomials is selected, and their inverses allow to build a parity check matrix of the Goppa code. The arithmetic in \(L[x]\) is a main tool in the design of efficient decoding algorithms for Goppa codes.

 

From an algebraic point of view, our proposal replaces, in the simplest case, the cyclic group of units of \(L[x]/\langle g \rangle\) by a general linear group, whose mathematical structure is more complex. In order to design an efficient decoding algorithm, this non-commutative group is presented as the group of units of Ore polynomials in \(L[x;\sigma,\partial]\) modulo a suitable invariant polynomial \(g\). The arithmetic of this non-commutative polynomial ring is used to design efficient decoding algorithms. Classic Goppa codes are instances of our construction. Therefore, the security of our cryptosystem is expected to be as strong as the original one.

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