Since from their inception, all most all the attacks on McEliece cryptosystem and its variant Niederreiter cryptosystem have been unsuccessful. They are believed to resist even quantum attacks. These two are code-based Public-key cryptosystem whose private key consists of a generator matrix of a Goppa code. All other modified versions of these cryptosystems which were proposed by replacing Goppa codes with other codes have been proven to be insecure. In view of this, what is the structural advantage that Goppa codes enjoy over other families of codes that makes them so strong for cryptographic applications?
What I heard is that they look random but enjoy a fast deterministic algorithm as opposed to a real random code...So they are pseudo random in some strong sense
umpredictability as opposed to statistical regularity..
Goppa code is an algebraic geometric code (AG-code). It is a general type of linear code constructed by using an algebraic curve X over a finite field Fq. Such codes were introduced by Valerii Denisovich Goppa. They should not be confused with Binary Goppa codes.
Binary Goppa code is an error-correcting code that belongs to the class of general Goppa codes originally described by Valerii Denisovich Goppa, but the binary structure gives it several mathematical advantages over non-binary variants. These provide a better fit for common usage in computers and telecommunication.
In cryptography, for a given security level, Goppa codes with smaller keys can be used.
@AR Reddy: the Goppa codes used in crypto are introduced by elementary means ie
explicit generator matrices. While they can be described by means of AG codes it is hardly relevant there. Historically they were introduced **before** general AG codes on curves. Also I dont understand your comment on the size of the key which depends only on the parameters n,k,q and not on the choice of the family of codes...
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