The original McEliece cryptosystem uses length-$n$ codes over $\rm{F}_2$ with dimension $\geq n-mt$ efficiently correcting t errors where $2^m \geq n$. This paper presents a generalized cryptosystem that uses length-$n$ codes over small finite fields $\rm{F}_q$ with dimension $\geq n-m(q-1)t$ efficiently correcting $\lfloor qt/2 \rfloor$ errors where $q^m \geq n$. Previously proposed cryptosystems with the same length and dimension corrected only $\lfloor (q-1)t/2 \rfloor$ errors for $q \geq 3$. This paper also presents list-decoding algorithms that efficiently correct even more errors for the same codes over $\rm{F}_q$. Finally, this paper shows that the increase from $\lfloor (q-1)t/2 \rfloor$ errors to more than $\lfloor qt/2 \rfloor$ errors allows considerably smaller keys to achieve the same security level against all known attacks.
Keywords: McEliece cryptosystem, Niederreiter cryptosystem, Goppa
codes, wild Goppa codes, list decoding.

Original language | English |
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Publisher | IACR |
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Number of pages | 17 |
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Publication status | Published - 2010 |
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Name | Cryptology ePrint Archive |
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Volume | 2010/410 |
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