Notice
2.6. Reducing the Key Size
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Descriptif
In the next three sessions,I will explain how to reduce the key size ofcode-based cryptosystem. Circulant matrices are thecentral point in many attempts to reduce the key size ofcode-based cryptosystems since they provideefficient representation. A circulant matrix isa square matrix, its rows are obtained bycyclically shifting the first row. An alternative representation of ann-tuple of elements is using polynomial. Thus, this matrix can bedescribed by a polynomial. And the i-th row of acirculant matrix can be expressed by this formula. Circulantmatrices are closed under product and sum. Thus, this operationpreserves cyclicity. So, we have thefollowing proposition. Circulant matrices of sizer, with elements in Fq are equivalent topolynomials in this quotient ring.Block-Circulant matricesare formed by concatenating circulant blocks of identical size. Quasi-cyclic codes havebeen defined as a linear code that admits a block-circulant matrix. We will considerquasi-cyclic codes that can be written in block-circulant systematic form. Quasi-cyclic subcodes of BCH codeswere proposed by Gaborit in 2005. Take notice that these codescan be efficiently decoded. Thus, there are suitablefamilies for code-based cryptosystem. This table presents theparameters suggested by the author. Note that the key size forthe same level of security drops considerably compared tothe original scheme with Goppa code. The minimum size forGoppa code is around 700 000 bits for 80 bits ofsecurity while with this security level with quasi-cyclicsubcode of BCH code, we just need 12 000 bits.
Intervention
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