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S-DES is a toy cipher intended for learning, defined by Edward F. Schaefer, A Simplified Data Encryption Standard Algorithm, in Cryptologia Volume 20, Issue 1, 1996 (paywalled with free preview). It has the same structure as DES, and uses the same notations: bits vectors (such as key and data) are numbered starting from 1 "on the left" functions of bit ...

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P10 and P8 are standard transposition keys that just need to be learnt. They are part of the S-DES standard.

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The function $L_a(x)=\langle a,x\rangle=a_1 x_1+\cdots+a_n x_n$ is a linear multivariate function of $(x_1,\ldots,x_n)$. The function $$f(x)+\langle a,x\rangle=f(x_1,\ldots,x_n)+a_1 x_1+\cdots+a_n x_n$$ equals $0$ mod 2 if $f(x)=\langle a,x\rangle$ and $1$ mod 2 otherwise. The sum  \hat{f}(a)=\sum_{(x_1,\ldots,x_n) \in \mathbb{F}_2^n} (-1)^{f(x)+\langle ...

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