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4

Let $\bf A$ be an $n \times n$ binary matrix. Let we want to check that whether $\bf A$ is an MDS matrix over the finite field $\mathbb{F}_{2^k}$ for some $k$? The necessary condition is that $k\mid n$ which means $n=km$ for some integer $m$. Now Let $\bf A$ be $km \times km$ binary matrix. The first step is that to consider the matrix $\bf A$ as a block ...

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The first step in verifying a signature is to hash the message, which takes a few thousand x86 instructions. Your language doesn't seem to support the logical operations that are common in modern hash functions - implementing those with arithmetic operations will cost you some overhead. So you're talking about hours before you even get to the public key ...

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You have a typo, $x$ is also in $\mathbb{F}_2^4,$ since the Present Sbox is $4\times 4$ bits. The quantities $S(x),a,b,x$ all lie in $\mathbb{F}_2^4,$ and the inner products such as $\langle b,S(x)\rangle$ etc lie in $\mathbb{F}_2.$ We now treat the inner product values as in $\mathbb{N},$ so we can calculate \$(-1)^{\langle b,S(x)\rangle+\langle a,x\rangle}....

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