The MAC scheme should not be using any primitives like PRG, PRF or PRP. Ok what could be possible solution if we ignore the statement "
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3$\begingroup$ Perhaps a Carter-Wegman Message Authentication Code using arithmetic in a binary field? But does this match the vague "not use any arithmetic operations" ? $\endgroup$– fgrieu ♦Oct 21 at 8:49
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3$\begingroup$ That "not use any arithmetic operations" is either really hard or meaningless. Everything computers do other than a load or a store is an arithmetic operation ("logical" operations are just arithmetic operations in the Boolean algebra) or nothing is (arithmetic operations are just applications of Boolean circuits). You could write a Carter-Wegman MAC algorithm with nothing but the multi-input boolean functions available in FPGA LUTs, after all, and it could be made quite efficient. But I doubt that's what the assignment wants. $\endgroup$– SAI PeregrinusOct 21 at 15:37
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$\begingroup$ Although the question is interesting I've closed it until the last sentence is cleared up, as I don't think it is possible to answer this without clarification. Please hit edit and clarify! $\endgroup$– Maarten Bodewes ♦Oct 21 at 18:14
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1$\begingroup$ We have to guess what the modified question wants to be ignored. Another reason to keep the question closed is that it is a straight dump of homework, without showing what was attempted, or even retyping it LaTex. The simplest that could be an answer goes: work in a binary field $(F_{2^n},+,\cdot)$, and use a "one-time Message Authentication Code" $m\mapsto a\cdot m+b$ with key $(a,b)$ in $\left(\{0,1\}^n\right)^2$. If the large key and MAC are an issue (e.g. contradict definitions given that require $n$ to grow faster than the MAC size), move to a "Carter-Wegman Message Authentication Code". $\endgroup$– fgrieu ♦Oct 22 at 9:11
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