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$\mathbf{Z}_2^b$ is the direct product of $b$ copies of $\mathbf{Z}_2$ ($\mathbf{Z}_2 \times\cdots \times \mathbf{Z}_2$, $b$ times). That is, its elements are $b$-tuples of elements of $\mathbf{Z}_2$, with both addition and multiplication defined componentwise. If $b > 1$, it is not a field. In fact it is not even an integral domain, because $(0,1)\times ...


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The definitions are not all universal and there is quite some overlap. The clearest definition is for PRG, which is the (synchronous) stream cipher model. A PRG maps a secret value to a long, random-looking keystream, so that an attacker cannot predict any part of the keystream from knowing other parts of it. For a precise definition, you can see the ...



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