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I can see many xor and rotation operations in cryptography. I just want to make some algorithms fast. So, I'm looking for a way to combine between xor rotation operations. I don't know if it is possible or not.

R3(A) xor R4(B) = A * B (R3,R4 means left rotate shift by 3 bits and 4 bits)

I'm looking for *. Is there any idea? If you have please, let me know. And if it is impossible, please show me why. Thanks.

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It seems to be asked for variations of the $*$ operation (over words of some bit size $n$) defined by $$A*B=(A\lll3)\oplus(B\lll4)$$ where $\lll$ designates left rotation, and $\oplus$ designates eXclusive-OR. We can rewrite this as $$A*B=(A\oplus(B\lll1))\lll3$$ and a number of other ways, using that $$\begin{align*} \forall X, \forall Y, \forall u,&&(X\oplus Y)\lll u &= (X\lll u)\oplus(Y\lll u)\\ \forall X, \forall u, \forall v,&&(X\lll u)\lll v &= X\lll(u+v\bmod n) \end{align*}$$ Is that what the question is about? –  fgrieu May 16 at 13:37
    
Thank you, fgrieu. this is exactly what i was looking for. And I also thought about this. But I actually wanted to ask about opposite directions to rotate such as (A<<<3)⊕(B>>>4). To make it simple. And I have a few questions. If you know, let me know. Is it possible to make them simple? ((A>>>3)+(B<<<4))<<<4 (+:addition mod 2^32) and (A⊕B+C⊕D)⊕E. Thank you so much for your answer anyways –  user140018 May 17 at 8:07
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1 Answer 1

The question asks for simple variations of the $*$ operations defined by formulas similar to $A*B=(A\lll3)\oplus(B\lll4)$ or $A*B=(A\lll3)\oplus(B\ggg4)$, where uppercase letters are words of $n=32$ bits, lowercase letters are integers, $\lll$ designates left rotation, $\ggg$ designates right rotation, $\oplus$ designates eXclusive-OR.

There are a number of properties we can use: $$\begin{align*} \forall X, \forall Y, &&X\oplus Y &=Y\oplus X\\ \forall X, \forall Y, \forall Z,&&(X\oplus Y)\oplus Z&=X\oplus(Y\oplus Z)\\ \forall X, \forall Y, \forall u,&&(X\oplus Y)\lll u &= (X\lll u)\oplus(Y\lll u)\\ \forall X, \forall u, \forall v,&&(X\lll u)\lll v &= X\lll((u+v)\bmod n)\\ \forall X, \forall u, &&X\ggg u &= X\lll((-u)\bmod n) \end{align*}$$ where $a\bmod n$ is defined to be the $r$ such that $0\le r<n$ and $n$ divides $a-r$.

It follows that $$\begin{align*} (A\lll3)\oplus(B\lll4)&=(A\oplus (B\lll1))\lll3\\ &=((B\lll1)\oplus A)\lll3\\ &=((A\ggg1)\oplus B)\lll4 \end{align*}$$ which can be considered a simplification (in particular, absent a smart compiler, some of these alternate forms will likely be faster on 8-bit CPUs, and 32-bit CPUs lacking a barrel shifter)

Similarly we can write $$\begin{align*} (A\lll3)\oplus(B\ggg4)&=((B\ggg7)\oplus A)\lll3\\ &=(A\oplus (B\ggg7))\lll3\\ &=((A\lll7)\oplus B)\ggg4\\ &=(((A\lll8)\ggg1)\oplus B)\ggg4 \end{align*}$$ with the last form geared for 8-bit CPUs, where rotation by multiples of 8 is relatively efficient.


There's no way to simplify $((A\ggg3)\boxplus(B\lll4))\lll4$, or $((A\oplus B)\boxplus(C\oplus D))\oplus E$, where $\boxplus$ is addition modulo $2^n$. Problem is, we have no simple exact properties of $\boxplus$ with respect to $\lll$, $\ggg$, and $\oplus$.

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