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EDIT: Rephrased.

I have, stored somewhere, the values $a$ , $Q$, $N_1$ (plus its factor) and $a^{2Q} \mod N_1$.

I also know $b$, $R$ and $N_2$ (but not its factors).

I want to know whether there is a calculation someone else can do (without knowing $b$, for example mixing it with $a$) such that the result will let me find $b^{2R} \mod N_2$ more easily than just exponentiating the usual way.

The other could have access to (I'll give to him in plain if necessary) the values $Q$, $N_1$ and $a^{2Q} \mod N_1$, but I must kept secret the value $a$ and $b$. Anyway, I can disclose any result made from an operation on $a$ and $b$ which is infeasible to invert.

Thank you

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I'm trying to make sense of this.... If your friend finds $y\equiv(ab)^{2R}\pmod{N_2}$, how do you get $b^{2R}$ out of $y/a$? don't you have to do $y/a^{2R}$? which means you have to do $a^{2R}$, which you're trying to avoid. Also, if you give him $ab$, and then in exchange for his work you give him $a$, haven't you given him $b$, which you say you don't want to reveal? –  Gerry Myerson May 17 '12 at 13:30
    
So let me ask whether this is your question: you know $a$ (or maybe you just know $a\pmod{N_1}$, and $a^{2Q}\pmod{N_1}$ (but do you know $Q$?), and you know primes $p,q$ with $pq=N_1$, and you know $b,R,N_2$, and you want to know whether there is a calculation someone else can do without knowing $b$ such that the result will let you find $b^{2R}\pmod{N_2}$ more easily than just exponentiating the usual way, and somewhere along the line (though it's not clear where) the other person gets to know $a$. Have I got it right? –  Gerry Myerson May 17 '12 at 13:37
    
I know Q, and your statement is right except last 2 lines: My friend isn't required to know a, I was trying to use it to exploit what I want (but it can be avoided, of course) –  Kronat May 17 '12 at 14:09
    
Then can you edit your question in the light of my comments and your response? It's better for all the information to be in the question than for people to have to go digging through the comments to get it. –  Gerry Myerson May 18 '12 at 0:19
    
Much better after the edit. I hope someone takes an interest in it. –  Gerry Myerson May 18 '12 at 12:43

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