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I'm trying to reverse engineer key exchange protocol and faced with the following problem: Protocol is based on Diffie-Hellman key exchange method:
g - generator
m - modulus
Alice:
a - private key
$A \equiv g^a\pmod{m}$ - public key
Bob:
b - private key
$B \equiv g^b\pmod{m}$ - public key

During exchange Bob generates password check signature:
$C = f(A,B,password)$
$D = f(m,g,password)$
where $f$ - cryptographic hash function $$X \equiv \left(\left(D+A\right)\pmod{m}\right)^\left(C+b\right)\pmod{m} $$ and then sends $ f(X)$ to Alice
Alice calculates X on her side without b and I can't figure out how.

Could somebody help me with the algorithm on Alice's side?

share|improve this question
    
Is my interpretation of "You don't know what $\hspace{.02 in}f$ is and hope that something about $\hspace{.02 in}f$ can be $\hspace{.9 in}$ worked out from the fact that Alice can calculate X" correct? $\;$ –  Ricky Demer Jun 10 '13 at 8:24
    
No, I know that $f$ is a cryptographic hash function e.g. sha1, and I'm sure that Alice can calculate X, but she doesn't know b and Bob doesn't send value of X he sends $f(X)$. I think $X = f_1(K,password)$ where K is the shared secret between Alice and Bob. They both know K and password, so there must be an expression equivalent to X on the Alice's side. –  dr.g100k Jun 10 '13 at 12:36
    
it maybe that: $X \equiv (DA)^{C+b} \pmod{m}$, then it can be rewritten $X \equiv g^{(d+a)(C+b)} \pmod{m}$, so Alice can get $X$ by this way $X \equiv (g^CB)^{d+a}\pmod{m} $ , you can check that –  Alex Sep 3 '13 at 1:17

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