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If $A$ is an efficient algorithm that solves the Computational Diffie-Hellman problem for $\frac{1}{2}$ of the inputs and returns a special symbol for the rest, how can I use $A$ to construct another algorithm B which solves $CDH$ with a higher probability ($1 -\frac{1}{2^k}$) ?

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    $\begingroup$ Hint: Can you perhaps transform a given CDH challenge into an independent-looking different one from a solution of which you can recover the original answer? $\endgroup$
    – SEJPM
    Jul 17 at 12:31
  • $\begingroup$ @SEJPM I am not able to understand how. Could you please elaborate a bit more? For any random instance how should B use A as a subroutine ? $\endgroup$
    – user766787
    Jul 17 at 14:55
  • $\begingroup$ I mean, how should B query A to get the correct answer ? $\endgroup$
    – user766787
    Jul 17 at 15:02
  • $\begingroup$ As A is unreliable, B needs to construct fresh, related CDH challenge inputs, can you think of ways to create $g^x,g^y$ from $g^a,g^b$ s.t. $g^{xy}$ helps you recover $g^{ab}$? $\endgroup$
    – SEJPM
    Jul 17 at 22:22
  • $\begingroup$ @SEJPM By taking some k multiple of a,b as x,y ? $\endgroup$
    – user766787
    Jul 18 at 4:40
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Suppose $C_k: (g, g^a, g^b) \mapsto g^{ab}$ is your CDH-solver, that solves it with probability $1 - \frac{1}{2^k}$ and the special symbol otherwise. Let's construct them from $C_1$ recursively.

Construction of $C_{k+1}$:

  1. Calculate $C_k(g, g^a, g^b)$. If it returns $g^{ab}$, output it (probability of this is $1 - \frac{1}{2^k}$). Otherwise, go to the second step.
  2. Generate two independent random numbers $x$ and $y$ uniformly distributed from 1 to $p-1$.
  3. Calculate $g^{ax}$ and $g^{by}$. Note, that they are independent from $g^a$ and $g^b$.
  4. Calculate $C_1(g, g^{ax}, g^{by})$. If it returns the special symbol, output it (probability of it is $\frac{1}{2^{k+1}}$). Otherwise it has calculated $g^{abxy}$. Go to the fifth step.
  5. Use the extended Euclidean algorithm to find $z$, such that $xyz \equiv 1 (\mod p-1)$.
  6. Calculate $g^{abxyz} = g^{ab}$ and output it (probability of it is $\frac{1}{2^{k+1}}$).

It is not hard to see, that the total probability of outputting $g^{ab}$ is $1 - \frac{1}{2^k}$ now.

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