I came across this problem, while currently reading "Introduction to Modern Cryptography" by Katz and Lindell. I am new to crypto and just trying to go through the book and solve the exercises, unfortunately, got stuck at this one.

Consider the following public-key encryption scheme. The public key is $(G, q, g, h)$ and the private key is $x$, generated exactly as in the El-Gamal encryption scheme. In order to encrypt a bit b, the sender does the following:

  • If $b = 0$ then choose a uniform $y ∈ Z_{q}$ and compute $c_{1} := g^{y}$ and $c_{2} := h^{y}$. The ciphertext is $(c_{1} , c_{2}) $.

  • If $b = 1$ then choose independent uniform $y,z ∈ Z_{q}$ , compute $c_1 := g^{y}$ and $c_{2} := g^{z}$ , and set the ciphertext equal to $(c_{1},c_{2})$.

Show that it is possible to decrypt efficiently given knowledge of $x$.

Would appreciate some help

  • $\begingroup$ Hint 1: take $x$-th power of $c_1$. $\endgroup$ – kelalaka Jan 16 '19 at 19:49
  • $\begingroup$ Thanks, that helped for b=0! Still not sure how to approach for the case b=1 though. $\endgroup$ – badoo Jan 16 '19 at 20:14
  • $\begingroup$ So you get equality right? $\endgroup$ – kelalaka Jan 16 '19 at 20:14
  • $\begingroup$ Yes, I do for b=0. $\endgroup$ – badoo Jan 16 '19 at 20:17
  • $\begingroup$ Then if there is no equality it means? $\endgroup$ – kelalaka Jan 16 '19 at 20:17

To determine the bit plaintext from $(c_1,c_2)$ take $x$-th power of the $c_1$. There are two options that we don't know

  1. $(c_1,c_2) = (g^y,h^y)$ then $(c_1^x,c_2) = (g^{xy},g^{xy})$
  2. $(c_1,c_2) = (g^y,g^z)$ then $(c_1^x,c_2) = (g^{xy},g^{z})$

Now check whether $c_1^x = c_2$ or not.

  • If they are equal then than bit $b=0$
  • If they are not equal than bit $b=1$.

There is a small chance that we will have $g^{xy}=g^{z}$ as false $b=0$. The probability of this event is $1/g$.

The cost of operation: if we use Left-to-right binary method for modular exponentiation than the cost is $\mathcal{O}(\log(x))$, therefore, it is efficient.

| improve this answer | |
  • $\begingroup$ Note: the question is solved by OP with the hints. $\endgroup$ – kelalaka Jan 16 '19 at 20:37

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