ElGamal signature - show that $(\gamma, \delta)$ signs $m$

We have as a given: $gcd(j, p − 1) = 1$.

$$γ = \alpha^i \beta^j \bmod p$$ $$\delta = −\gamma j^{-1} \bmod (p−1)$$ $$m = i \delta \bmod (p − 1)$$

I want to show that $(\gamma; \delta)$ is an ElGamal signature for message $m$.

In this book, they show an example:

How can I take the proof that $(\gamma, \delta)$ signs $m$?

• The image you posted is actually the proof! Did you want help walking through it?
– user47922
Jun 12 '17 at 0:02
• I'm editing the post because the original formula provided for $m$ is incorrect.
– user47922
Jun 12 '17 at 14:40

A valid ElGamal signature satisfies, in the notation from the book you posted:

$$\beta^\gamma \gamma^\delta = \alpha^m \bmod p$$

That's just the definition of an ElGamal signature. The snippet you posted is not an example, but precisely the proof that $(\gamma, \delta)$ is a valid signature of $m = i \delta \bmod (p-1)$. On page 68 of the book, it says

($\gamma, \delta$) este o semnatura valida pentru x. Intr-adevar, se verifica...

(In English: "$(\gamma,\delta)$ is a valid signature for $x$. Indeed, one verifies...")

So let's step through the proof. Everything is $\bmod p$:

$$\beta^\gamma \gamma^\delta = \beta^{\alpha^i \beta^j } (\alpha^i \beta^j)^{-\alpha^i \beta^j j^{-1} }$$

(Substitution of values)

$$... = \beta^{\alpha^i \beta^j } \alpha^{-i j^{-1} \alpha^i \beta^j } \beta^{-\alpha^i \beta^j}$$

(Just distribute the exponent across $\alpha^i$ and $\beta^j$. In the $\beta$ exponent, you get a $j j^{-1} = 1$)

$$...= \alpha^{-i j^{-1} \alpha^i \beta^j }$$

(The $\beta$ terms cancel out)

$$...= \alpha^{-\gamma i j^{-1} }$$

(definition of $\gamma$)

$$...= \alpha^m$$

(definition of $m$)

Here, $i$ and $j$ are such that $0 \leq i,j \leq p-2$. The proof you are showing is that this scheme is existentially forgeable (see page 381, Theorem 16). This is a two-parameter forgery (the $(i,j)$).

• Thanks, yes. But if $$m=−iδmod(p−1)$$ than $α^m = α^-m$. Is that ok? What should we do with that minus? Jun 12 '17 at 13:51
• That formula is wrong; the confusion is happening because: $$\delta = -\gamma j^{-1} \mod (p-1)$$ and $$m = -\gamma i j^{-1} \bmod (p-1) = i \delta \mod (p-1)$$. The minus sign is "in the $\delta$". Example 3.5 shows this in the link you provided.
– user47922
Jun 12 '17 at 14:35
• Also, $-x \bmod p \equiv (p-x) \bmod p$. Again, from Example 3.5, $$-117 \cdot 151 \bmod 466 \equiv (466 - 117) \cdot 151 \mod 466 \equiv 52699 \bmod 466 \equiv 41 \bmod 466$$ So negative numbers aren't a problem either, mod $p$.
– user47922
Jun 12 '17 at 15:45