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Let's consider a discrete logarithm

$\beta \equiv \alpha ^{x} \bmod \,\, p$

We can solve it using Pohlig-Hellman algorithm. And, if $p-1 = tq$ where $q$ is a large prime factor, we can avoid any leakages by choosing $\beta=a^{t}$.

This is what my professor said at lesson, however i don't succeed in understanding why it avoids any leakages. Can you explain to me why ? Also with calculations if it's possible

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Probably what your professor meant is that you start with any group element $\alpha$, and then use $g := \alpha^t$ as the generator for a cryptosystem such as Schnorr signatures, as long as $g$ is not itself the identity.

Why? If $g \ne 1$, then $g$ is guaranteed to have prime order $q$, because $g^q = (\alpha^t)^q = \alpha^{tq} = \alpha^{\phi(p)} = 1$, and since $q$ is prime there are no smaller orders possible for $g$ (just start over with a different $\alpha$ if it is).

Then when you choose a public key $\beta := g^x$ for secret $x$, you're guaranteed that $\log_\alpha \beta \equiv 0 \pmod t$. (Of course, you could have equivalently chosen $\beta := \alpha^{tx}$, but precomputing $g$ may be cheaper.)

Now since the group generated by $g$ has prime order, Pohlig–Hellman has no advantage over any other DLOG algorithms.

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  • $\begingroup$ Can you explain me why $\log_\alpha \beta \equiv 0 \pmod t$ ? $\endgroup$ – AleWolf Nov 18 '19 at 7:51
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    $\begingroup$ $\log_\alpha \beta = \log_\alpha g^x = \log_\alpha \alpha^{tx} = tx$ $\endgroup$ – Squeamish Ossifrage Nov 18 '19 at 8:09

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