Solving for $x$ in discrete logarithm

Question

$$9 = 2^x \text{ mod } 11$$

How do you use a calculator to obtain this value?

The $x$ is an integer. Used in Diffie–Hellman key exchange algorithm.

In pari/gp calculator (https://pari.math.u-bordeaux.fr/gp.html) enter m=Mod(2,11);znlog(9,m,znorder(m)) and get 6. Verify m=Mod(2,11);m^6 and get Mod(9, 11). — Dmitry Ezhov, Jan 05 '22 at 13:38
You can do it counting powers of $2$ on your fingers (reduce mod $11$ as you go). — paw88789, Jan 05 '22 at 13:52
Square $\ 2^3\equiv -3.\ $ More generally you can use Shanks Baby Giant Step — Bill Dubuque, Jan 05 '22 at 14:03

score 1 · Accepted Answer · answered Jan 07 '22 at 04:45

1

We check by hand and get $2^{6} \equiv 9 \pmod{11}.$

By Fermat's little theorem, $2^{10} \equiv 1 \pmod{11}.$

$2^{10n+6}\equiv 2^{6} (2^{10})^{n}\equiv2^{6}\equiv9 \pmod{11}.$

Hence $x=10n+6$ for all $n\in\Bbb Z.$

answered Jan 07 '22 at 04:45

Tomita

1 Answers1