4

I am currently trying to find a primitive element of the multiplicative group of field $GF(p)$. Since the numbers are relatively small, I know the factorization of

$$\phi(p)=p-1 = {p_1}^{k_1} {p_2}^{k_2} ... {p_n}^{k_n}$$

Wikipedia says that $m\in GF(p)$ is a generator if

$$m^{\phi(n)/p_k} \not\equiv 1 \mod p \quad \forall k=1..n$$

However, there is no clear explanation why it is the case. Could you please help me to understand this?

wh1t3cat1k
  • 945

1 Answers1

2

Can you see that every proper divisor of $\phi(p)$ is a divisor of $\phi(p)/p_k$ for some $k$? So if the order of $m$ isn't $\phi(p)$, then it's a divisor of some $\phi(p)/p_k$?

Gerry Myerson
  • 179,216
  • Could you please get into a bit more detail in here? I can't figure out what these divisors are about. Maybe I lack something fundamental here – wh1t3cat1k May 06 '12 at 12:30
  • Well, that's the question: what do you know, and what don't you know, about raising numbers to powers modulo a prime? It's hard to give an answer that will help you when you volunteer no information. Anyway, here are a few facts you'll find proved in any intro number theory text: $m$ in $F_p$, $m\ne0$ implies $m^{p-1}=1$; smallest positive $d$ such that $m^d=1$ is a divisor of $p-1$; $m$ is a primitive element (i.e., generator) is the same as $d=p-1$; if $d=p-1$ then $d$ doesn't divide $(p-1)/p_k$; if $d$ divides $p-1$ but $d\ne p-1$ then $d$ divides some $(p-1)/p_k$. Continued.... – Gerry Myerson May 06 '12 at 13:03
  • Continuation. That last bit: if $d$ divides $p-1$ but $d\ne p-1$ then $d=p_1^{r_1}\cdots p_n^{r_n}$ where $r_i\le k_i$ for all $i$ and $r_i\lt k_i$ for at least one $i$; for that $i$, $d$ divides $(p-1)/p_i$. – Gerry Myerson May 06 '12 at 13:07