Proof of Fermat's Little Theorem: g Generator of Fp*

In summary, Fermat's Little Theorem is a mathematical theorem that states that for any prime number p, a^(p-1) ≡ 1 (mod p) for all a that are relatively prime to p. Its proof involves mathematical induction and modular arithmetic, and a generator of Fp* is an element that, when repeatedly multiplied by itself, produces all elements of the group Fp*. To prove that g is a generator of Fp*, we need to show that g^(p-1) ≡ 1 (mod p) and that for any a that is relatively prime to p, there exists an integer k such that g^k ≡ a (mod p). This proof is important in cryptography as it
  • #1
jacquelinek
3
0
Prove that:
g is a generator of Fp* if and only if g^(p-1) = 1 (mod p) and gq ≠ 1 (mod p) for all prime divisors q of (p – 1).

I am thinking about applying Fermat's theorem...but don't know how...
Request help, thanks.
 
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  • #2
Should that be gq?

Fermat's theorem may be useful for part of the proof. It certainly cannot prove this theorem all by itself.

What do you know about finite fields? And about their unit groups?
 

FAQ: Proof of Fermat's Little Theorem: g Generator of Fp*

What is Fermat's Little Theorem?

Fermat's Little Theorem is a mathematical theorem that states that if p is a prime number, then for any integer a, a^p - a is divisible by p.

What is the proof of Fermat's Little Theorem?

The proof of Fermat's Little Theorem involves using mathematical induction and modular arithmetic. It can be shown that for any prime number p, a^(p-1) ≡ 1 (mod p) for all a that are relatively prime to p.

What does it mean for g to be a generator of Fp*?

A generator of Fp* is an element g that, when repeatedly multiplied by itself, produces all the elements of the group Fp*. In other words, g is a primitive root modulo p.

How do you prove that g is a generator of Fp*?

To prove that g is a generator of Fp*, we need to show that g^(p-1) ≡ 1 (mod p) and that for any a that is relatively prime to p, there exists an integer k such that g^k ≡ a (mod p). This can be done using the proof of Fermat's Little Theorem and the fact that g is a primitive root modulo p.

Why is the proof of Fermat's Little Theorem important in cryptography?

The proof of Fermat's Little Theorem is important in cryptography because it is the basis for the RSA encryption algorithm. This algorithm relies on the fact that it is difficult to compute the discrete logarithm, which is related to Fermat's Little Theorem. Therefore, understanding the proof of this theorem is crucial for understanding the security of RSA encryption.

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