Solving the Primitive Root Problem for g & -g Modulo p

In summary, the Primitive Root Problem for g & -g Modulo p is a mathematical problem that seeks to find a primitive root for a given prime number p. Finding primitive roots is important in number theory and cryptography, and the problem is solved using a combination of number theory concepts and algorithms. Solving this problem has practical applications in cryptography and theoretical importance in number theory. While there are known solutions, ongoing research is being conducted to find more efficient and scalable solutions.
  • #1
b0mb0nika
37
0
let g be a primitive root of the odd prime p
show that -g is a primitive root or not according as
p==1 ( mod 4) or p==3(mod4)

how would i start in solving this problem
thanks
:cool:
 
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  • #2
For p==1 Mod 4, there exists an X such that X^2 ==-1 Mod p. This is not true for p==3 Mod 4. If x is a primitive root then x^((p-1)/2) = -1 since the smallest power of x that is 1 Mod p is x^(p-1). So, those are the facts needed.
 
  • #3


To start solving this problem, we first need to understand what a primitive root is and how it relates to the given conditions of p.

A primitive root, also known as a primitive element, is an integer g that generates the multiplicative group of integers modulo p. This means that for any integer a coprime to p, there exists an integer k such that g^k ≡ a (mod p). In other words, g raised to different powers can produce all the possible remainders when divided by p.

Now, let's consider the two cases given: p ≡ 1 (mod 4) and p ≡ 3 (mod 4).

Case 1: p ≡ 1 (mod 4)
In this case, p can be expressed as p = 4q + 1 for some integer q. This means that p is congruent to 1 modulo 4.

Using this information, we can show that -g is also a primitive root modulo p.
Since g is a primitive root, we know that g^q ≡ -1 (mod p).
Multiplying both sides by -1, we get (-g)^q ≡ 1 (mod p).
This shows that -g is also a primitive root modulo p.

Case 2: p ≡ 3 (mod 4)
In this case, p can be expressed as p = 4q + 3 for some integer q. This means that p is congruent to 3 modulo 4.

Using this information, we can show that -g is not a primitive root modulo p.
Assume that -g is a primitive root, then (-g)^q ≡ 1 (mod p) for some integer q.
But we know that g^q ≡ -1 (mod p) from the definition of primitive root.
Multiplying both sides by -1, we get (-g)^q ≡ 1 (mod p).
This means that (-g)^q ≡ g^q ≡ -1 (mod p).
However, this contradicts the fact that g is a primitive root.
Therefore, -g is not a primitive root modulo p in this case.

In conclusion, the statement holds true and the solution depends on the congruence of p modulo 4. If p ≡ 1 (mod 4),
 

1. What is the Primitive Root Problem for g & -g Modulo p?

The Primitive Root Problem for g & -g Modulo p is a mathematical problem that seeks to find a primitive root for a given prime number p. A primitive root is an integer that, when raised to all possible powers modulo p, generates all non-zero residues. In other words, it is an integer whose powers cover all the numbers between 1 and p-1.

2. Why is finding primitive roots important?

Finding primitive roots is important in number theory and cryptography. In number theory, primitive roots are used to study the properties of prime numbers. In cryptography, primitive roots are used to create public key encryption systems, which are crucial for secure communication over the internet.

3. How is the Primitive Root Problem solved?

The Primitive Root Problem is solved by using a combination of number theory concepts and algorithms. One approach is to use the primitive root theorem, which states that for a prime number p, the number of primitive roots modulo p is equal to φ(p-1), where φ is Euler's totient function. Another approach is to use the index calculus algorithm, which involves finding a set of small primes that, when combined, yield a primitive root.

4. What is the significance of solving the Primitive Root Problem for g & -g Modulo p?

Solving the Primitive Root Problem for g & -g Modulo p has practical applications in cryptography, as it allows for the creation of secure public key encryption systems. It also has theoretical importance in number theory, as it helps to better understand the properties of prime numbers.

5. Are there any known solutions to the Primitive Root Problem for g & -g Modulo p?

Yes, there are known solutions to the Primitive Root Problem for g & -g Modulo p. However, these solutions are not always efficient or practical, as the problem becomes increasingly difficult for larger prime numbers. Therefore, ongoing research is being conducted to find more efficient and scalable solutions.

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