- #1
sid_galt
- 502
- 1
Is there an easy (by which I mean an algorithm polynomial in size of input) way to know whether in the multiplicative group of integers mod P (P is a prime), whether an element is a generator or not?
Is there a simple way to determine if an element is a primitive root modulo n?
In summary, there is no known easy method for determining if an element in the multiplicative group of integers mod P (where P is a prime) is a generator or not. However, there is a method that involves factoring P-1 and checking for congruence, but this is not a simple task. Primitive roots are defined as elements that are not congruent to 1 for any value less than P, and Gauss provided a table of primitive roots in his work, showing that there is no easy solution to this problem.
- #2
ialbrekht
- 7
- 0
Hi, there is one method but it required knowing expansion of the number P-1 on the simple factors.
So, if [tex] P-1 = p_{1}^{a_1} \times \cdots \times p_{n}^{a_n} [/tex] where each [tex]p_i[/tex] is prime and for each [tex]i=1...n : \frac{g^{P-1}}{p_i} \equiv 1 (mod P)[/tex], than [tex]g[/tex] - is primitive root (generator) by modulo P.
Unfortunately, exponential factorization algorithms not known yet.
So, if [tex] P-1 = p_{1}^{a_1} \times \cdots \times p_{n}^{a_n} [/tex] where each [tex]p_i[/tex] is prime and for each [tex]i=1...n : \frac{g^{P-1}}{p_i} \equiv 1 (mod P)[/tex], than [tex]g[/tex] - is primitive root (generator) by modulo P.
Unfortunately, exponential factorization algorithms not known yet.
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- #3
robert Ihnot
- 1,059
- 1
ialbekht: Hi, there is one method
THAT! You are submitting as an EASY method? Anyway you got the exponent wrong. Should be:
[tex]i=1...n : g^\frac{P-1}{p_i} \equiv 1 (mod P)[/tex] Also, to be a primitative root, it would NOT--by definition--be congruent to 1 for any value less than[tex] \varphi({P})[/tex].
THAT! You are submitting as an EASY method? Anyway you got the exponent wrong. Should be:
[tex]i=1...n : g^\frac{P-1}{p_i} \equiv 1 (mod P)[/tex] Also, to be a primitative root, it would NOT--by definition--be congruent to 1 for any value less than[tex] \varphi({P})[/tex].
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- #4
ialbrekht
- 7
- 0
robert Ihnot:
Thanks for correcting me.
It's the first method I've remembered. The hardest thing is to factorize P-1, and It's not a problem for a not large P and if you know factorization of P-1 you may check g to be primitive root for polynomial time.
And I did not say that it's EASY way. Unfortunately, there are not much easy solutions in number-theory problems
Thanks for correcting me.
It's the first method I've remembered. The hardest thing is to factorize P-1, and It's not a problem for a not large P and if you know factorization of P-1 you may check g to be primitive root for polynomial time.
And I did not say that it's EASY way. Unfortunately, there are not much easy solutions in number-theory problems
- #5
robert Ihnot
- 1,059
- 1
From Wickipedia: Gauss defines primitive root in Article 57 of the Disquisitiones Arithmeticae (1801), where he credits Euler with coining the term.
Gauss then went ahead and gave the reader a table of primitive roots, so that he was aware that there was no really easy way to find them. It then seems, from the responses we are getting here, that's about where it is today!
Gauss then went ahead and gave the reader a table of primitive roots, so that he was aware that there was no really easy way to find them. It then seems, from the responses we are getting here, that's about where it is today!
Related to Is there a simple way to determine if an element is a primitive root modulo n?
1. What is a primitive root modulo n?
A primitive root modulo n is an integer that, when raised to certain powers, can generate all the numbers in the set of integers from 1 to n-1. This means that every integer in the set can be expressed as a power of the primitive root modulo n.
2. How do I find the primitive root modulo n?
Finding the primitive root modulo n can be a complex mathematical problem. One method is to use the primitive root theorem, which states that for any prime number p, there will always be at least one primitive root modulo p. Another method is to use brute force and test different numbers until one is found that satisfies the criteria of a primitive root.
3. What is the significance of the primitive root modulo n?
The primitive root modulo n is significant in number theory and cryptography. It is used in various algorithms and protocols, such as the Diffie-Hellman key exchange, which is used for secure communication over a public channel. The primitive root modulo n also has applications in solving certain mathematical problems.
4. Can there be more than one primitive root modulo n?
No, there can only be one primitive root modulo n. This is because the primitive root theorem states that for any prime number p, there will always be exactly φ(p-1) primitive roots modulo p, where φ is Euler's totient function. So, for example, if p=7, there can only be 6 primitive roots modulo 7.
5. How is the primitive root modulo n related to Euler's totient function?
The primitive root modulo n and Euler's totient function are closely related. The primitive root theorem states that for any prime number p, there will always be exactly φ(p-1) primitive roots modulo p. This means that Euler's totient function can be used to calculate the number of primitive roots modulo n for any given prime p.
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