Thursday, February 27, 2014

Primitive Roots modulo a prime number

Finding primitive roots

First off, it has to be noted that the constraint that the modulo value is a prime number is just for simplicity (hint: all values from 1 to p-1 are coprime to p for a prime number p).
Now, what EVEN is a primitive root modulo p you may ask. This is just a number g which generates all numbers from 1 to p-1 with \(g^{t} \: mod \: p\). That is, for every positive integer x < p there exists a t, so that \(g^{t} \equiv x \pmod{p}\).
Properties of primitive roots:
  • for a given p there are phi (phi(p))=phi(p-1) primitive roots
  • g is a primitive root \( \Leftrightarrow\) for every divisor d of \((p-1)=phi(p)\), \(g^{d} \not\equiv 1\pmod{p}\) holds \( \Leftrightarrow\) for every prime \(x\) from the prime factorization of \(p-1\), \(g^{(p-1)/x} \not\equiv 1\pmod{p}\) holds.
  • (comment: this is essentially there to check for repeating sequences in \(g^{t}\) for \(t \in {1,..,p-1} \pmod{p}\).
Some research: I've ran this code to look at the distribution of the primitive roots for primes between 100 and 1,000,000,000. By running the code we see this from the output:
  • the lower bound for the number of primitive roots is \(0.15 \cdot p\). Therefore, 15% of the numbers smaller than \(p\) are primitive roots for a given p (in worst case!). By selecting a random positive integer < p we have a 15% chance of selecting a primitive root!
  • given a random p, the chance of hitting a primitive root on random is even ~40%.

Therefore, in order to find a primitive root, we may simply pick up random numbers and check the above property which has to hold for primitive roots. Code which does that: link.

Now to the discrete logarithm (the index):

for a given x, how to find a \(y=ind_{g}(x)\) with \(g^{y} \equiv x \pmod{p}\). The answer is: you can't. Many encryption algorithms are based on this fact. Though, they have some nice properties as normal logarithms:
  • \(ind_{g}(ab) \equiv ind_{g}(a) + ind_{g}(b)\pmod{p-1} \)
  • \(ind_{g}(a^{k}) \equiv k \cdot ind_{g}(a)\pmod{p-1} \)
That should bundle the core things about this topic. One is advised to check the references for a better understanding.

Useful links:
- Wiki - Wiki(discrete logratihm) - Finding primitive roots - Index properties

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