Elliptic Curve y^2 = x^3 +17; show N_p = p

**chiph588@** · March 31st 2010, 12:12 PM

Originally Posted by vernongetzler

Hat’s off. Well done, as we know that “hard work always pays off”, after a long struggle with sincere effort it’s done.
========
vernon
Exercises

Huh? The link confuses me. Is your post about the link, or about what's been done on this thread?

**kingwinner** · March 31st 2010, 10:35 PM

double posted by mistake...

**kingwinner** · March 31st 2010, 10:39 PM

Originally Posted by chiph588@

Let's assume $b_1,b_2\not|p$ and $g$ is a primitive root ( $b_1\equiv0\mod{p}$ is trivial).

Why are you assuming that $b_1,b_2\not|p$ ? Do you actually mean $p\not|b_1,b_2$ ?

Then $b_1\equiv g^n,\;b_2\equiv g^m \mod{p}$ , so $g^{3n}\equiv g^{3m}\mod{p}\implies 3n\equiv 3m\mod{\phi(p)}$ .

Why $b_1\equiv g^n,\;b_2\equiv g^m \mod{p}$ ? What are n and m?

Since $(3,p-1)=1 \implies n\equiv m\mod{\phi(p)} \implies m=n+a(p-1)$

So $b_2\equiv g^m=g^{n+a(p-1)} \equiv 1\cdot g^n\equiv b_1\mod{p}$ .

On the last line, I think you're using Fermat's little theorem, but how do you know that $p\not|g$ ?

Thanks for clarifying this!

**kingwinner** · April 2nd 2010, 03:57 PM

This is my version of the definition of primitive root:
a is a primitive root mod m iff the order of a mod m is equal to phi(m) (where the order of a mod m is the smallest positive integer h such that a^h is congruent to 1 mod m.)

But from here I really don't see how we can get b1=g^n (mod p)...

**chiph588@** · April 2nd 2010, 05:20 PM

Originally Posted by kingwinner

This is my version of the definition of primitive root:
a is a primitive root mod m iff the order of a mod m is equal to phi(m) (where the order of a mod m is the smallest positive integer h such that a^h is congruent to 1 mod m.)

But from here I really don't see how we can get b1=g^n (mod p)...

A primitive root will 'generate' every element in the set $\{1,2,\cdots,p-1\}$ .

i.e. $\{g,g^2,\cdots, g^{p-1}\} = \{1,2,\cdots,p-1\}$

Thread: Elliptic Curve y^2 = x^3 +17; show N_p = p

LinkBack

Thread Tools

Display

Similar Math Help Forum Discussions

negative of point on elliptic curve

Elliptic Curve Group / Multiplication

The circle as a degenerate elliptic curve

Elliptic Curve

torsion for elliptic curve

Search Tags