On the number of points on an algebraic curve in a ring of residues

Project Description

In our paper we obtained the following results:

1. We introduce a new function R_(f,I) (n), which is the number of all possible combinations of giving coordinates of the solutions of the congruence f(x_1,…,x_k )≡0 mod n, and proved that this function is multiplicative.

2. We obtain the exact number of points on a curve x^m-y^k≡0 modn.

3. We introduce a new definition m/k-power residue modulo n and found an exact formula for their number modulo n. From this formula as a corollary we obtained the full results about m-power residues.

4. We calculated the number of points on a curve ax^2+bxy+cy^2≡0modulo a prime number.

5. We found an exact formula for the number of all possible values of quadratic polynomial mod n.

These results can be useful algebraic geometry and asymmetric cryptography.