Consider an expression of the form .

ax2 + bx + c 0 (mod N)

1. Show that the 's of the following form are solutions of the above system:.

x (b b2 4ac) (2a) (mod N)

when and is a Quadratic Residue mod . (Here is an integer square root of a quadratic residue .)

.)2. Give all the necessary and sufficient conditions for existence of solutions to the above system and for any tuple of parameters specify how many solutions exist ? Be as exhaustive as