# While generating a random Elliptic curve what are the conditions i have to considerd?

1. I want to generate a random elliptic curve over a prime field. What are the conditions I should satisfy?

2. For the NIST recommended standard ECC-224 bit curve with prime $p=2^{224}-2^{96}+1$, a reduction technique is given by $(z_1+z_2+z_3-z_4-z_5) \bmod{p}$, what is the logic behind in this?

• Just checking, you want to generate a curve or a point on a curve? A curve over a finite field or infinite or either? Commented Feb 28, 2013 at 14:04
• i want to generate a random curve over prime field... Commented Mar 1, 2013 at 15:43
• I don't understand question #2 at all. Can you help me understand? Commented Mar 1, 2013 at 15:59
• NIST recommended standard ECC-224 bit, prime value is 2^224-2^96+1. 224 bit * 224 bit multiplication results 448 bit output. which can be converted back over field 224 bit, NIST standard has given one formula ((z1+z2+z3-z4-z5) mod prime )). what is mathematical logic beind in this formula. Commented Mar 2, 2013 at 14:36
• I have updated the question to make #2 a little better. Can you make sure I didn't change the meaning? Also, what are the $z$ values? Commented Mar 3, 2013 at 13:43

An elliptic curve can be written as $y^2=x^3+ax+b$. To generate a random curve over a prime field, choose $a,b$ at random from the prime field.