Reduced Elliptic Curve Groups

This chapter provides notes and tutorials on reduced elliptic curve groups. Topics include elliptic curve on in integer space; elliptic curves and the addition operation reduced by modular arithmetic; elliptic curve groups and examples.

Converting Elliptic Curve Groups

Elliptic Curves in Integer Space

Python Program for Integer Elliptic Curves

Elliptic Curves Reduced by Modular Arithmetic

Python Program for Reduced Elliptic Curves

Point Pattern of Reduced Elliptic Curves

Integer Points of First Region as Element Set

Modular Arithmetic Reduction on Rational Numbers

What Is Reduced Elliptic Curve Group

Reduced Elliptic Curve Group - E23(1,4)

Reduced Elliptic Curve Group - E97(-1,1)

Reduced Elliptic Curve Group - E127(-1,3)

Reduced Elliptic Curve Group - E1931(443,1045)

What Is Hasse's Theorem

Finite Elliptic Curve Group, Eq(a,b), q = p^n

Conclusion:

• There are infinite number of integer points on an elliptic curve.
• An elliptic curve can be reduced by modular arithmetic of a prime number p.
• There many more points on a reduced elliptic curve than the non-reduced elliptic curve.
• Points on a reduced elliptic curve can be divided into regions with the first region as [(0,0), (p-1,p-1)].
• Every region is a carbon copy of the first region, because if a point P=(x,y) is on the elliptic curve reduced by p, Q=(x+iP,y+jP) is also on the curve for any integers of i and j.
• The addition operation based on the rule of chord can also be reduced by modular arithmetic of a prime number p.
• The first region of point on a reduced elliptic curve and the reduced addition operation with the same prime number p is an Abelian group.
• Modular arithmetic reduction of a rational number, i/j (mod p), is equivalent to i*(1/j) (mod p).
• k = (1/j) (mod p) is called modular multiplicative inverse, which is to find the least positive integer k, such that j*k = 1 (mod p).
• Hasse's Theorem, which states that the order, n, of a reduced elliptic curve group, Ep(a,b), is bounded in the range of [p+1 - 2*sqrt(p), p+1 + 2*sqrt(p)].
• Finite elliptic curve groups can also be constructed with modular arithmetic reduction of prime power numbers, p^n.