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
Reduced Point Additive Operation
Modular Arithmetic Reduction on Rational Numbers
Reduced Point Additive Operation Improved
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
- 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.
Table of Contents
About This Book
Geometric Introduction to Elliptic Curves
Algebraic Introduction to Elliptic Curves
Abelian Group and Elliptic Curves
Discrete Logarithm Problem (DLP)
Generators and Cyclic Subgroups
►Reduced Elliptic Curve Groups
Elliptic Curve Subgroups
tinyec - Python Library for ECC
EC (Elliptic Curve) Key Pair
ECDH (Elliptic Curve Diffie-Hellman) Key Exchange
ECDSA (Elliptic Curve Digital Signature Algorithm)
ECES (Elliptic Curve Encryption Scheme)
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