**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.00, by Dr. Herong Yang

Reduced Elliptic Curve Group - E23(1,4)

This section provides an example of a reduced Elliptic Curve group E23(1,4). A detailed calculation of reduced point doubling operation on (0,2) is also provided.

Let's take a look at our first reduced elliptic curve group, E_{23}(1,4),
as discussed in "Implementation Of Elliptic Curve Diffie-Hellman and EC Encryption Schemes by Kefa Rabah
at
docsdrive.com/pdfs/ansinet/itj/2005/132-139.pdf:

The reduced elliptic curve: y^{2}= x^{3}+ x + 4 (mod 23) The group elements: ( 0, 2) ( 1,11) ( 4, 7) ( 7, 3) ( 8, 8) ( 9,11) (10, 5) ( 0,21) ( 1,12) ( 4,16) ( 7,20) ( 8,15) ( 9,12) (10,18) (11, 9) (13,11) (14, 5) (15, 6) (17, 9) (18, 9) (22, 5) (11,14) (13,12) (14,18) (15,17) (17,14) (18,14) (22,18) ( ∞, ∞)

As the first verification case, we can perform the point doubling operation of P = (0,2) again using the reduced additive operation:

Given: P = (x_{P}, y_{P}) = (0, 2) Find: 2P = P + P = R = (x_{R}, y_{R}) Where: x_{R}= m^{2}- 2x_{P}(mod p) (11) y_{R}= m(x_{P}- x_{R}) - y_{P}(mod p) (12) 2m(y_{P}) = 3(x_{P})^{2}+ a (mod p) (19) Calculation: 2 * m * 2 = 3 * 0 * 0 + 1 (mod 23) 4 * m = 1 (mod 23) m = 1/4 (mod 23) m = 6 x_{R}= 6*6 - 2*0 = 36 (mod 23) x_{R}= 13 y_{R}= 6*(0 - 13) - 2 = -78 - 2 = -80 (mod 23) y_{R}= 12 Result: 2P = R = (x_{R}, y_{R}) = (13, 12)

Yes, the result of (0,2) + (0,2) = (13, 12) is in the group element set! We Abelian group "Closure" condition is satisfied.

We can also redo the same calculation by using the original equation for the parameter m. We should get the same resulting point:

Given: P = (x_{P}, y_{P}) = (0, 2) Find: 2P = P + P = R = (x_{R}, y_{R}) Where: x_{R}= m^{2}- 2x_{P}(mod p) (11) y_{R}= m(x_{P}- x_{R}) - y_{P}(mod p) (12) 3(x_{P})^{2}+ a m = --------- (6) 2(y_{P}) Calculation: m = (3*0*0 + 1)/(2*2) = 1/4 x_{R}= (1/4)*(1/4) - 2*0 = 1/16 (mod 23) x_{R}= 13 x_{R}= (1/4)*(0 - 1/16) - 2 = -1/64 - 2 = -129/64 (mod 23) x_{R}= 9 * 1/18 (mod 23) = 9 * 9 (mod 23) x_{R}= 12 Result: 2P = R = (x_{R}, y_{R}) = (13, 12)

Cool. This demonstrates that keep m as a rational number without modular reduction is also provides the same resulting point.

Here is a diagram of all non-infinite points of the E_{23}(1,4) group:

Table of Contents

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

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)

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

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)