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, E23(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:
   y2 = x3 + 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 = (xP, yP) = (0, 2)

Find:
   2P = P + P = R = (xR, yR)

Where:    
   xR = m2 - 2xP (mod p)         (11)
   yR = m(xP - xR) - yP (mod p)  (12)   
   2m(yP) = 3(xP)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
   
   xR = 6*6 - 2*0 = 36 (mod 23) 
   xR = 13 
   
   yR = 6*(0 - 13) - 2 = -78 - 2 = -80 (mod 23)
   yR = 12

Result: 
   2P = R = (xR, yR) = (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 = (xP, yP) = (0, 2)

Find:
   2P = P + P = R = (xR, yR)

Where:    
   xR = m2 - 2xP (mod p)         (11)
   yR = m(xP - xR) - yP (mod p)  (12)   

       3(xP)2 + a
   m = ---------                 (6)
         2(yP) 

Calculation: 
   m = (3*0*0 + 1)/(2*2) = 1/4

   xR = (1/4)*(1/4) - 2*0 = 1/16 (mod 23)
   xR = 13 
   
   xR = (1/4)*(0 - 1/16) - 2 = -1/64 - 2 = -129/64 (mod 23)
   xR = 9 * 1/18 (mod 23) = 9 * 9 (mod 23)
   xR = 12

Result: 
   2P = R = (xR, yR) = (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 E23(1,4) group:

Elliptic Curve Group E23(1,4)
Elliptic Curve Group E23(1,4)

Last update: 2019.

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)

 Finite Fields

 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)

 What Is Hasse's Theorem

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

 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)

 Terminology

 References

 Full Version in PDF/EPUB