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

This section provides an example of a reduced Elliptic Curve group E97(-1,1). Some example points on the curve are is also provided.

Let's take a look at our second reduced elliptic curve group, E97(-1,1), as discussed in "A (relatively easy to understand) primer on elliptic curve cryptography" by Nick Sullivan at arstechnica.com/information-technology/2013/10 /a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography/2/.

Here is the elliptic curve in the real number space:

   y2 = x3 - x + 1
Elliptic Curve E(-1,1) in Real Number Space
Elliptic Curve E(-1,1) in Real Number Space

Here is the reduced elliptic curve group using modular arithmetic of prime number 97, E97(-1,1):

   y2 = x3 - x + 1 (mod 97)
Elliptic Curve Group E97(-1,1)
Elliptic Curve Group E97(-1,1)

Example points on the curve:

Verify (x,y) = (70,6) is on the curve:
   y2 = x3 - x + 1 (mod 97)
   6*6 = 70*70*70 - 70 + 1 (mod 97)
   36 = 342931 (mod 97)
   36 = 36
   
Verify (x,y) = (76,48) is on the curve:
   y2 = x3 - x + 1 (mod 97)
   48*48 = 76*76*76 - 76 + 1 (mod 97)
   2304 = 438901 (mod 97)
   73 = 73
   
Verify (x,y) = (82,6) is on the curve:
   y2 = x3 - x + 1 (mod 97)
   6*6 = 82*82*82 - 82 + 1 (mod 97)
   36 = 551287 (mod 97)
   36 = 36
   
Verify (x,y) = (69,22) is on the curve:
   y2 = x3 - x + 1 (mod 97)
   22*22 = 69*69*69 - 69 + 1 (mod 97)
   484 = 328441 (mod 97)
   96 = 96

An example of the addition operation using the rule of chord is also illustrated on the diagram presented above. But the straight line is wrapped around because of the modular arithmetic reduction:

Given two points on the curve: 
   A = (17,12) 
      12*12 = 17*17*17 - 17 + 1 (mod 97)
      144 = 4897 (mod 97)
      47 = 47
   B = (63,93)
      93*93 = 63*63*63 - 63 + 1 (mod 97)
      8649 = 249985 (mod 97)
      16 = 16

Draw a straight line passing through A and B, 
And wrap the line around when it reaches the boundary of the region.
It will reach another point -C on the curve. 
Take the symmetric point C of -C:
   C = A + B 
   C = (4,35)
      35*35 = 4*4*4 - 4 + 1 (mod 97)
      1225 = 61 (mod 97)
      61 = 61

However, when I tried to calculate C = A+B using algebraic equations given by the reduced elliptic curve group definition, I got C = (89,51):

For any two given points on the curve:
   A = (xP, yP) = (17,12)
   B = (xQ, yQ) = (63,93)
  
C = A + B is a third point on the curve: 
   C = (xR, yR)

Where:    
   xR = m2 - xP - xQ (mod p)     (11)
   yR = m(xP - xR) - yP (mod p)  (12)   
   m(xP - xQ) = yP - yQ (mod p)  (18)
   
Calculation: 
   m*(17-63) = 12-93 (mod 97)
   m*(-46) = -81 (mod 97)
   51*m = 16 (mod 97)
   m = 16 * 1/51 (mod 97)
   m = 16 * 78 (mod 97)
   m = 1248 (mod 97)
   m = 84
   
   xR = 84*84 - 17 - 63 = 6976 (mod 23) 
   xR = 89
   
   yR = 86*(17 - 89) - 12 = -6060 (mod 23)
   yR = 51
   
   C = (89,51)

May be the illustration of C=A+B on the diagram is not accurate.

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