EC Cryptography Tutorials - Herong's Tutorial Examples - v1.02, by Dr. Herong Yang
Elliptic Curves Reduced by Modular Arithmetic
This section describes elliptic curves reduced by modular arithmetic of prime numbers. We can find lots of more integer points on those reduced elliptic curves.
From previous sections, we see that finding integer points on elliptic curves are difficult. For example, there are only 4 valid integer points on the elliptic curve of (a,b) = (1,4):
Elliptic curve: y2 = x3 + x + 4 Integer points on the curve in the range of < (1000000, 31622303) (x, y) = (0, 2) (x, y) = (0, -2) (x, y) = (4128, 265222) (x, y) = (4128, -265222)
One way to increase the number of valid integer points is to relax the elliptic curve equation with the modular arithmetic reduction, which can be defined as:
Element Set: All P = (x,y), such that: y2 = x3 + ax + b (mod p) where: a and b are integers p is a prime number 4a3 + 27b2 != 0 x and y are integers
Now let's see if we can find more integer points on this relaxed elliptic curve using (a,b) = (1,4) and p = 23 as an example:
Elliptic curve: y2 = x3 + x + 4 (mod 23) Or let l be the left hand side value, and r be the right hand side value: l = y2 (mod 3) r = x3 + x + 4 (mod 23) l = r
To find all integer points, we can use the following algorithm:
1. Build the y-to-l map to find all unique values for the left hand side l, using y in (0, 1, 2, ..., 22) only. Other values of y will not generate any new l values, because of the modular reduction of 23.
Possible values of l = y2 (mod 23): l = 0*0 mod 23 = 0 l = 1*1 mod 23 = 1 l = 2*2 mod 23 = 4 l = 3*3 mod 23 = 9 l = 4*4 mod 23 = 16 l = 5*5 mod 23 = 2 l = 6*6 mod 23 = 13 l = 7*7 mod 23 = 3 l = 8*8 mod 23 = 18 l = 9*9 mod 23 = 12 l = 10*10 mod 23 = 8 l = 11*11 mod 23 = 6 l = 12*12 mod 23 = 6 l = 13*13 mod 23 = 8 l = 14*14 mod 23 = 12 l = 15*15 mod 23 = 18 l = 16*16 mod 23 = 3 l = 17*17 mod 23 = 13 l = 18*18 mod 23 = 2 l = 19*19 mod 23 = 16 l = 20*20 mod 23 = 9 l = 21*21 mod 23 = 4 l = 22*22 mod 23 = 1
So, we have a list of unique l values of: 0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18.
2. Build the x-to-r map to find all unique values for the left hand side r, using x in (0, 1, 2, ..., 22) only. Other values of x will not generate any new l values, because of the modular reduction of 23.
Possible values of r = x3 + x + 4 (mod 23): r = (0*0*0 + 0 + 4 ) mod 23 = 4 r = (1*1*1 + 1 + 4 ) mod 23 = 6 r = (2*2*2 + 2 + 4 ) mod 23 = 14 r = (3*3*3 + 3 + 4 ) mod 23 = 11 r = (4*4*4 + 4 + 4 ) mod 23 = 3 r = (5*5*5 + 5 + 4 ) mod 23 = 19 r = (6*6*6 + 6 + 4 ) mod 23 = 19 r = (7*7*7 + 7 + 4 ) mod 23 = 9 r = (8*8*8 + 8 + 4 ) mod 23 = 18 r = (9*9*9 + 9 + 4 ) mod 23 = 6 r = (10*10*10 + 10 + 4 ) mod 23 = 2 r = (11*11*11 + 11 + 4 ) mod 23 = 12 r = (12*12*12 + 12 + 4 ) mod 23 = 19 r = (13*13*13 + 13 + 4 ) mod 23 = 6 r = (14*14*14 + 14 + 4 ) mod 23 = 2 r = (15*15*15 + 15 + 4 ) mod 23 = 13 r = (16*16*16 + 16 + 4 ) mod 23 = 22 r = (17*17*17 + 17 + 4 ) mod 23 = 12 r = (18*18*18 + 18 + 4 ) mod 23 = 12 r = (19*19*19 + 19 + 4 ) mod 23 = 5 r = (20*20*20 + 20 + 4 ) mod 23 = 20 r = (21*21*21 + 21 + 4 ) mod 23 = 17 r = (22*22*22 + 22 + 4 ) mod 23 = 2
So, we have a list of unique r values of: 2, 3, 4, 5, 6, 9, 11, 12, 13, 14, 17, 18, 19, 20, 22.
3. Find all common values of l and r:
Possible common values of l and r: l = r in {2, 3, 4, 6, 9, 12, 13, 18}
4. Find all integer points (x,y), where x and y in {0, 1, 2, ..., 22} for each common value of l = r using the y-to-l map and the x-to-r map:
For l = r = 2: (x, y) = (10, 5) (x, y) = (10, 18) (x, y) = (14, 5) (x, y) = (14, 18) (x, y) = (22, 5) (x, y) = (22, 18) For l = r = 3: (x, y) = (4, 7) (x, y) = (4, 16) For l = r = 4: (x, y) = (0, 2) (x, y) = (0, 21) For l = r = 6: (x, y) = (1, 11) (x, y) = (1, 12) (x, y) = (9, 11) (x, y) = (9, 12) (x, y) = (13, 11) (x, y) = (13, 12) For l = r = 9: (x, y) = (7, 3) (x, y) = (7, 20) For l = r = 12: (x, y) = (11, 9) (x, y) = (11, 14) (x, y) = (17, 9) (x, y) = (17, 14) (x, y) = (18, 9) (x, y) = (18, 14) For l = r = 13: (x, y) = (15, 6) (x, y) = (15, 17) For l = r = 18: (x, y) = (8, 8) (x, y) = (8, 15) Total of 28 points!
Comparing this results with previous tutorial, we can see that reduced elliptic curves have many more integer points than original elliptic curves.
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)