Some ECC Types and Operations

Some ECC Types and Operations

We have a number of possible types of elliptic curve methods. These have a field (the prime number used), the order (the number of elliptic curve points), an a value, a b value, and a generator point (G). For a Weierstrass curve the standard form is y²=x³+ax+b (such as secp256k1). With a twisted Edwards curve (such as Ed25519) we have the form of ax²+y²=1+dx²y². A Montgomery curve has the form of by²=x³+ax²+x.

With ECC, we typically take a base point (G) and multiply it with our private key (sk), and then generate our public key point (sk.G). A sample run for secp256k1 and for 1,000G is [here]:

Curve: secp256k1
====================
G: (0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798 , 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8)
Bit size: 256
Order: 115792089237316195423570985008687907852837564279074904382605163141518161494337
Field: 115792089237316195423570985008687907853269984665640564039457584007908834671663
a: 0
b: 7
====================

1000G: (0x4a5169f673aa632f538aaa128b6348536db2b637fd89073d49b6a23879cdb3ad , 0xbaf1e702eb2a8badae14ba09a26a8ca7cb1127b64b2c39a1c7ba61f4a3c62601)

Point add (add 1G on)

1001G: (0x9d1abaec9f5715a15c7628244170951e0f85e87f68ca5393d3f9fc3fa23a69c8 , 0xf21ee70050dbb61c238c89e62942353871b010e798867bdd149ad28b3f28cadf)

Point subtraction (take 1G off)

1000G: (0x4a5169f673aa632f538aaa128b6348536db2b637fd89073d49b6a23879cdb3ad , 0xbaf1e702eb2a8badae14ba09a26a8ca7cb1127b64b2c39a1c7ba61f4a3c62601)

The first 10 points ...
(1,29896722852569046015560700294576055776214335159245303116488692907525646231534)
(2,46580984542418694471253469931035885126779956971428003686700937153791839982430)
(3,94471189679404635060807731153122836805497974241028285133722790318709222555876)
(4,75283998438183369598817001785077409976464767920444208230223825605154713824156)
(5,None)
(6,19112057249303445409876026535760519114630369653212530612662492210011362204224)
(7,None)
(8,91736135629086734185706894124002126994554994840140056297753929940646699135966)
(9,None)
(10,None)

The field is the prime number used, and the order is the number of points on the curve. We can see we have the form of y²=x³+7. The outline code is [here]:

from ecpy.curves     import Curve

import sys

o=3
k=3
if (len(sys.argv)>1):
  o=int(sys.argv[1])
if (len(sys.argv)>2):
  k=int(sys.argv[2])

cv = Curve.get_curve('Curve25519')

if (o==1): cv = Curve.get_curve('Curve25519')
if (o==2): cv = Curve.get_curve('Curve448')
if (o==3): cv = Curve.get_curve('Ed25519')
if (o==4): cv = Curve.get_curve('Ed448')
if (o==5): cv = Curve.get_curve('secp160r2')
if (o==6): cv = Curve.get_curve('secp256k1')
if (o==7): cv = Curve.get_curve('secp521r1')
if (o==8): cv = Curve.get_curve('NIST-P192')
if (o==9): cv = Curve.get_curve('NIST-P224')
if (o==10): cv = Curve.get_curve('NIST-P256')

print (f"Curve: {cv.name}")
print ("====================")
print (f"G: {cv.generator}")
print (f"Bit size: {cv.size}")
print (f"Order: {cv.order}")
print (f"Field: {cv.field}")
print (f"a: {cv.a}")
if (o==3 or o==4): print (f"b: {cv.d}")
else: print (f"b: {cv.b}")
print ("====================")

G=cv.generator

Q  = k*G

print (f"\n{k}G: {Q}")

print ("\nPoint add (add 1G on)")
P  = G
R=P+Q

print (f"\n{k+1}G: {R}")

print ("\nPoint subtraction (take 1G off)")
R=R-G

print (f"\n{k}G: {R}")

print ("\nThe first 10 points ...")
for i in range(1,11):
  if (o==3 or o==4): 
    try:
      x=cv.x_recover(i)
      print (f"({x},{i})")
    except:
      print (f"(NO POINT,{i})")
  else: 
    y=cv.y_recover(i)
    print (f"({i},{y})")

You can run this here: