I'm trying to perform so kind of calculation. I think you can help me. Task :
Rounding up to the nearest integer that’s congruent to k mod N during halfing by 2
We have curve: The elliptic curve domain parameters over Fp associated with a Koblitz curve secp256k1 are specified by the sextuple T = (p,a,b,G,n,h) where the finite field Fp is defined by:
p = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F
= 2256 - 232 - 29 - 28 - 27 - 26 - 24 - 1
The curve E: y2 = x3+ax+b over Fp is defined by:
a = 0
b = 7
The base point G form is:
G = (Gx,Gy)
Gx = 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798
Gy = 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8
Finally the order n of G :
n = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141
and the cofactor is:
h = 01
Explanation of task:
Example and code:
Code:
p_string = 'FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F'
p = ZZ( '0x' + p_string.replace(' ', ''))
p
E = EllipticCurve( GF(p), [0, 7] )
E
n_string = 'FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141'
n = ZZ( '0x' + n_string.replace(' ', '') )
half_mod_n = 1 / GF(n)(2)
half_mod_n = ZZ(half_mod_n)
#first point on curve and Generator of curve
xP = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
yP = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
P = E.point( (xP, yP) )
#Point no 10 of the curve is multiply Generator and 10
P_10 = P * 10
#Point no 9 of the curve is multiply Generator and 9
P_9 = P * 9
# test halfing Point by 2
P_10_half = P_10 * half_mod_n
print("is P_10 after halfed by 2 is equal P5 : ",(P*5)==P_10_half)
P_9_half = P_9 * half_mod_n
# it is P_9_half = P_4 + P_05
P_4 = P*4
P_05 = P * half_mod_n
print("is P_9 after halfed by 2 is equal P4 + P_05 : ",(P_4+P_05)==P_9_half)
print(" it is possibiliy mult generator with float point P *1.5 : ",P*1.5)
question: how to round Point after halfing to full integer ? example : P_9 / 2 = P*5 not P_4 + P_05?
