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p = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
E = EllipticCurve(GF(p), [0, 7])
G = E.point( (0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798,0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8)) # Base point
def egcd(a, b):
if a == 0:
return (b, 0, 1)
else:
g, y, x = egcd(b % a, a)
return (g, x - (b // a) * y, y)
def modinv(a, m):
g, x, y = egcd(a, m)
if g != 1:
raise Exception('modular inverse does not exist')
else:
return x % m
def verify(r, s,z,public_key):
w = int(modinv(s, n))
u1 = int((z * w) % n)
u2 = int((r * w) % n)
D=u1*G + u2*public_key
x,y=D.xy()
x=int(x)
if (r % n) == (x % n):
print( "signature matches")
return 1
else:
print("invalid signature",r,x%n,hex(int(x%n)))
return -1
def sign(privkey,nonce,message):
z=message
nonceG=nonce*G
rx,rye=nonceG.xy()
rx=int(rx)
s=(z+int(rx)*privkey)*modinv(nonce,n)%n
return int(rx), int(s), int(z)
def make_minus(k_minus,r,s,z,pub):
w = int(modinv(s, n)%n)
u1 = int((z * w) % n)
u2 = int((r * w) % n)
D=u1*G + u2*pub
du= D-k_minus*G
x,y=du.xy()
r_new=Integer(x)
#c = E.lift_x(Integer(r)) - k_minus * G
#print("c",du)
s_new = s * modinv(r,n) * r_new % n
h_new = (z - s * k_minus) * modinv(r,n) * r_new % n
return int(r_new),int(s_new),int(h_new)
def make_divide(r,s,z,how_much,pub):
w = int(modinv(s, n)%n)
u1 = int((z * w) % n)
u2 = int((r * w) % n)
D=u1*G + u2*pub
R=D*modinv(how_much,n)
r_new=int(R.xy()[0])
#with the same pubkey
dsa=int(s*modinv(r,n)%n) # dsa = s * 1/r
dma= int(z*modinv(r,n)%n) # dma = z * 1/m
dsa = dsa*how_much *r_new%n
m=dma*r_new%n
s_new=dsa
z_new=m
return r_new,int(s_new),int(z_new)
# input values:
private=100
pub=private*G
nonce=200
message=11111111111111111111111111111111
# first transaction
r,s,z = sign(private,nonce,message)
#test 1
#test for plus and minus nonce in transaction
k_minus = 100
#output must be nonce=100 with the same pubkey
r1,s1,z1=make_minus(k_minus,r,s,z,pub)
#verify new transaction
verify(r1,s1,z1,pub)
k=(r1*private+z1)*modinv(s1,n)%n
print("new nonce==",k)
# test 2
#test for divide nonce in transaction
#output must be nonce=100 with the same pubkey
how_much=2 #we divide by 2 nonce in transaction
r2,s2,z2=make_divide(r,s,z,how_much,pub)
#verify new transaction
verify(r2,s2,z2,pub)
k=(r2*private+z2)*modinv(s2,n)%n
print("new nonce==",k)
