-rw-r--r-- 2365 lib25519-20240928/crypto_mGnP/ed25519/ref10/point26.py raw
#!/usr/bin/env python3
q = 2**255 - 19
l = 2**252 + 27742317777372353535851937790883648493
def expmod(b,e,m):
if e == 0: return 1
t = expmod(b,e//2,m)**2 % m
if e & 1: t = (t*b) % m
return t
def inv(x):
return expmod(x,q-2,q)
d = -121665 * inv(121666)
I = expmod(2,(q-1)//4,q)
def xrecover(y):
xx = (y*y-1) * inv(d*y*y+1)
x = expmod(xx,(q+3)//8,q)
if (x*x - xx) % q != 0: x = (x*I) % q
if x % 2 != 0: x = q-x
return x
def isoncurve(P):
x,y = P
return (-x*x+y*y-1-d*x**2*y**2) % q == 0
By = 4 * inv(5)
Bx = xrecover(By)
B = Bx%q,By%q
assert isoncurve(B)
def edwards(P,Q):
x1,y1 = P
x2,y2 = Q
x3 = (x1*y2+x2*y1) * inv(1+d*x1*x2*y1*y2)
y3 = (y1*y2+x1*x2) * inv(1-d*x1*x2*y1*y2)
return x3%q,y3%q
def scalarmult(P,e):
assert e >= 0
if e == 0: return 0,1
Q = scalarmult(P,e//2)
Q = edwards(Q,Q)
if e & 1: Q = edwards(P,Q)
return Q
assert scalarmult(B,l) == (0,1)
Cy = 26 # minimum integer >1 giving C on curve and order l
Cx = xrecover(Cy)
C = Cx%q,Cy%q
assert isoncurve(C)
assert scalarmult(C,l) == (0,1)
# sanity checks:
# C is not a small multiple of B
# namely (0,1) or (+-...,4/5) or (+-...,-985469/985549) or ...
# and B is not a small multiple of C
# namely (0,1) or (+-...,10) or (+-...,-596219233/596219333) or ...
# and similarly for other small relations: e.g., 2B = 3C
# could do more serious discrete-log check
def hassmallrelation(A,B):
jAyset = set()
jA = 0,1
for j in range(1,1000):
jA = edwards(jA,A)
jAyset.add(jA[1])
jB = 0,1
for j in range(1,1000):
jB = edwards(jB,B)
if jB[1] in jAyset: return True
return False
testy = 333922976373947162567639162515495974004 * inv(333933611346170188668120256727948534525)
testx = xrecover(testy)
test = testx%q,testy%q
assert hassmallrelation(test,B)
assert not hassmallrelation(C,B)
def radix255(x):
x = x % q
if x + x > q: x -= q
x = [x,0,0,0,0,0,0,0,0,0]
bits = [26,25,26,25,26,25,26,25,26,25]
for i in range(9):
carry = (x[i] + 2**(bits[i]-1)) // 2**bits[i]
x[i] -= carry * 2**bits[i]
x[i + 1] += carry
result = ""
for i in range(9):
result = result+str(x[i])+","
result = result+str(x[9])
return result
print('static const fe point26_x = {')
print('//',C[0])
print(radix255(C[0]))
print('} ;')
print('static const fe point26_y = {')
print('//',C[1])
print(radix255(C[1]))
print('} ;')