def madd(p1,p2,A,B,p):
"""Adds p1 and p2 on a Montgomery curve"""
if B*(A**2 - 4) == 0 % p: #checks Montgomery requirements
print("Singular Curve")
return
if math.isinf(p1[0]) and math.isinf(p2[0]):
return math.inf,1 #O + O = O
if math.isinf(p1[0]): return p2 #O + p2 = p2
if math.isinf(p2[0]): return p1 #p1 + O = p1
else:
x1,y1 = p1 #assigns coordinates
x2,y2 = p2 #assigns coordinates
if B*y1**2 % p != (x1**3 + A*x1**2 + x1) % p: #checks if point is on curve
return False #otherwise reject
if B*y2**2 % p != (x2**3 + A*x2**2 + x2) % p: #checks if point is on curve
return False #otherwise reject
if x1==x2 and -y2 % p == y1:
return math.inf,1 #(x1,y1) + (x1,-y1) = O
elif x1==x2:
#point doubling
sl = ((3 * x1**2 + 2*A*x1 + 1)*mminv(2*B*y1,p)) % p #slope
x = (B*(sl**2) - 2*x1 - A) % p
return x, (sl*(x1 - x) - y1) % p
else:
#point addition
sl = (y2-y1)*mminv(x2-x1,p) % p #slope
x = (B*(sl**2) - x1 - x2 - A) % p
return x, (sl*(x1 - x) - y1) % p
def w2check(a, b, p):
"""Checks for point of order two on WNF curve"""
"""Returns x coordinate alpha if exists"""
dis = -16 * (4 * a**3 + 27 * b**2) % p #calculates discriminant
ldis = legendres(dis, p) #finds Legendre Symbol of discriminant
if dis == 0:
print("Discriminant = 0") #rejects if discriminant is 0
return
if ldis % 2 == 1:
#1 root if -1, 3 if 1, doing both cases here
r = ((b**2) * mminv(4, p) +
(a**3) * mminv(27, p)) % p #using cubic formulae
rt = modroot(r, p) #sqrt of r
if not rt:
print(
"No transformation") #if sqrt does not exist, no alpha exists
return
else:
rt = rt[0] #assign one root to use in algebra
bf = -b * mminv(2, p) % p #cubic formulae
alpha = (modcube(bf + rt, p) + modcube(bf - rt, p)) #cubic formulae
if (alpha**3 + a * alpha + b) % p == 0:
return alpha % p #double checking value of alpha
else:
print("error") #reject otherwise, output alpha for error checking
return alpha % p
else:
print("No point of order two")
return
def wadd(p1,p2,a,b,p):
"""Adds p1 and p2 on a WNF curve"""
if math.isinf(p1[0]) and math.isinf(p2[0]):
return math.inf,1 #O + O = O
if math.isinf(p1[0]): return p2 #O + p2 = p2
if math.isinf(p2[0]): return p1 #p1 + O = p1
else:
x1,y1 = p1 #assigns coordinates
x2,y2 = p2 #assigns coordinates
if y1**2 % p != (x1**3 + a*x1 + b) % p: #checks if point is on curve
return False #otherwise reject
if y2**2 % p != (x2**3 + a*x2 + b) % p: #checks if point is on curve
return False #otherwise reject
if x1==x2 and -y2 % p == y1:
return math.inf,1 #(x1,y1) + (x1,-y1) = O
elif x1==x2:
#point doubling
sl = ((3 * x1**2 + a)*mminv(2*y1,p)) % p #slope
x = (sl**2 - x1 - x2) % p
return x, -(sl * (x - x1) + y1) % p
else:
#point addition
sl = (y2-y1)*mminv(x2-x1,p) % p #slope
x = (sl**2 - x1 - x2) % p
return x, -(sl * (x - x1) + y1) % p
def pmtoe(p1, A, B, p):
"""Transforms a Montgomery point to a twisted Edwards point"""
start = time.process_time() * 1000
if A < 3 % p and A > -2 % p: #checks requirements on A for curve mapping
print("A in {-2,2}") #if fails, reject
return False
elif B == 0 % p: #checks requirements on B for curve mapping
print("B = 0") #if fails, reject
return False
x1, y1 = p1 #assigns coordinates
if math.isinf(x1): return 0, 1 #inf -> (0,1)
if B * y1**2 % p != (x1**3 + A * x1**2 +
x1) % p: #checks if point is on curve
return False #otherwise reject
#exceptional points
elif y1 == 0 % p:
print(y1)
if x1 == 0 % p: return 0, -1 % p #(0,0) -> (0,-1)
elif 2 * x1 % p == (-A + modroot((A + 2) * (A - 2), p)[0]) % p:
return math.inf, 2 #point of order two on M -> inf ('positive' root)
elif 2 * x1 % p == (-A - modroot((A + 2) * (A - 2), p)[0]) % p:
return math.inf, -2 #point of order two on M -> inf ('negative' root)
else:
return
elif x1 == -1 % p:
if y1 == modroot((A - 2) * mminv(B, p), p):
return math.inf, 4 #point of order four on M -> inf ('positive' root)
elif y1 == -modroot((A - 2) * mminv(B, p), p) % p:
return math.inf, -4 #point of order four on M -> inf ('negative' root)
#general points
else:
print(time.process_time() * 1000 - start)
return x1 * mminv(y1, p) % p, (x1 - 1) * mminv(x1 + 1, p) % p
def petow(p1, a, d, p):
"""Transforms a twisted Edwards point to a Weierstrass point"""
if a == d: #checks curve mapping requirements
print("a=d") #if fails, reject
return
elif a == 0 or d == 0:
print("Must be non zero")
return
x1, y1 = p1 #assigns coordinates
#exceptional points
if math.isinf(x1):
#computes inf seperately, since may be (inf,4) or (inf,2)
p2 = petom(p1, a, d,
p) #computes point at infinity -> point on montgomery
A, B = (2 * (a + d) * mminv(a - d, p)) % p, (
4 * mminv(a - d, p)) % p #calculates A and B
p3 = pmtow(p2, A, B, p) #computes point on montgomery -> point on W
return p3
if (a * x1**2 + y1**2) % p != (1 + d * (x1**2) *
(y1**2)) % p: #checks point is on curve
return False #otherwise reject
elif x1 % p == 0 and y1 % p == 1:
return math.inf, 1 #(0,1) -> inf -> inf
else:
p2 = petom(p1, a, d, p) #computes point on Montgomery
A, B = (2 * (a + d) * mminv(a - d, p)) % p, (
4 * mminv(a - d, p)) % p #calculates A and B
p3 = pmtow(p2, A, B, p) #computes point on W
return p3
def mtow(A, B, p):
"""Converts Montgomery curve to WNF"""
#conversion always exists - no need to check
start = time.process_time() * 1000
a = ((3 - A**2) * mminv(3 * B**2, p)) % p #calculates a
b = ((2 * A**3 - 9 * A) * mminv(27 * B**3, p)) % p #calculates b
print("y**2 = x**3 + ", a, "x +", b) #prints curve equation
print("An Isomorphism!") #all existing W to M mappings are isomorphisms
print(time.process_time() * 1000 - start)
return a, b
def pmtow(p1, A, B, p):
"""Transforms a Montgomery point to a Weierstrass point"""
start = time.process_time() * 1000
x1, y1 = p1 #assigns coordinates
if B * y1**2 % p != (x1**3 + A * x1**2 +
x1) % p: #checks if point is on curve
return False #otherwise reject
#exceptional points
if math.isinf(x1):
print(time.process_time() * 1000 - start)
return math.inf, 1 #inf -> inf
#general points
else:
print(time.process_time() * 1000 - start)
return (x1 * mminv(B, p) + A * mminv(3 * B, p)) % p, y1 * mminv(B,
p) % p
def etom(a, d, p):
"""Converts twisted Edwards to Montgomery"""
"""Checks for conversion"""
#start = time.process_time() *1000
if a == d: #checks parameters distinct
print("a=d") #otherwise reject
return
elif a == 0 or d == 0: #checks parameters non zero
print("Must be non zero") #otherwise reject
return
A = (2 * (a + d) * mminv(a - d, p)) % p #calculates A
B = (4 * mminv(a - d, p)) % p #calculates B
print(B, "y**2 = x**3 +", A, "x**2 + x") #prints curve equation
if legendres(a, p) == legendres(d, p):
print("Not an Isomorphism :("
) #not isomorphic if a & d both square/not square
elif legendres(a, p) == 1:
print("An Isomorphism!") #isomorphic if a square, d not square
#print(time.process_time()*1000 - start)
return A, B
def mtoe(A, B, p):
"""Converts Montgomery curve to twisted Edwards"""
"""Checks for conversion"""
#start = time.process_time()*1000
if A < 3 % p and A > -2 % p: #checks requirement of A
print("A in {-2,2}") #if fails, reject
return
elif B == 0 % p: #checks B non zero
print("B = 0") #otherwise reject
return
a = ((A + 2) * mminv(B, p)) % p #calculates a
d = ((A - 2) * mminv(B, p)) % p # calculates d
print(a, "x**2 + y**2 = 1 + ", d, "x**2y**2") #prints curve equation
if legendres(a, p) == legendres(d, p):
print("Not an Isomorphism :("
) #not isomorphic if a & d both square/not square
elif legendres(a, p) == 1:
print("An Isomorphism!") #isomorphic if a square, d not square
elif legendres(a, p) == -1:
print("Not an Isomorphism :(") #not isomorphic if a not square
# print(time.process_time()*1000 - start)
return a, d
def eadd(p1,p2,a,d,p):
"""
Adds p1 and p2 on a twsited Edwards curve
We assume that there exists a pt of order 4, so identity is (0,1)
"""
if d*(1 - d) == 0 % p: #checks Edwards requirements
print("Singular Curve")
return
x1,y1 = p1
x2,y2 = p2 #check if in curve here
if (a*x1**2 + y1**2) % p != (1 + d*(x1**2)*(y1**2)) % p: #checks point is on the curve
return False #otherwise reject
if (a*x2**2 + y2**2) % p != (1 + d*(x2**2)*(y2**2)) % p: #checks point is on the curve
return False #otherwise reject
if p1 == (0,1) and p2 == (0,1):
return 0,1 #O + O = O
if p1 == (0,1): return p2 #O + p2 = p2
if p2 == (0,1): return p1 #p1 + O = p2
if y1==y2 and -x2 % p == x1: #(x1, y1) + (-x1, y1) = O
return 0,1
else:
#point addition
dxy = d*x1*x2*y1*y2 % p
return (x1*y2 + y1*x2)*mminv(1+dxy,p) % p, (y1*y2 - a*x1*x2)*mminv(1-dxy,p) % p
def petom(p1, a, d, p):
"""Transforms a twisted Edwards point to a Montgomery point"""
if a == d: #checks curve mapping requirements
print("a=d") #if fails, reject
return False
elif a == 0 or d == 0:
print("Must be non zero")
return False
x1, y1 = p1 #assigns coordinates
#exceptional points
if math.isinf(x1):
A = (2 * (a + d) * mminv(a - d, p)) % p #calculates A
B = (4 * mminv(a - d, p)) % p #calculates B
if y1 % p == 2:
x = (-A + modroot((A + 2) * (A - 2), p)[0]) * mminv(
2, p) % p #point of order two on M ('positive' root)
return x, 0
elif y1 % p == -2 % p:
x = (-A - modroot((A + 2) * (A - 2), p)[0]) * mminv(
2, p) % p #point of order two on M ('negative' root)
return x, 0
elif y1 % p == 4:
y = modroot((A - 2) * mminv(B, p),
p) #point of order four on M ('positive' root)
return -1, y
elif y1 % p == -4 % p:
y = -modroot((A - 2) * mminv(B, p),
p) % p #point of order four on M ('negative' root)
return -1, y
else:
return #no other points at infinity should exist on a twisted Edwards curve
if (a * x1**2 + y1**2) % p != (1 + d * (x1**2) *
(y1**2)) % p: #checks point is on the curve
return False #otherwise reject
elif x1 % p == 0 and y1 % p == 1:
return math.inf, 1 #(0,1) -> inf
elif x1 % p == 0 and y1 % p == -1 % p:
return 0, 0 #(0,-1) -> (0,0)
#general points
else:
return (1 + y1) * mminv(1 - y1, p) % p, (1 + y1) * mminv(
x1 * (1 - y1), p) % p
def wtom(a, b, p):
"""Converts WNF curve to Montgomery"""
"""Checks for conversion"""
start = time.process_time() * 1000
alpha = w2check(a, b, p) #finds point of order two on W
if not alpha:
print("No conversion") #if no such point, reject
return
else:
if not modroot(3 * alpha**2 + a, p): #checks if square root exists
print("No conversion") #otherwise, reject
return
B = mminv(max(modroot(3 * alpha**2 + a, p)), p) #calulates B
A = 3 * alpha * B % p #calulates A
print(B, "y**2 = x**3 +", A, "x**2 + x") #prints curve equation
print(
"An Isomorphism!") #all existing W to M mappings are isomorphisms
print(time.process_time() * 1000 - start)
return A, B, alpha
def pwtom(p1, a, b, p, A, B):
"""Transforms a Weierstrass point to a Montgomery point"""
x1, y1 = p1 #assigns coordinates
if math.isinf(p1[0]): return math.inf, 1 #inf -> inf
if y1**2 % p != (x1**3 + a * x1 + b) % p: #checks if point is on curve
return False #otherwise reject
if not A: #if user does not input A, calulates it
s, alpha = wtom(a, b, p)[1], wtom(a, b,
p)[2] #finds B, alpha from wtom
if not alpha: #checks if a point of order two exists
print("No conversion") #otherwise reject
return
else:
alpha = A * mminv(3 * B,
p) % p #if user inputs A,B calculate alpha from them
s = B #as above
#exceptional points
if p1 == (alpha, 0):
return 0, 0 #if point has order two, map to 0,0
#general points
else:
return s * (x1 - alpha) % p, s * y1 % p
