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verify.sage
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verify.sage
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import os
import sys
from errno import ENOENT, EEXIST
from sortedcontainers import SortedSet
def readfile(fn):
fd = open(fn,'r')
r = fd.read()
fd.close()
return r
def writefile(fn,s):
fd = open(fn,'w')
fd.write(s)
fd.close()
def expand2(n):
s = ""
while n != 0:
j = 16
while 2**j < abs(n): j += 1
if 2**j - abs(n) > abs(n) - 2**(j-1): j -= 1
if abs(abs(n) - 2**j) > 2**(j - 10):
if n > 0:
if s != "": s += " + "
s += str(n)
else:
s += " - " + str(-n)
n = 0
elif n > 0:
if s != "": s += " + "
s += "2^" + str(j)
n -= 2**j
else:
s += " - 2^" + str(j)
n += 2**j
return s
def requirement(fn,istrue):
writefile(fn,str(istrue) + '\n')
return istrue
def verify():
try:
os.mkdir('proof')
except OSError as e:
if e.errno != EEXIST: raise
try:
s = set(map(Integer, readfile('primes').split()))
except IOError as e:
if e.errno != ENOENT: raise
s = set()
needtofactor = SortedSet()
V = SortedSet() # distinct verified primes
verify_primes(V, s, needtofactor)
verify_pass(V, needtofactor)
old = V
needtofactor.update(V)
while len(needtofactor) > len(old):
k = len(needtofactor) - len(old)
sys.stdout.write('Factoring %d integer%s' % (k, '' if k == 1 else 's'))
sys.stdout.flush()
for x in needtofactor:
if x not in old:
for (y, z) in factor(x):
s.add(y)
sys.stdout.write('.')
sys.stdout.flush()
print('')
old = needtofactor.copy()
verify_primes(V, s, needtofactor)
writefile('primes', '\n'.join(map(str, s)) + '\n')
writefile('verify-primes', '<html><body>\n' +
''.join(('2\n' if v == 2 else
'<a href=proof/%s.html>%s</a>\n' % (v,v)) for v in V) +
'</body></html>\n')
verify_pass(V, needtofactor)
def verify_primes(V, s, needtofactor):
for n in sorted(s):
if not n.is_prime() or n in V: continue
needtofactor.add(n-1)
if n == 2:
V.add(n)
continue
for trybase in primes(2,10000):
base = Integers(n)(trybase)
if base^(n-1) != 1: continue
proof = 'Primality proof for n = %s:\n' % n
proof += '<p>Take b = %s.\n' % base
proof += '<p>b^(n-1) mod n = 1.\n'
f = factor(1)
for v in reversed(V):
if f.prod()^2 <= n:
if n % v == 1:
u = base^((n-1)/v)-1
if u.is_unit():
if v == 2:
proof += '<p>2 is prime.\n'
else:
proof += '<p><a href=%s.html>%s is prime.</a>\n' % (v,v)
proof += '<br>b^((n-1)/%s)-1 mod n = %s, which is a unit, inverse %s.\n' % (v,u,1/u)
f *= factor(v)^(n-1).valuation(v)
if f.prod()^2 <= n: continue
if n % f.prod() != 1: continue
proof += '<p>(%s) divides n-1.\n' % f
proof += '<p>(%s)^2 > n.\n' % f
proof += "<p>n is prime by Pocklington's theorem.\n"
proof += '\n'
writefile('proof/%s.html' % n,proof)
V.add(n)
break
def verify_pass(V, needtofactor):
p = Integer(readfile('p'))
k = GF(p)
kz.<z> = k[]
l = Integer(readfile('l'))
x0 = Integer(readfile('x0'))
y0 = Integer(readfile('y0'))
x1 = Integer(readfile('x1'))
y1 = Integer(readfile('y1'))
shape = readfile('shape').strip()
rigid = readfile('rigid').strip()
safefield = True
safeeq = True
safebase = True
saferho = True
safetransfer = True
safedisc = True
saferigid = True
safeladder = True
safetwist = True
safecomplete = True
safeind = True
pstatus = 'Unverified'
if not p.is_prime(): pstatus = 'False'
needtofactor.add(p)
if p in V: pstatus = 'True'
if pstatus != 'True': safefield = False
writefile('verify-pisprime',pstatus + '\n')
pstatus = 'Unverified'
if not l.is_prime(): pstatus = 'False'
needtofactor.add(l)
if l in V: pstatus = 'True'
if pstatus != 'True': safebase = False
writefile('verify-lisprime',pstatus + '\n')
writefile('expand2-p','= %s\n' % expand2(p))
writefile('expand2-l','<br>= %s\n' % expand2(l))
writefile('hex-p',hex(p) + '\n')
writefile('hex-l',hex(l) + '\n')
writefile('hex-x0',hex(x0) + '\n')
writefile('hex-x1',hex(x1) + '\n')
writefile('hex-y0',hex(y0) + '\n')
writefile('hex-y1',hex(y1) + '\n')
gcdlpis1 = gcd(l,p) == 1
safetransfer &= requirement('verify-gcdlp1',gcdlpis1)
writefile('verify-movsafe','Unverified\n')
writefile('verify-embeddingdegree','Unverified\n')
if gcdlpis1 and l.is_prime():
u = Integers(l)(p)
d = l-1
needtofactor.add(d)
for v in V:
while d % v == 0: d /= v
if d == 1:
d = l-1
for v in V:
while d % v == 0:
if u^(d/v) != 1: break
d /= v
safetransfer &= requirement('verify-movsafe',(l-1)/d <= 100)
writefile('verify-embeddingdegree','<font size=1>%s</font><br>= (l-1)/%s\n' % (d,(l-1)/d))
t = p+1-l*round((p+1)/l)
if l^2 > 16*p:
writefile('verify-trace','%s\n' % t)
f = factor(1)
d = (p+1-t)/l
needtofactor.add(d)
for v in V:
while d % v == 0:
d //= v
f *= factor(v)
writefile('verify-cofactor','%s\n' % f)
else:
writefile('verify-trace','Unverified\n')
writefile('verify-cofactor','Unverified\n')
D = t^2-4*p
needtofactor.add(D)
for v in V:
while D % v^2 == 0: D /= v^2
if prod([v for v in V if D % v == 0]) != -D:
writefile('verify-disc','Unverified\n')
writefile('verify-discisbig','Unverified\n')
safedisc = False
else:
f = -prod([factor(v) for v in V if D % v == 0])
if D % 4 != 1:
D *= 4
f = factor(4) * f
Dbits = (log(-D)/log(2)).numerical_approx()
writefile('verify-disc','<font size=1>%s</font><br>= <font size=1>%s</font><br>≈ -2^%.1f\n' % (D,f,Dbits))
safedisc &= requirement('verify-discisbig',D < -2^100)
pi4 = 0.78539816339744830961566084581987572105
rho = log(pi4*l)/log(4)
writefile('verify-rho','%.1f\n' % rho)
saferho &= requirement('verify-rhoabove100',rho.numerical_approx() >= 100)
twistl = 'Unverified'
d = p+1+t
needtofactor.add(d)
for v in V:
while d % v == 0: d /= v
if d == 1:
d = p+1+t
for v in V:
if d % v == 0:
if twistl == 'Unverified' or v > twistl: twistl = v
writefile('verify-twistl','%s\n' % twistl)
writefile('verify-twistembeddingdegree','Unverified\n')
writefile('verify-twistmovsafe','Unverified\n')
if twistl == 'Unverified':
writefile('hex-twistl','Unverified\n')
writefile('expand2-twistl','Unverified\n')
writefile('verify-twistcofactor','Unverified\n')
writefile('verify-gcdtwistlp1','Unverified\n')
writefile('verify-twistrho','Unverified\n')
safetwist = False
else:
writefile('hex-twistl',hex(twistl) + '\n')
writefile('expand2-twistl','<br>= %s\n' % expand2(twistl))
f = factor(1)
d = (p+1+t)/twistl
needtofactor.add(d)
for v in V:
while d % v == 0:
d //= v
f *= factor(v)
writefile('verify-twistcofactor','%s\n' % f)
gcdtwistlpis1 = gcd(twistl,p) == 1
safetwist &= requirement('verify-gcdtwistlp1',gcdtwistlpis1)
movsafe = 'Unverified'
embeddingdegree = 'Unverified'
if gcdtwistlpis1 and twistl.is_prime():
u = Integers(twistl)(p)
d = twistl-1
needtofactor.add(d)
for v in V:
while d % v == 0: d /= v
if d == 1:
d = twistl-1
for v in V:
while d % v == 0:
if u^(d/v) != 1: break
d /= v
safetwist &= requirement('verify-twistmovsafe',(twistl-1)/d <= 100)
writefile('verify-twistembeddingdegree',"<font size=1>%s</font><br>= (l'-1)/%s\n" % (d,(twistl-1)/d))
rho = log(pi4*twistl)/log(4)
writefile('verify-twistrho','%.1f\n' % rho)
safetwist &= requirement('verify-twistrhoabove100',rho.numerical_approx() >= 100)
precomp = 0
joint = l
needtofactor.add(p+1-t)
needtofactor.add(p+1+t)
for v in V:
d1 = p+1-t
d2 = p+1+t
while d1 % v == 0 or d2 % v == 0:
if d1 % v == 0: d1 //= v
if d2 % v == 0: d2 //= v
# best case for attack: cyclic; each power is usable
# also assume that kangaroo is as efficient as rho
if v + sqrt(pi4*joint/v) < sqrt(pi4*joint):
precomp += v
joint /= v
rho = log(precomp + sqrt(pi4 * joint))/log(2)
writefile('verify-jointrho','%.1f\n' % rho)
safetwist &= requirement('verify-jointrhoabove100',rho.numerical_approx() >= 100)
x0 = k(x0)
y0 = k(y0)
x1 = k(x1)
y1 = k(y1)
if shape in ('edwards', 'tedwards'):
d = Integer(readfile('d'))
a = 1
if shape == 'tedwards':
a = Integer(readfile('a'))
writefile('verify-shape','Twisted Edwards\n')
writefile('verify-equation','%sx^2+y^2 = 1%+dx^2y^2\n' % (a, d))
if a == 1:
writefile('verify-shape','Edwards\n')
writefile('verify-equation','x^2+y^2 = 1%+dx^2y^2\n' % d)
a = k(a)
d = k(d)
elliptic = a*d*(a-d)
level0 = a*x0^2+y0^2-1-d*x0^2*y0^2
level1 = a*x1^2+y1^2-1-d*x1^2*y1^2
if shape == 'montgomery':
writefile('verify-shape','Montgomery\n')
A = Integer(readfile('A'))
B = Integer(readfile('B'))
equation = '%sy^2 = x^3<wbr>%+dx^2+x' % (B,A)
if B == 1:
equation = 'y^2 = x^3<wbr>%+dx^2+x' % A
writefile('verify-equation',equation + '\n')
A = k(A)
B = k(B)
elliptic = B*(A^2-4)
level0 = B*y0^2-x0^3-A*x0^2-x0
level1 = B*y1^2-x1^3-A*x1^2-x1
if shape == 'shortw':
writefile('verify-shape','short Weierstrass\n')
a = Integer(readfile('a'))
b = Integer(readfile('b'))
writefile('verify-equation','y^2 = x^3<wbr>%+dx<wbr>%+d\n' % (a,b))
a = k(a)
b = k(b)
elliptic = 4*a^3+27*b^2
level0 = y0^2-x0^3-a*x0-b
level1 = y1^2-x1^3-a*x1-b
writefile('verify-elliptic',str(elliptic) + '\n')
safeeq &= requirement('verify-iselliptic',elliptic != 0)
safebase &= requirement('verify-isoncurve0',level0 == 0)
safebase &= requirement('verify-isoncurve1',level1 == 0)
if shape in ('edwards', 'tedwards'):
A = 2*(a+d)/(a-d)
B = 4/(a-d)
x0,y0 = (1+y0)/(1-y0),((1+y0)/(1-y0))/x0
x1,y1 = (1+y1)/(1-y1),((1+y1)/(1-y1))/x1
shape = 'montgomery'
if shape == 'montgomery':
a = (3-A^2)/(3*B^2)
b = (2*A^3-9*A)/(27*B^3)
x0,y0 = (x0+A/3)/B,y0/B
x1,y1 = (x1+A/3)/B,y1/B
shape = 'shortw'
try:
E = EllipticCurve([a,b])
numorder2 = 0
numorder4 = 0
for P in E(0).division_points(4):
if P != 0 and 2*P == 0:
numorder2 += 1
if 2*P != 0 and 4*P == 0:
numorder4 += 1
writefile('verify-numorder2',str(numorder2) + '\n')
writefile('verify-numorder4',str(numorder4) + '\n')
completesingle = False
completemulti = False
if numorder4 == 2 and numorder2 == 1:
# complete edwards form, and montgomery with unique point of order 2
completesingle = True
completemulti = True
# should extend this to allow complete twisted hessian
safecomplete &= requirement('verify-completesingle',completesingle)
safecomplete &= requirement('verify-completemulti',completemulti)
safecomplete &= requirement('verify-ltimesbase1is0',l * E([x1,y1]) == 0)
writefile('verify-ltimesbase1',str(l * E([x1,y1])) + '\n')
writefile('verify-cofactorbase01',str(((p+1-t)//l) * E([x0,y0]) == E([x1,y1])) + '\n')
except:
writefile('verify-numorder2','Unverified\n')
writefile('verify-numorder4','Unverified\n')
writefile('verify-ltimesbase1','Unverified\n')
writefile('verify-cofactorbase01','Unverified\n')
safecomplete = False
montladder = False
for r,e in (z^3+a*z+b).roots():
if (3*r^2+a).is_square():
montladder = True
safeladder &= requirement('verify-montladder',montladder)
indistinguishability = False
elligator2 = False
if (p+1-t) % 2 == 0:
if b != 0:
indistinguishability = True
elligator2 = True
safeind &= requirement('verify-indistinguishability',indistinguishability)
writefile('verify-ind-notes','Elligator 2: %s.\n' % ['No','Yes'][elligator2])
saferigid &= (rigid == 'fully rigid' or rigid == 'somewhat rigid')
safecurve = True
safecurve &= requirement('verify-safefield',safefield)
safecurve &= requirement('verify-safeeq',safeeq)
safecurve &= requirement('verify-safebase',safebase)
safecurve &= requirement('verify-saferho',saferho)
safecurve &= requirement('verify-safetransfer',safetransfer)
safecurve &= requirement('verify-safedisc',safedisc)
safecurve &= requirement('verify-saferigid',saferigid)
safecurve &= requirement('verify-safeladder',safeladder)
safecurve &= requirement('verify-safetwist',safetwist)
safecurve &= requirement('verify-safecomplete',safecomplete)
safecurve &= requirement('verify-safeind',safeind)
requirement('verify-safecurve',safecurve)
originaldir = os.open('.',os.O_RDONLY)
for i in range(1,len(sys.argv)):
os.fchdir(originaldir)
os.chdir(sys.argv[i])
verify()