diff --git a/verify.sage b/verify.sage index d9d9bd8..96caf5e 100644 --- a/verify.sage +++ b/verify.sage @@ -1,5 +1,8 @@ import os import sys +from errno import ENOENT, EEXIST +from sortedcontainers import SortedSet + def readfile(fn): fd = open(fn,'r') @@ -42,6 +45,82 @@ def requirement(fn,istrue): return istrue def verify(): + + try: + s = set(map(Integer, readfile('primes').split())) + except IOError, 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', '\n' + + ''.join(('2\n' if v == 2 else + '%s\n' % (v,v)) for v in V) + + '\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 += '

Take b = %s.\n' % base + proof += '

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 += '

2 is prime.\n' + else: + proof += '

%s is prime.\n' % (v,v) + proof += '
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 += '

(%s) divides n-1.\n' % f + proof += '

(%s)^2 > n.\n' % f + proof += "

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. = k[] @@ -51,7 +130,6 @@ def verify(): x1 = Integer(readfile('x1')) y1 = Integer(readfile('y1')) shape = readfile('shape').strip() - s = readfile('primes') rigid = readfile('rigid').strip() safefield = True @@ -66,48 +144,16 @@ def verify(): safecomplete = True safeind = True - V = [] # distinct verified primes - for line in s.split(): - n = Integer(line) - if not n.is_prime(): continue - if n == 2: - if not n in V: V += [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 += '

Take b = %s.\n' % base - proof += '

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(): - proof += '

%s is prime.\n' % (v,v) - proof += '
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 += '

(%s) divides n-1.\n' % f - proof += '

(%s)^2 > n.\n' % f - proof += "

n is prime by Pocklington's theorem.\n" - proof += '\n' - writefile('../../../proof/%s.html' % n,proof) - if not n in V: V += [n] - break - - writefile('verify-primes',''.join('%s\n' % (v,v) for v in V)) - 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') @@ -130,6 +176,7 @@ def verify(): 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: @@ -146,6 +193,7 @@ def verify(): 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 @@ -156,6 +204,7 @@ def verify(): 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: @@ -178,6 +227,7 @@ def verify(): twistl = 'Unverified' d = p+1+t + needtofactor.add(d) for v in V: while d % v == 0: d /= v if d == 1: @@ -201,6 +251,7 @@ def verify(): writefile('expand2-twistl','
= %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 @@ -214,6 +265,7 @@ def verify(): 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: @@ -231,6 +283,8 @@ def verify(): 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