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# generates a random Sophie-Germaine prime (p=2*q+1 with q prime) # we use these to make Fp^times as close to a cyclic group of prime order as possible # this means a Pohlig-Hellman attack on Fp^* will have very little benefit def random_sg_prime(bits): while true: p=random_prime(2^bits,2^(bits-1)) if is_prime(2*p+1): break return p def L(a,c,N): z = ceil(exp(c*log(N)^a*log(log(N))^(1-a))) if z%2: z+=1 return ceil(z) def trial_factor(x,B): f=[] lastp=0 while x > 1: p=x.trial_division(B) if p > B: return [] if p == lastp: f[-1][1] += 1 else: f.append([p,1]) x = x.divide_knowing_divisible_by(p) lastp = p return f def dlog(a,b,p): B=ceil(2*L(1/2,1/2,p)) print "smoothness bound is ", B pi=[q for q in primes(B)] pimap=[0 for i in range(0,B)] i=0 for q in primes(B): pimap[q] = i i += 1 n=len(pi) print "factor base has %d elements, searching for relations..."%n M=matrix(ZZ,n+2,n+2,sparse=true) t = cputime() bi = 1/b i = 0 while i < M.nrows(): e = randint(1,p-1) f=trial_factor(ZZ(a^e*bi),B) if f and f[-1][0] < B: for q in f: M[i,pimap[q[0]]]=q[1] M[i,n]=1; M[i,n+1]=e i += 1 print "Found %d relations in %.2f s, attempting to solve system"%(i,cputime()-t) q = ZZ((p-1)/2) print M Mq=M.change_ring(GF(q)).echelon_form() M2=M.change_ring(GF(2)).echelon_form() print Mq print M2 i = n; j = n while not Mq[i,n]: i-=1 while not M2[j,n]: j-=1 print i,j,n return ((q+1)*ZZ(Mq[i,n+1])+q*ZZ(M2[j,n+1]))%(2*q)
%time p=2*random_sg_prime(50)+1 print p F=GF(p) a=F.random_element() while a.multiplicative_order() != p-1: a=F.random_element() x=randint(1,p-1) b=a^x print b, "=", a, "^", x y=dlog(a,b,p) print "dlog is", y assert x==y
1004098078754747 83469179667162 = 970128661762077 ^ 666556705276658 smoothness bound is 508 factor base has 96 elements, searching for relations... Found 98 relations in 7.24 s, attempting to solve system
WARNING: Output: 153860 truncated by MAX_STDOUT_SIZE to 40000. Type 'smc?' to learn how to raise the output limit.
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