def pollard_pm1(N,B=0):
    if not B: B=ceil(sqrt(N))
    a = Integers(N).random_element()
    b = a
    for ell in primes(B):
        q = 1
        while q < N:
            q *= ell
        b = b^q
        if b == 1:
            return 0
        d = gcd(b.lift()-1,N)
        if d > 1: return d
    return 0
    
def random_unsafe_prime(bits):
    while true:
        a=randint(0,bits)
        b=randint(0,floor((bits-a)/log(3,2)))
        c=randint(0,floor((bits-a-floor(b*log(3,2)))/log(5,2)))
        d=floor((bits-a-floor(b*log(3,2))-floor(c*log(5,2)))/log(7,2))
        p = 2^a*3^b*5^c*7^d+1
        if is_prime(p): return p

factor(random_unsafe_prime(2048)-1)

b = 4096
t=cputime()
p1=random_unsafe_prime(b//2)
p2=random_prime(2^(b//2),proof=True)
print(p1,p2,p1*p2)
print("Generated %d-bit RSA modulus in %.3fs\n"%(b,cputime()-t))
t=cputime()
print(pollard_pm1(p1*p2))
print("Factored %d-bit RSA modulus in %.3fs\n"%(b,cputime()-t))