18.783 Lecture 11 Pollard p-1.ipynb

Kernel: SageMath 9.2

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(512)-1)

2^208 * 3^48 * 5^4 * 7^78

p1=random_unsafe_prime(512)
p2=random_prime(2^512,2^511)
print(p1,p2)
%time print(pollard_pm1(p1*p2))

15233457733737344526540743467037883976191131688801290976111437578683757334559566682309858873270788500743177592868343644160000000000000000000000000000000001 10805811561697681649612317523735012268289842506849046934843189811833967445914708434992672188003562368615706302208602535704643748161444047008827305924614487
15233457733737344526540743467037883976191131688801290976111437578683757334559566682309858873270788500743177592868343644160000000000000000000000000000000001
CPU times: user 3.29 ms, sys: 0 ns, total: 3.29 ms
Wall time: 3.27 ms