Modular Factorial
December 13, 2013
This is a trick question. There is no better algorithm, regardless whether the modulus is prime or composite, than to calculate the modulus after each multiplication:
(define (mod-fact n m)
(let loop ((k 2) (p 1))
(if (< n k) p
(loop (+ k 1) (modulo (* p k) m)))))
If it was possible to compute n! (mod m) in polynomial time, then it would be simple to factor integers in polynomial time: Given an integer m, the smallest f such that gcd(f! (mod m), m) > 1 is the smallest prime factor of m. For every n > f, every gcd(n! (mod m), m) is greater than 1, so we can use binary search to find f, reducing the task of factoring integers to the task of computing n! (mod m). Of course, we don’t do that because there is no polynomial-time algorithm to compute modular factorizations.
> (mod-fact 1000000 1001001779)
744950559
You might be amused to search for this task on the internet and see the complicated solutions that people have developed in the name of efficiency. You can run the program at http://programmingpraxis.codepad.org/HZwVjALJ.
In Python. I used a few small optimizations. Of course, if n >= p (or k) than the factorial will be zero. If the mod is not for a prime, than time can be saved by checking the intermediate result. There is no point of continuing, if the intermediate result is zero. The functions return the factorial plus the number of multiplications done. For the case shown below the number of calculations can be much less than n-1 if the end result is zero.
def fac_mod(n, p): multiplications = 0 if n >= p: return 0 f = 1 for i in xrange(1, n + 1): f = f * i % p multiplications += 1 return f, multiplications def fac_mod2(n, k): multiplications = 0 if n >= k: return 0 f = 1 for i in xrange(1, n + 1): f = f * i % k multiplications += 1 if f == 0: break return f, multiplications k = 1000005 with Timer(1): print fac_mod(1000000, k) with Timer(1): print fac_mod2(1000000, k) # (0, 1000000) # elapsed time: 135 msec # (0, 409) # elapsed time: 81.6 usecHaskell:
There is a way to reduce the multiplications by factoring n!
eg; 100!
2: 50+25+12+6+3+1 = 97 = 64 + 32 + 1 –> M=8
3: 33+11+3+1= 48 = 32 + 16 –> M= 6 TOT=15
5: 20 +4 = 24 =16 +8 –> M=5, tot = 21
7 14+2=16 -> M=4 , tot =26
11 9 M=4, tot = 31
13 7 M=6 tot = 38
17 5 M=3 tot= 42
19 5 M=3 , tot = 46
23 4 M=2 , tot = 49
29 3 M=2, tot = 52
31 3 M=2 , tot = 55
37 2 M=1,tot = 57
41 2 M=1,tot =59
43 2 M=1,tot=61
47 2 M=1, tot=63
53 1 M=0, tot=64
59 1 M=0, tot=65
61 1 M=0, tot=66
67 1 M=0, tot=67
71 1 M=0, tot=68
73 1 M=0, tot=69
79 1 M=0, tot=70
83 1 M=0, tot=71
89 1 M=0, tot=72
93 1 M=0, tot=73
97 1 M=0, tot=74
so by factoring you can reduce the amount of modulo multiplications , on the other hand you have to do a lot divisions to calculate the factors
Haskell, similar to @svenningsson, but incorporating a few optimizations from @Paul:
facMod :: Integer -> Integer -> Integer facMod n m = loop n 1 where loop 0 acc = acc loop k acc | k >= m = 0 | acc == 0 = 0 | otherwise = loop (k - 1) ((acc * k) `mod` m) main :: IO () main = print $ 1000000 `facMod` 1001001779Regarding the discussion on the second page: the last time I checked, the complexity class of integer factoring was an open problem. It’s certainly known to be in NP, but I don’t think it’s been proven NP-complete, so it’s conceivable that it is in P even if P≠NP.
I didn’t say that quite right, did I. Perhaps it would be better if I said we don’t do that “because it is not possible to compute n! (mod m) in polynomial time.”