biection's blog

By biection, history, 23 months ago, translation, In English

Hello, Codeforces! Today I've faced the following problem: is it possible to calculate N! modulo M (M is prime) in polynomial (relative to length of M) time? I thought it would be pretty popular problem, but I couldn't google anything useful. Does anybody know such algorithm or proofs that it does not exist?

 
 
 
 
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23 months ago, # |
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You can try this algo, if M is not too large.

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    23 months ago, # ^ |
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    Thanks, but I'm just curious if the polynomial algorithm exists.

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23 months ago, # |
Rev. 2   Vote: I like it +8 Vote: I do not like it

Actually, you can calculate $$$n! \bmod p$$$ in $$$O(\sqrt{p} \log p)$$$.

See this paper Linear recurrences with polynomial coefficients and application to integer factorization and Cartier-Manin operator for more details.

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23 months ago, # |
Rev. 3   Vote: I like it 0 Vote: I do not like it

wrong

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23 months ago, # |
  Vote: I like it +31 Vote: I do not like it

polynomial (relative to length of M) time.

if $$$N>=M$$$ then $$$N!$$$ $$$mod$$$ $$$M$$$ $$$=$$$ $$$0$$$.
Else calculate it in $$$O(N)$$$ :P
Am I missing something?

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    23 months ago, # ^ |
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    Maybe your's is not in Length of m complexity.

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      23 months ago, # ^ |
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      Then my questions is what is length of m? Can you please define it?
      Is this no of digits in M?
      Maybe saying $$$O(logM) == O(logM/log10)$$$ would have been better.

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23 months ago, # |
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This is the problem you are describing, maybe with few extra constraints: https://www.spoj.com/problems/BORING/en/

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    23 months ago, # ^ |
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    maybe with few extra constraints
    Extra constraints sometimes make life easier.

    Notice the constraint Abs(N-P) < 10^4.

    My Soln with one unproven lemma.
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    23 months ago, # ^ |
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    If I'm not mistaken this shall be the problem (even though the constrains are solvable with approach suggested by zimpha ==> so not polynomial in length ^_^ )