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Re: Modulo order

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  • joseph_osiecki
    Can someone give me some more details on working out the order of an element mod p when we have a factorization of p-1? Please use the following example p =
    Message 1 of 6 , Jul 6, 2004
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      Can someone give me some more details on working out the order of an
      element mod p when we have a factorization of p-1?
      Please use the following example p = 176963110569601 =
      2^7*3*5^2*151*1367*89303 = 1. The order of 2 is 22120388821200.
      But why? How would I compute this from the factorization of p-1?


      --- In primenumbers@yahoogroups.com, Joseph Moore <jtpk@y...> wrote:
      > --- Jose_Ramón_Brox <ambroxius@t...> wrote:
      > > Hi:
      > >
      > > The order of a number n modulo p (with GCD(n,p)=1)
      > > is defined as the smallest exponent e such that n^e
      > > == 1 (mod p).
      > >
      > > Is there a straight way to find the order different
      > > from doing 1/n in base p and counting the length of
      > > the period?
      >
      > Yes, somewhat. The order of n mod p is in the set of
      > the proper divisors of p-1. However, finding *which*
      > of the proper divisors of p-1 is an order of n can be
      > a challenge. For instance, 4 has order 3 mod 7, while
      > 5 has order 6 mod 7.
      >
      > >
      > > Jose Brox
      > >
      >
      >
      >
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    • Joseph Moore
      First, I was partly wrong. *Any* factor of p-1 could be the order of n mod p. One thing to do is cycle through the factors of p-1. In this case,
      Message 2 of 6 , Jul 6, 2004
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        First, I was partly wrong. *Any* factor of p-1 could
        be the order of n mod p.

        One thing to do is cycle through the factors of p-1.
        In this case, 22120388821200*8+1=p, so we see that 2
        has order 2^4*3*5^2*151*1367*89303. Note that the
        multiplicative inverse of n mod p has the same order
        as n mod p.

        I'm sure there's a more efficient way to get the order
        than by cycling through all the factors, but that's a
        start.

        Joseph.


        --- joseph_osiecki <osieckis@...> wrote:
        > Can someone give me some more details on working out
        > the order of an
        > element mod p when we have a factorization of p-1?
        > Please use the following example p = 176963110569601
        > =
        > 2^7*3*5^2*151*1367*89303 + 1. The order of 2 is
        > 22120388821200.
        > But why? How would I compute this from the
        > factorization of p-1?




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      • elevensmooth
        ... an ... The best way I know is to start with N=p-1 and trial divide by each prime factor q of N. If 2^(N/q) = 1 mod p, then replace N by N/q and
        Message 3 of 6 , Jul 7, 2004
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          --- In primenumbers@yahoogroups.com, "joseph_osiecki" <osieckis@n...>
          wrote:
          > Can someone give me some more details on working out the order of
          an
          > element mod p when we have a factorization of p-1?
          > Please use the following example p = 176963110569601 =
          > 2^7*3*5^2*151*1367*89303 = 1. The order of 2 is 22120388821200.
          > But why? How would I compute this from the factorization of p-1?

          The best way I know is to start with N=p-1 and trial divide by each
          prime factor "q" of N. If 2^(N/q) = 1 mod p, then replace N by N/q
          and continue. In your example, trial division by 2 works the first 3
          times. Trial division by the other primes fails.

          I do this whenever the ElevenSmooth Special Project finds a new
          divisor. These factors are found as divisors of 2^3326400-1, so I
          know the order is a divisor of GCD(3326400, p-1). I use this process
          to find the order of 2, which tells me for which Mersenne number the
          prime is a primitive factor. I then tell Will Edgington and post it
          on the ElevenSmooth factors page at
          http://ElevenSmooth.com/ElevenFactors.html

          William
          ==================
          ElevenSmooth http://ElevenSmooth.com
          Distributed Factoring of 2^3326400-1
        • Kevin Acres
          Hi, I have two experimental methods under investigation for this problem. One is the reverse algorithm as mentioned a week or so back. A second method
          Message 4 of 6 , Jul 7, 2004
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            Hi,

            I have two experimental methods under investigation for this problem.

            One is the 'reverse algorithm' as mentioned a week or so back.

            A second method involves growing a 'binary tree'. This works from
            knowing that the final modulo result in a trial factorisation is 1.
            From this you can grow a tree, which will eventually give you the
            number that you want.

            The trick with the tree is to spot non-fruitful branches and to 'prune
            them'. You also need to control the growth of the tree, since there
            are many possible results, which are multiples of the one that you
            want. The terminal condition for any branch of the tree is to trial
            factor the number that you just generated and to check for a modulo 1
            result.

            Once you have grown a few trees then you will start to see the 1's and
            0's of the binary representation of a number as a series of decision
            points. From that you can start to appreciate that the final order 2
            is as much a result of the spacings of 1's and 0's of the original
            number as anything else.

            The whole trick with this approach is to grow each branch one level at
            a time. This prevents uncontrolled growth of any one branch. Another
            thing is to watch out for repeating sequences, these need weeding out
            too.

            Kevin.


            > --- In primenumbers@yahoogroups.com, "joseph_osiecki" <osieckis@n...>
            > wrote:
            > > Can someone give me some more details on working out the order of
            > an
            > > element mod p when we have a factorization of p-1?
            > > Please use the following example p = 176963110569601 =
            > > 2^7*3*5^2*151*1367*89303 = 1. The order of 2 is
            22120388821200.
            > > But why? How would I compute this from the factorization of p-1?
            >
            > The best way I know is to start with N=p-1 and trial divide by each
            > prime factor "q" of N. If 2^(N/q) = 1 mod p, then replace N by N/q
            > and continue. In your example, trial division by 2 works the first 3
            > times. Trial division by the other primes fails.
            >
            > I do this whenever the ElevenSmooth Special Project finds a new
            > divisor. These factors are found as divisors of 2^3326400-1, so I
            > know the order is a divisor of GCD(3326400, p-1). I use this process
            > to find the order of 2, which tells me for which Mersenne number the
            > prime is a primitive factor. I then tell Will Edgington and post it
            > on the ElevenSmooth factors page at
            > http://ElevenSmooth.com/ElevenFactors.html
            >
            > William
            > ==================
            > ElevenSmooth http://ElevenSmooth.com
            > Distributed Factoring of 2^3326400-1
            >
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