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24957Re: [PrimeNumbers] Infallible prime tests for n<2^64 and n<2^32, C code

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  • Phil Carmody
    Mar 18, 2013
      --- On Mon, 3/18/13, WarrenS <warren.wds@...> wrote:
      > Note my Lehmer test actually ran in "1.60 Selfridges"
      > experimentally
      >   while(j>0){
      >     A = mulmod64(A, A, m);
      >     A = mulmod64(A, A, m);
      >     A = mulmod64(A, A, m);
      >     j -= 3;
      >     es = (e>>j)&7;
      >     //prefacing this line with "if(es)" is
      > optional; the "if" gives 2-3% speedup on my machine: 
      >     if(es){ A = mulmod64(A, gpow[es], m); }
      >   }

      For the fixed g==2 case, this is suboptimal, running at about 1.3x (31/24 = (3+7/8)/3) the cost it could have. And 1.3 * 1.6 = 2, as expected.

      Use balanced residues (-p/2 .. +p/2), and your multiply/squaring can be accompanied by a doubling (multiplication by g=2) almost for free. Then at each single bit step you either do a mulmod() or a mulmoddouble().

      Balanced residues work best when using the mixed floating-point/fixed-point mulmod algorithm. (Find a*b*(1/p) on the FPU, round to nearest, multiply by p in integers, subtract its bottom bits from the bottom bits of integer (a*b).)

      Note, however, that if you're interested in speed, you'll want a sqmod() operation as well as a mulmod() operation, there's no point in creating unnecessary duplicates. A good C compiler might be able to perform this optimisation for you if all the functions are inline and written in a compiler-friendly way, but sometimes it's just best to keep things explicitly simple.

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