Re^2: Random 1-1 mapping

Update: Your solution is cool but $max is not guaranteed to be a power of 2. In fact, it almost certainly isn't.

I just scribbled this on a piece of paper, I think I heard something about it to do with the RSA encryption algorithm once...

$result = ($prime_number * $input + $seed) % $max;
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Example for $max = 10, $prime_number = 7 and $seed = 5..

shuffle(5,10,0) == (7 * 0 + 5) % 10 == 5 % 10 == 5
shuffle(5,10,1) == (7 * 1 + 5) % 10 == 12 % 10 == 2
shuffle(5,10,2) == (7 * 2 + 5) % 10 == 19 % 10 == 9
shuffle(5,10,3) == (7 * 3 + 5) % 10 == 26 % 10 == 6
shuffle(5,10,4) == (7 * 4 + 5) % 10 == 33 % 10 == 3
shuffle(5,10,5) == (7 * 5 + 5) % 10 == 40 % 10 == 0
shuffle(5,10,6) == (7 * 6 + 5) % 10 == 47 % 10 == 7
shuffle(5,10,7) == (7 * 7 + 5) % 10 == 54 % 10 == 4
shuffle(5,10,8) == (7 * 8 + 5) % 10 == 61 % 10 == 1
shuffle(5,10,9) == (7 * 9 + 5) % 10 == 68 % 10 == 8
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So that works for 7 and 10. Does it work for any $max and any $prime_number? If not what conditions will it work for? I am no good with proofs.

-Andrew.

Andrew Tomazos | andrew@tomazos.com | www.tomazos.com

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Re^3: Random 1-1 mapping
by blokhead (Monsignor) on May 28, 2006 at 21:26 UTC

f(x) = (a*x + b) % m
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I would suggest splitting your "seed" value into the a & b coefficients in the following way:

a = largest number less than seed satisfying gcd(a,m)=1
b = seed - a

sub gcd { $_[1] ? gcd($_[1], $_[0] % $_[1]) : $_[0] }

sub shuffle {
    my ($seed, $max, $i) = @_;

    my $ca = $seed;
       $ca-- until gcd($ca, $max) == 1;
    my $cb = $seed - $ca;

    ($ca * $i + $cb) % $max;
}

for my $seed (1 .. 10) {
    my @result = map shuffle($seed, 15, $_), 0 .. 14;
    print "seed=$seed ==> @result\n";
}
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only

seed=1 ==> 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
seed=2 ==> 0 2 4 6 8 10 12 14 1 3 5 7 9 11 13
seed=3 ==> 1 3 5 7 9 11 13 0 2 4 6 8 10 12 14
seed=4 ==> 0 4 8 12 1 5 9 13 2 6 10 14 3 7 11
seed=5 ==> 1 5 9 13 2 6 10 14 3 7 11 0 4 8 12
seed=6 ==> 2 6 10 14 3 7 11 0 4 8 12 1 5 9 13
seed=7 ==> 0 7 14 6 13 5 12 4 11 3 10 2 9 1 8
seed=8 ==> 0 8 1 9 2 10 3 11 4 12 5 13 6 14 7
seed=9 ==> 1 9 2 10 3 11 4 12 5 13 6 14 7 0 8
seed=10 ==> 2 10 3 11 4 12 5 13 6 14 7 0 8 1 9
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blokhead

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