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April 3rd, 2009, 09:32 PM   #1
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Help with a mod calculation

Hi All,

I have this little problem with a mod calculation that I need some advice on.

given:

s = k-1(m-ar) mod (p-1); where k-1 is the inverse of k, the "-1" is supposed to be superscripted; k=20519, m=649, a=407, r=4205, p=83773

how do I calculate the value of s? I've having difficulty with the mod calulation. Thanks
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April 5th, 2009, 06:18 AM   #2
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Re: Help with a mod calculation

http://world.std.com/~reinhold/BigNumCalc.html

Setting modulo to 83772 and pressing the buttons, the answer is: s = 20730

If you have to show the work for doing it by hand, there are two parts.

Firstly you have to find 649 - 407.4205 mod 83772.

20.4205 = 84100 = 83772 + 328
so 400.4205 ? 20.328 or 6560
and 407.4205 ? 35995
etc

Secondly you have to find the inverse of 20519, mod 83772.

Use the extended Euclidean algorithm:

1, 0 | 83772, _
0, 1 | 20519, 4

The 1,0,0,1 are your starting points. The 4 is the number of times 20519 goes into 83772.

Now write down the third row. The first number is the number two rows directly above, minus 4 times the number one row above. The second number is found the same way. The third number is the remainder from dividing 83772 by 20519, and the fourth number is the number of times this remainder goes into 20519.

1, -4 | 1696, 12

Write down the fourth row. The first number is the number two rows directly above, minus 12 times the number one row above... etc. The final answer will be the second number in the last row: -22573, or 61199.

So the problem becomes s ? 61199.48426 mod 83772.

There may be a quicker way to do it but this will work.
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