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Old 2020-10-02, 20:22   #1
bhelmes
 
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Default calculation of modulo Mp

A peaceful and pleasant day for you,


I do not understand how the calculation modulo a Mersenne prime is made:


https://en.wikipedia.org/wiki/Mersenne_prime:
"Arithmetic modulo a Mersenne number is particularly efficient on a binary computer"


Perhaps this is an interesting question also for others.


Greetings from the modulo operator

Bernhard
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Old 2020-10-02, 21:43   #2
Batalov
 
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Read https://www.mersenne.org/various/math.php first ?

Quote:
Originally Posted by https://www.mersenne.org/various/math.php#lucas-lehmer
...and it performs the mod 2P-1 step for free.
In very simple terms:
Fact 1. When doing mod operations, (a * b) (mod Mp) = a (mod Mp) * b (mod Mp). Consequence: you never need to keep "the real value"; always keep only the value (mod Mp). This means that it will never be more than p bits long.

Fact 2. if we multiply one value by another and both are less than p bits long, you will get a result that is < 2p bits long.

Fact 3 (assuming mult or square is already done; there was no question about that part).
Mod operation becomes this: cut the bits above p-th bit. Slide them down, align with lower part. Add. Only if 1 bit carry sticks out above p bits, cut it again and add 1 to lower part. Done!

The DWT actually doesn't need this operation literally. But even if it did, this is how cheap computationally it would have been. Essentially - there is no division. Division by 2p-1 is luckily this elegant and easy.
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Old 2020-10-03, 07:18   #3
LaurV
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Probably easier for him, you have a number a which you want to reduce (mod m), so you are looking for a number b such as a=b (mod m). From the definition of the modulus, if a=b (mod m), this means that there is a number k, such as a=k*m+b. Now, if m=2^p-1, then you have a=k*(2^p-1)+b, or other way written, a=k*(2^p)-k+b. When you represent this is binary, on 2p bits, the first most-significant p bits (MSB) of the representation contain the k value (because multiplying with 2^p just moves k, p bits to the left), and the last p LSB bits (least significant bits) contain the value b-k. If you add the two halves together, you get b. That's all**.

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(**Edit: except, sometimes you may get it all 1, by addition, and then the result is zero, or you may need to repeat the procedure once, but those "trifles" won't be discussed here.)

Edit 2, as I have some more time, and there is no reply yet, say you want to reduce 107 (mod 31), now 107 is 3*32+11, i.e. 3*2^5+11, therefore when you represent it in binary on 10 bits (5+5) you have 00011 01011. The MSB contains 3, and the LSB contain 11 (decimal, sorry for the confusion, I will "go advanced" now to change the font for the binary numbers here, and color all of them nicely). This is 3*32+11=3*(31+1)+11=3*31+3+11. The 3*31 part won't matter for the modulus. So, all you have to do, is to add the two halves to get 11+3=14. Which is the right result of 107 (mod 31).

Last fiddled with by LaurV on 2020-10-05 at 07:56
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Old 2020-10-03, 08:49   #4
paulunderwood
 
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2^p - 1 == 0 mod Mp
2^p == 1 mod Mp
k*2^p == k mod Mp
That is k shifted to the left p-bits is equivalent to k.

If a and b each have no more than p bits then a*b is at most 2*p-1 bits, you can just add the number composed of the top most bits (above p-1 counting from 0) to the number composed of the bottom p bits, to get the result mod Mp.

Last fiddled with by paulunderwood on 2020-10-03 at 09:35
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Old 2020-10-03, 17:03   #5
bhelmes
 
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Most people know the 9-rule in the decimalsystem
which based on the fact that 10 = 1 mod 9


For Mersenne numbers it is the same reflection
that 2^p = 1 mod 2^p -1




Thanks for all replies,

now we know one advantage from the Mersenne numbers.

Last fiddled with by bhelmes on 2020-10-03 at 17:04
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