Math Is Fun Forum / Weird modular multiplicative inverse

calccrypto · 2010-07-24 05:19:54

im trying to get the value of mu as shown in the stuff below, but im getting a completely different value

from http://liris.cnrs.fr/~ohasan/pprs/paillierdemo/index.html?pt1=10&pt2=21 (dont bother going there. the values change everytime)

# Select two random and distinct primes p and q of length k/2
p = 33581 <-- given
q = 53299 <-- given

# Compute n = p*q
n = 1789833719 <-- got this

# Compute lambda = lcm(p-1,q-1) = (p-1)*(q-1) / gcd(p-1,q-1)
lambda = 894873420 <-- got this

# Select a random integer g in Z*_{n^2}
g = 3082513844323755387 <-- given

# Compute mu = (L(g^lambda mod n^2))^-1 mod n, where L(u) = (u-1)/n <-- im getting the inside stuff to be 1323971239912197708, L(u) = 739717453
mu = 1430230768

im getting mu to be 148619716. yet, the values produced work, while mine dont. what am i doing wrong?

bobbym · 2010-07-24 06:21:39

Hi cal;

His mu is correct.

calccrypto · 2010-07-24 07:18:23

edit: okay. i fail. i reversed the values. sorry for wasting your time

Last edited by calccrypto (2010-07-24 07:20:33)

bobbym · 2010-07-24 07:24:40

No problem! Good to see ya!

calccrypto · 2010-07-24 08:35:04

Thanks

bobbym · 2010-07-24 09:05:24

What are you working on?

calccrypto · 2010-07-24 09:38:00

im trying to write the Paillier cryptosystem. it seems easy enough, but its not working for some reason or another

calccrypto · 2010-07-24 10:27:05

quick thing: what does it mean when a integer symbol has a little star next to it? i cant seem to remember and cant think of how to search for it

bobbym · 2010-07-24 14:21:29

What's the integer symbol? You mean Z or N?

calccrypto · 2010-07-25 02:28:33

the z, but this one also has a * next to it

bobbym · 2010-07-25 02:36:09

I think it is a multiplicative group modulo n.

http://en.wikipedia.org/wiki/Multiplica … s_modulo_n

calccrypto · 2010-07-25 03:11:13

Thanks! now... how do i chose random integers from the set? i cant list out all the possible values every time. the O(n) would be ridiculous

bobbym · 2010-07-25 03:33:41

To produce random numbers from any set is not difficult. Either you have the rule or law that creates the set or if the set is finite. You create it and store it once as an array or whatever random access container your language supports.

P = { 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47} This is the the set P such that n is prime and n<50. To draw numbers from this set, you would choose a random number from 1 to 15 and access the array element using that number:

Begin loop:

k = random(1,15): 10 - >k

P[k] is P[10] which yields 29.

loop

Next iteration:

k = random(1,15): 3 - >k

P[k] is P[3] which yields 5.

etc.

calccrypto · 2010-07-25 03:43:30

yes, but what if the upper limits are large?

do you know java?
http://liris.cnrs.fr/~ohasan/pprs/paillierdemo/Paillier.java

Last edited by calccrypto (2010-07-25 03:45:19)

bobbym · 2010-07-25 03:54:55

If k is large than you will need the rule that creates the set.

S = { 1,3,5,7,9,11,13,17,19,...1000000000001} This set contains the odd numbers [1,1000000000001]

We certainly don't want to create an array that big. But there is a rule that generates those numbers it is 2n-1 for n = 1 to 500 000 000 001.

in pseudocode:

k = random(1,500 000 000 001): 512 - >k

P[k] is P[512] which yields 1023

Next iteration:

k = random(1,500 000 000 001): 1000000 - >k

P[k] is P[1000000] which yields 2(1000000) - 1 = 1999999

etc

calccrypto · 2010-07-25 04:01:48

interesting...

i think i found another way: im going to get a random number and add 1 until the value is coprime to k

bobbym · 2010-07-25 04:03:59

How will you test for coprime?

calccrypto · 2010-07-25 04:19:22

i have a small function i wrote in my personal math library.

Code:

def pair_coprime(numberlist):
    for x in numberlist:
        for y in xrange(len(numberlist)):
            if x != numberlist[y]:
                out = gcd(x,numberlist[y])
                if out != 1:
                    return False #Some values are not coprime
    return True #List is pairwise coprime

it could be faster, i think, but for now, its alright

i have another question:
can (a^b * c^d) mod e be done as [(a^b mod e) * (c^d mod e)] mod e? if not, whats an efficient way to do this on a computer? the exponentiation by squaring thing will still go through the second equation

bobbym · 2010-07-25 04:29:01

Hi;

can (a^b * c^d) mod e be done as [(a^b mod e) * (c^d mod e)] mod e?

Yes.

To do it efficiently you use modular exponentiation. Described here:

http://webcache.googleusercontent.com/s … =firefox-a

calccrypto · 2010-07-25 04:55:50

cool! thanks

bobbym · 2010-07-25 05:12:05

Your welcome!

calccrypto · 2010-07-25 05:23:56

scratch that. i messed up again

Last edited by calccrypto (2010-07-25 05:25:53)

bobbym · 2010-07-25 06:52:33

Did you get the point of modular exponentiation? It pops up a lot in computational number theory.

calccrypto · 2010-07-25 07:44:47

yeah, but something is just not working

bobbym · 2010-07-25 07:49:52

Nothing in computational math ever works, you have to force it.

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#1 2010-07-24 05:19:54

Weird modular multiplicative inverse

#2 2010-07-24 06:21:39

Re: Weird modular multiplicative inverse

#3 2010-07-24 07:18:23

Re: Weird modular multiplicative inverse

#4 2010-07-24 07:24:40

Re: Weird modular multiplicative inverse

#5 2010-07-24 08:35:04

Re: Weird modular multiplicative inverse

#6 2010-07-24 09:05:24

Re: Weird modular multiplicative inverse

#7 2010-07-24 09:38:00

Re: Weird modular multiplicative inverse

#8 2010-07-24 10:27:05

Re: Weird modular multiplicative inverse

#9 2010-07-24 14:21:29

Re: Weird modular multiplicative inverse

#10 2010-07-25 02:28:33

Re: Weird modular multiplicative inverse

#11 2010-07-25 02:36:09

Re: Weird modular multiplicative inverse

#12 2010-07-25 03:11:13

Re: Weird modular multiplicative inverse

#13 2010-07-25 03:33:41

Re: Weird modular multiplicative inverse

#14 2010-07-25 03:43:30

Re: Weird modular multiplicative inverse

#15 2010-07-25 03:54:55

Re: Weird modular multiplicative inverse

#16 2010-07-25 04:01:48

Re: Weird modular multiplicative inverse

#17 2010-07-25 04:03:59

Re: Weird modular multiplicative inverse

#18 2010-07-25 04:19:22

Re: Weird modular multiplicative inverse

Code:

#19 2010-07-25 04:29:01

Re: Weird modular multiplicative inverse

#20 2010-07-25 04:55:50

Re: Weird modular multiplicative inverse

#21 2010-07-25 05:12:05

Re: Weird modular multiplicative inverse

#22 2010-07-25 05:23:56

Re: Weird modular multiplicative inverse

#23 2010-07-25 06:52:33

Re: Weird modular multiplicative inverse

#24 2010-07-25 07:44:47

Re: Weird modular multiplicative inverse

#25 2010-07-25 07:49:52

Re: Weird modular multiplicative inverse

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