Question

Discrete product logarithm fun question (slightly wrong)

Answer 1

The definition for the discrete product logarithm I came up with is: \[\Large y \equiv a*k^a \mod n \\ \Large W_k(y) \equiv W_k(ak^a) \equiv a \mod n\] So what this is saying is "product log base k of ak^a is a". Of course this is multivalued often times.

However there is always at least one value x such that for all k that the product log will return its own value other than the trivial answer of x=0. \[\Large W_k(x) \equiv x \mod n\]

Answer 2

So what's x?

Answer 3

Note that W_k(x)=x doesn't mean that W_k(x) can't also have other values as well since it's multivalued. But this has a pretty satisfying answer I think. =)

Answer 4

Here's a couple examples of the discrete product log in action so you can understand how it works better: \[\Large W_3(3) \equiv W_3(1*3^1) \equiv 1 \mod 5\]\[\Large 3 \equiv 18 \equiv 2*3^2 \mod 5 \\ \Large W_3(3) \equiv 2 \mod 5\] So here we see the multivalueness of the product log coming through, however W_3(3) only has these two answers in mod 5. An example where it is single valued is \[\Large W_2(3) \equiv W_2(2*2^2) \equiv 2 \mod 5\]
Feel free to discuss or ask questions haha. =P

Answer 5

i'm from 6th grade so yea...

Answer 6

I just realized that it's really only true when \[\Large \gcd(k,n)=1\]