Pohlig-Hellman Algorithm posted December 2014
I'm reading through A Key Recovery Attack on Discrete Log-based Schemes Using a Prime Order Subgoup which is a Small subgroup confinement attack.
It deals with stuff I had no knowledge of, like Schnorr's Signature that I talk about in a previous post, or like what I'm going to talk about now:
The Pohlig-Hellman Algorithm is a method to compute a Discrete Logarithm (which is a difficult problem) on a multiplicative group whose order is a smooth number (also called friable). Meaning its order can be factorized into small primes.
y = g^x mod p ord_p(g) = p - 1 p - 1 = q_1^(i_1) * ... * q_j^(i_j)
Here y is the public key,
xis the secret key we're trying to compute.
The order of
g, our generator, is
p - 1since p is prime.
p - 1 is smooth so it can be factorized into something like 2^2 * 3 * 7 (extremely convenient example!)
Following is an overview of the method, if you read an equation and feel like it comes from nowhere (and it should feel like that), I posted a very short paper containing the simple proofs of those bellow.
The idea that should come to your mind, if you're used to seeing that kind of problem, is that there might be a way to use the Chinese Remainder Theorem (abbreviated CRT) to our advantage. What if we could write
x modulo the factors of
p - 1 and then reconstruct the real
x with the CRT? Well, let's do just that!
x modulo a factor
p - 1 we can write
y^((p-1) / q) which we know and can compute, and is also equal to
g^(x_q * (p-1) / q)
(If you have a factor that appears multiple times in the prime decomposition of
p - 1 (for example
p - 1 = 2^5, then there is also a way to ease the computations by finding multiple unknowns (5 unknowns in our example))
We then have a discrete logarithm to compute, but a small one, that we can compute efficiently thanks to Shanks' method (baby-step giant-step) or Pollard's rho algorithm.
Then we have multiple ways of writing our
x (modulo the factors of
p - 1) and we find out what is
x with CRT (I explained this step in my airbus write-up here).
Proofs + Video
You can read through the Algorithm (or watch the video bellow), but I don't really like to do that since I'm not really good at memorizing something if I don't understand the nut and bolts of it. So here's a really good paper written by D. R. Stinson that demonstrate where the equations of the algorithm come from. And here's an explanation + example of the algorithm:
Thank you for this post, it is really interesting and clear. The paper written by D. R. Stinson is also very helpful if you want to code the algorithm.
leave a comment...