Suppose that you were merrily encrypting your data and sending them to your friend whose public key he gave you and you happen to find some data that remains unchanged after exponentiation. That is, you have found a number $x$ such that $x^e=x \mod n$. You can now factor their number. Now if you’re like me, you …

So given a collection of linearly independent vectors $$v_1, v_2, \ldots, v_n$$ we can form the n dimensional vector space with the above collection of $v_i$ as our basis. In fact, we basically define the dimension of a Vector space by the size of the smallest set of vectors that span the space. Importantly, given any …

Alright, so it turns out that Pari doesn’t use just pohlig helman, or rather it doesn’t proceed use Baby Step Giant Step or Pohlard Rho for the discrete log step on the reduced prime (I’ll probably do a write up for those later, they’re cool!)

Rather, it uses the Index Calculus.