## SIV Research Errata

### Neff 2001:
- pg 4, eq 14: Product of y_i's should say from `i = 1`, not `i - 1`.
  -> Simple typographical error.

### Neff 2004:
- pg 17, eq 49: `b_j` and `c_i` positions appear accidentally swapped, if we want those equations to match definition of `r_i` and `s_i`.
  -> Simple typographical error. Algebra breaks if not corrected, which is easy.

- pg 18, eq 50, and 51: Missing `\gamma` factor on right side of both equations to match eq 48.
  -> Simple typographical error. Algebra breaks if not corrected, which is easy.

- pg 5 / pg 19: Lemma 4 restricts its `r_vector` to unique non-zero random values in `Z_q`, but eq 64's `b_i`'s, which are supposed to be analogous to lemma 4's `r_vector`, are simply random in `Z_q`, but do not share the same restrictions to be non-zero nor unique.
  -> This does not weaken Soundness, but makes the proof needlessly more complicated. We guess that this is an artifact from an earlier version of the paper. We can remove the Lemma's restrictions and still prove a stronger version (see next note).

- pg 5, Lemma 4: Unproven. He gives a probability of `< 2/q`, this appears to depend upon the uniqueness restriction on the `r_vector`, which as described in previous errata is not applicable when used in the full proof.
  -> Here is a proof for why the probability becomes `1/q`, by dropping the uniqueness and non-zero restrictions on the `r_vector`.
    1. Sort `v_vector` such that the last item is non-zero.
    2. Define `v_bar` as the sorted `v_vector` up to `k-1`, without the last item, and define `r_bar` as the matching re-sorted `r_vector` up to `k-1`, without the last item.
    3. Take the dot-product of `v_bar` and `r_bar`, and subtract from constant `a`, to result in a new constant `a_bar`.
    4. We're left with `v_k` * `r_k` = `a_bar`.
    5. We multiply both sides by the inverse of `v_k` (which must be non-zero by step 1), leaving:
       `r_k` = `c`, where c is some number in `Z_q`.
    6. This holds with a `1 / q` chance.
