13.78126
there’s apparently technical guidelines^{1} that talk about turning random bits into random numbers between 0 and something other than powers of two. unfortunately it’s lacking a bit of nuance.
they provide two algorithms for generating random numbers in the set ${1,…,n1}$, both use $k = \lceil\log_2 n\rceil$:

This algorithm maintains uniform distribution but has probabilistic runtime.
 $r = \mathrm{RNG}({0,1,2,…,2^k1})$
 If $(r < n)$ and $(r > 0)$, output $r$ else goto 1.

This algorithm has deterministic runtime but does not fully maintain uniform distribution.
 $r = \mathrm{RNG}({0,1,2,…,2^{k+64}1})$
 Output $(r \mod (n1)) + 1$
they include this helpful note:
Note: The usage of a nonuniformly distributed $\textrm{RNG}({1,2,…,n−1})$ can enable an attack on signature algorithms (cf. Bleichenbacher’s attack on DSA, described e.g. in [28]). Algorithm 2 does not provide uniform distribution. It is however assumed that the deviation from uniform distribution produced by Algorithm 2 is too small to be exploited by an attacker.
Ed25519, for whatever reason, chose to do algorithm 2 anyway:
EdDSA samples r from the interval $[0, 2^{2b} − 1]$, ensuring almost uniformity of the distribution modulo $\ell$. The guideline [2, Section 4.1.1, Algorithm 2] specifies that the interval should be of size at least $[0, 2^{b+61} − 1]$, i.e., 64 bits more than $\ell$; for Ed25519 there are 259 extra bits.
aside from the odd typo, this is very comforting, we’re 195 bits better than the guideline!
but how much deviation from a uniform distribution does the guideline guarantee, and what do those 195 bits get us?
i’m going to measure deviation as the sum of the absolute values of the difference between the uniform probability and the actual probability over the entire target set. that is:
$$d = \sum_{i=0}^{n1} P(i)\frac{1}{n}$$
given that were reducing our input ${0,…,r}$ modulo $n$, there are only two different probabilities:
$$P(i) = \begin{cases} \frac{c + 1}{r}, & \text{if $i < r, \mod n$} \\ \frac{c}{r}, & \text{if $i ≥ r, \mod n$} \end{cases}, \text{where $c = \lfloor\frac{r}{n}\rfloor$}$$
so our whole deviation function becomes, switching to Haskell:
deviation :: Integer > Integer > Rational
deviation n r = low + high where
(c, r') = r `divMod` n
low = r' * abs (((c+1) % r)  (1 % n))
high = (n  r') * abs ((c % r)  (1 % n))
time to make graphs!
very tidy, with a peak deviation of 0.343 right around $n = \frac{r}{\sqrt 2}$…
this one is a lot more scattered. peak deviation of $2.62\cdot10^{20} \approx 2^{65}$ almost at the righthand side. that’s great! we’ve reduced our maximum deviation by a factor of $2^{64}$ by sampling from an extra 64 bits! so that should work the same for curve25519 scalars too, right?
as it turns out, when your numbers get big, it really matters what range you’re reducing to. reducing a random 253 bit number modulo $2^{252} + 7237005577332262213973186563042994240857116359379907606001950938285454250989$ (the group order of curve25519), is more uniform than reducing a random 317 bit number modulo $2^{252}+2^{251}$. and if we can assume the deviation of $2^{65.6}$ is too small to be exploited, does going from $2^{126.6}$ to $2^{261.5}$ really get us anything?
 ← 13.78042
i’ve been using a pair of (relatively) cheap ar glasses as my primary display for a few weeks. overall i like it a lot, still need to get lens inserts to correct my elliptical (and a little interpupiary distance difference). but it’s only 1080p per eye, so of course i’m looking at how to build something better. which end me up looking at datasheets for ti’s mems display chips…
 13.78283 →
had a strange idea for a magic system…