[3.10] gh-95778: Correctly pre-check for int-to-str conversion (GH-96537) by gpshead · Pull Request #96563 · python/cpython

…96563)

Summary:
cherry-picked the upstream 3.10 backport

```
git cherry-pick 8f0fa4b eace09e
```

this one is python/cpython#96563

original commit message below

--------

Converting a large enough `int` to a decimal string raises `ValueError` as expected. However, the raise comes _after_ the quadratic-time base-conversion algorithm has run to completion. For effective DOS prevention, we need some kind of check before entering the quadratic-time loop. Oops! =)

The quick fix: essentially we catch _most_ values that exceed the threshold up front. Those that slip through will still be on the small side (read: sufficiently fast), and will get caught by the existing check so that the limit remains exact.

The justification for the current check. The C code check is:
```c
max_str_digits / (3 * PyLong_SHIFT) <= (size_a - 11) / 10
```

In GitHub markdown math-speak, writing $M$ for `max_str_digits`, $L$ for `PyLong_SHIFT` and $s$ for `size_a`, that check is:
$$\left\lfloor\frac{M}{3L}\right\rfloor \le \left\lfloor\frac{s - 11}{10}\right\rfloor$$

From this it follows that
$$\frac{M}{3L} < \frac{s-1}{10}$$
hence that
$$\frac{L(s-1)}{M} > \frac{10}{3} > \log_2(10).$$
So
$$2^{L(s-1)} > 10^M.$$
But our input integer $a$ satisfies $|a| \ge 2^{L(s-1)}$, so $|a|$ is larger than $10^M$. This shows that we don't accidentally capture anything _below_ the intended limit in the check.

<!-- gh-issue-number: gh-95778 -->
* Issue: gh-95778
<!-- /gh-issue-number -->

(cherry picked from commit b126196)

Reviewed By: alexmalyshev

Differential Revision: D39369517

fbshipit-source-id: 750f9c3

Co-authored-by: Gregory P. Smith [Google LLC] <greg@krypto.org>
Co-authored-by: Mark Dickinson <dickinsm@gmail.com>