Description

Ensure that no overflow occurs in euclidian division.

Unfortunately the added check is not compatible with our truncation so I had to re-work it.
This could be avoided if we disallow truncation for field elements. In doing so this would:

• break modulo operation for fields
• break casting of field elements into sized integers

The first point is fine IMHO, since we already forbid field comparison because it is not consistent across field elements (0<3 but 3<p for instance).
Indeed, modulo is also inconsistent across field elements (0%2=0 but p%2=1 for instance)

For the second point, we could change the semantic of casting and make x as u32 becomes simply a 32 bits range check.

Problem*

Euclidian division was not checking for overflow. This is a big issue since it means noir is producing under-constrained code. For instance, the prime order of the noir native field is odd, so p=2k+1, which means that p%2=1. But we can also have p=0*2+0, which means that p%2=0. The latter does not overflow.

Related to #1604

Summary*

We add a check in euclidian division to ensure that the result 'qb+r' does not overflow.
But this also means that euclidian division cannot be performed on unbounded integers anymore, which has impact on the truncation operation.
This is resolved by reducing the input to a bit size less than the field modulus (with some margin).
As discussed in the description, this can be avoided if we change the semantic of the casting in noir.

Documentation

• This PR requires documentation updates when merged.

  • I will submit a noir-lang/docs PR.
  • I will request for and support Dev Rel's help in documenting this PR.

Additional Context

PR Checklist*

• I have tested the changes locally.
• I have formatted the changes with Prettier and/or cargo fmt on default settings.