Can I use `div_mod` on an arbitrary field element?

[mmagician]

In your tests the element that I'm dividing is known to have a fixed number of bits.

However, I would like to apply this function to an arbitrary field element - the problem is that the number of bits can be up to F::NUM_BITS, so I end up with something like:

let small_mod: usize = 1 << 16;
let (_, r) = range_gate.div_mod(ctx, arbitrary_field_element, small_mod, F::NUM_BITS as usize);

which doesn't actually work (aside from not being efficient, if it did) as I get an index out of bounds error. By increasing the param a_num_bits to F::NUM_BITS + 2 circumvents this error but then it panics with some low level assert failure.

Perhaps the specific errors aren't very relevant, because I suspect that using div_mod for this purpose is simply not the best approach in the first place. If you're wondering why I care about arbitrary elements, it's because these elements comes from a hash output, and at a later stage I want to treat them as indices to an array.

Let me know if you have any better ideas for modular reduction of F to "usize" and constraining that the resulting F is indeed less than the modulus - many thanks!

[jonathanpwang]

This has to do with our implementation of range_check, which does not quite work all the way up to F::NUM_BITS.

Let's say you call div_mod(ctx, a, b, num_bits). It will call check_big_less_than_safe to range check a // b to num_bits - log2(b) + 1 bits. In your case that's F::NUM_BITS - 15 bits.

Now check_big_less_than_safe will make several range checks, all to range_bits = F::NUM_BITS - 15 bits. Our requirement on range_bits is that ceil(range_bits / lookup_bits) * lookup_bits <= F::NUM_BITS - 1 due to some optimizations in range check.

So it seems that if you make lookup_bits >= 16, then the above assumption will fail for range_bits = F::NUM_BITS - 15. However it will probably work if you use a smaller lookup_bits.

In the long term we could consider enhancing range_check so for range_bits very close to F::NUM_BITS, it doesn't have this ceiling requirement. In the short term if you are ok using slightly smaller lookup_bits, that's likely the most immediate solution.

[mmagician]

Thanks so much for a quick and detailed reply @jonathanpwang!
I've tried out your suggestion and indeed that seems to solve the panic issue.

I've moved on to the next error now :) 'circuit was not satisfied'. I've prepared a concise code piece that's an extended poseidon example from your scaffold:
https://github.com/HungryCatsStudio/halo2-scaffold/tree/div-mod-panic where this panic is shown on running the example.

Help is greatly appreciated, but also let me know if I can assist in fixing this!