Support for Large Prime Field

[Wen-Ding Li]

Hi,

I'd like to work with some large prime field where the prime is ~ 256 bits integer. I tried

```

i36 : GF(2^256-189)

stdio:36:1:(3): error: expected argument 1 to be a small integer
```
and it seems that it is too big.

Does Macaulay2 support large prime fields? Could we use the Grobner basis solver with large prime fields?

Thanks!

Wen-Ding Li

[Mahrud Sayrafi]

I don't think M2 currently supports primes this large.

My understanding is that Flint supports Galois fields of cardinality at most 64 bits and Givaro much smaller. It's also possible to use multiprecision big integers for computations using either library and reduce after every step, but I don't think it's straightforward to adapt M2's Grobner basis computations this way.

Mahrud

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[Daniel R. Grayson]

You could try something like this:

i3 : ZZ[x,y,z]/(2^256-189)

                                        ZZ[x..z]
o3 = ------------------------------------------------------------------------------
     115792089237316195423570985008687907853269984665640564039457584007913129639747

o3 : QuotientRing


It won't be as efficient, but it might work for you.

[Mahrud Sayrafi]

Ah, I forgot about this trick!

Would it be worth adding an option or check to ZZp to do this automatically?

Mahrud

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[Daniel R. Grayson]

Maybe it would be worth it.  If p is not small, then ZZ/p would return ZZ[ ]/p instead.  Mike might have an opinion.

[Mahrud Sayrafi]

It should probably return toField(ZZ[]/p) instead, so that the degree length is 0.

On Mon, Dec 12, 2022 at 11:24 AM Daniel R. Grayson <danielrich...@gmail.com> wrote:

Maybe it would be worth it.  If p is not small, then ZZ/p would return ZZ[ ]/p instead.  Mike might have an opinion.

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[Daniel R. Grayson]

Good idea.

[Daniel R. Grayson]

Maybe we should verify that p is actually prime before applying toField to ZZ[ ]/p.

We could also do ZZ[DegreeRank=>0]/p instead to get the degree length to be 0.