Add GLV for simple curves: BW6-767, Grumpkin, secp256k1, secq256k1

[weikengchen]

Description

This PR adds GLV curves for those simple cases where a = 0 and the curve is a SW curve.

The parameters are generated as follows. First of all, you get beta. Another beta would be beta^2.

beta  = mod(1, q).nth_root(3)

Then, you get lambda. There are two candidates. One can just test it out.

root_of_minus_three = mod(-3, r).sqrt()
mod(-1 + root_of_minus_three, r) / mod(2, r)
# or
mod(-1 - root_of_minus_three, r) / mod(2, r)

Use an algorithm to find out the scalars

def find_basis(a, b): 
    res = []
    x1 = 1
    y1 = 0
    u = a
    x2 = 0
    y2 = 1
    v = b
    res.append((x2, y2, v, v * v > b))
    res.append((x1, y1, u, u * u > b))
    while u != 0:
        q = v // u
        r = v - q * u
        x = x2 - q * x1
        y = y2 - q * y1
        v = u
        u = r
        x2 = x1
        x1 = x
        y2 = y1
        y1 = y
        res.append((x1, y1, u, u * u > b))
    return res

res = find_basis(793479390729215512621379701633421447060886740281060493010456487427281649075476305620758731620350, r)

# find the first l
for i in range(len(res)):
    if res[i][3] == False: 
        l = i - 1
        break
l
a1 = res[l + 1][2]
a2 = -res[l + 1][0]
rtl_size = res[l][2] ^ 2 + res[l][0] ^ 2
rtl2_size = res[l + 2][2] ^ 2 + res[l + 2][0] ^ 2
if rtl_size < rtl2_size:
    a2 = res[l][2]
    b2 = -res[l][0]
else:
    a2 = res[l + 2][2]
    b2 = -res[l + 2][0]

The implementation here follows this Hackmd https://hackmd.io/@drouyang/glv by @drouyang

This PR is related to #718 but does not solve it.

---

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[Pratyush]

Can we reuse standard settings of beta and lambda that are also used by other implementations?

Also, can we add the relevant sage scripts to the curve folders?

[weikengchen]

Let me ask a clarifying question. What do you mean by standard settings?

So, I cross check with BW6-761 and I can reproduce their parameters, although the scalar decomposition is a little bit different---BW6-761's scalar decomposition coeffs are true, false, true, true. I, locally, have false, true, false, false, which would be equivalent but just not sure why BW6-761's coeffs prefer negations.

(If you have any info about why previous implementation tends to prefer negations, or whether we should ping @simonmasson for help...)

For the rest, there is an issue. BLS12-381, BLS12-377, BN254 don't use the standard GLV parameters. They use a special trick that is tailored for them (as you can see, their scalar decomposition is fairly simple). The curves that we are implementing here, however, does not have those special parameters. So the references for those would be a specific paper.

[weikengchen]

Let me organize my script and can include one. My current script is mostly doing a trial-and-error to figure out which lambda should be used, which is clearly not ideal.

Another thing pending for this PR is that I may want to run the benchmark to see the improvement.

[Pratyush]

@weikengchen I assume that secp256k1 uses particular parameters, especially in libsecp256, right? Similarly, for Grumpkin, other implementations might have a particular choice. Best to be compatible with them. That's all I meant

[weikengchen]

Got it. Let me look around.

By the way, the GLV implementation is already different from the standard GLV. The standard GLV rounds numbers. We floor. Per the original proof, this may imply that, for a 256-bit number, the decomposed scalar could be 129-bit instead of 128-bit.

[weikengchen]

allow me to do additional review about the parameters. The problem is that the positive sign and negative sign are crucial, and I think the current test template is insufficient in that it does not detect "correct but not short vector GLV".

[mmagician]

@weikengchen indeed, the tests are only checking correctness.
I would bench each individual curve with the proposed parameters before making it the default. See e.g. the PR and description here.