Generalize field in Circlepoint

[atgrosso]

Depends on #98

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This change is 

[andrewmilson]

Reviewed all commit messages.
Reviewable status: 0 of 9 files reviewed, 14 unresolved discussions (waiting on @atgrosso)

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stwo_cairo_verifier/src/circle.cairo line 8 at r1 (raw file):

pub const M31_CIRCLE_GEN: CirclePoint<M31> =
    CirclePoint::<M31> { x: M31 { inner: 2 }, y: M31 { inner: 1268011823 }, };

Nit: can this be implicit

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stwo_cairo_verifier/src/circle.cairo line 31 at r1 (raw file):

    // Returns the neutral element of the circle.
    fn zero() -> CirclePoint<F> {
        CirclePoint::<F> { x: One::<F>::one(), y: Zero::<F>::zero() }

Same here?

Suggestion:

CirclePoint { x: One::one(), y: Zero::zero() }

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stwo_cairo_verifier/src/circle.cairo line 50 at r1 (raw file):

        };
        result
    }

Why does this have to change so much? Can't it be the same as before just change scalar to u128?

Code quote:

    fn mul(
        self: @CirclePoint<F>, initial_scalar: u128
    ) -> CirclePoint<
        F
    > {
        let mut scalar = initial_scalar;
        let mut result: CirclePoint<F> = Self::zero();
        let mut cur: CirclePoint<F> = *self;
        while scalar > 0 {
            if scalar & 1 == 1 {
                result = result + cur;
            }
            cur = cur + cur;
            scalar = scalar / 2;
        };
        result
    }

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stwo_cairo_verifier/src/circle.cairo line 54 at r1 (raw file):

impl CirclePointAdd<F, +Add<F>, +Sub<F>, +Mul<F>, +Drop<F>, +Copy<F>> of Add<CirclePoint<F>> {
    // The operation of the circle as a group with additive notation.

I know unrelated to your changes but mind chageing to docstring here please?

Suggestion:

    /// Performs the operation of the circle as a group with additive notation.

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stwo_cairo_verifier/src/circle.cairo line 56 at r1 (raw file):

    // The operation of the circle as a group with additive notation.
    fn add(lhs: CirclePoint<F>, rhs: CirclePoint<F>) -> CirclePoint<F> {
        CirclePoint::<F> { x: lhs.x * rhs.x - lhs.y * rhs.y, y: lhs.x * rhs.y + lhs.y * rhs.x }

Same here?

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stwo_cairo_verifier/src/circle.cairo line 72 at r1 (raw file):

        CirclePoint { x: self.x.complex_conjugate(), y: self.y.complex_conjugate() }
    }
}

Suggestion:

#[generate_trait]
pub impl ComplexConjugateImpl of ComplexConjugateTrait {
    fn complex_conjugate(self: CirclePoint<QM31>) -> CirclePoint<QM31> {
        CirclePoint { x: self.x.complex_conjugate(), y: self.y.complex_conjugate() }
    }
}

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stwo_cairo_verifier/src/circle.cairo line 74 at r1 (raw file):

}

/// Represents the coset initial + <step>.

Suggestion:

/// Represents the coset `initial + <step>`.

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stwo_cairo_verifier/src/circle.cairo line 112 at r1 (raw file):

    }

    fn at(self: @Coset, index: usize) -> CirclePoint::<M31> {

Suggestion:

CirclePoint<M31> 

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stwo_cairo_verifier/src/circle.cairo line 122 at r1 (raw file):

    /// Creates a coset of the form G_2n + \<G_n\>.
    /// For example, for n=8, we get the point indices \[1,3,5,7,9,11,13,15\].

Suggestion:

    /// Creates a coset of the form `G_2n + <G_n>`.
    ///
    /// For example, for `n=8`, we get the point indices `[1,3,5,7,9,11,13,15]`.

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stwo_cairo_verifier/src/circle.cairo line 124 at r1 (raw file):

    /// For example, for n=8, we get the point indices \[1,3,5,7,9,11,13,15\].
    fn odds(log_size: u32) -> Coset {
        //CIRCLE_LOG_ORDER

Remove

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stwo_cairo_verifier/src/circle.cairo line 131 at r1 (raw file):

    /// Creates a coset of the form G_4n + <G_n>.
    /// For example, for n=8, we get the point indices \[1,5,9,13,17,21,25,29\].
    /// Its conjugate will be \[3,7,11,15,19,23,27,31\].

Suggestion:

    /// Creates a coset of the form `G_4n + <G_n>`.
    ///
    /// For example, for `n=8`, we get the point indices `[1,5,9,13,17,21,25,29]`.
    /// Its conjugate will be `[3,7,11,15,19,23,27,31]`.

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stwo_cairo_verifier/src/circle.cairo line 156 at r1 (raw file):

    #[test]
    fn test_add_1() {
        let i = CirclePoint::<M31> { x: m31(0), y: m31(1) };

Same here? and below?

Suggestion:

        let i = CirclePoint { x: m31(0), y: m31(1) };

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stwo_cairo_verifier/src/circle.cairo line 160 at r1 (raw file):

        let expected_result = CirclePoint::<M31> { x: -m31(1), y: m31(0) };

        assert_eq!(result, expected_result);

I'd prefer no variable for expected_result.
Same below

Suggestion:

        let result = i + i;

        assert_eq!(result, CirclePoint { x: -m31(1), y: m31(0) });

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stwo_cairo_verifier/src/circle.cairo line 285 at r1 (raw file):

            x: qm31(1, 0, 478637715, 513582971),
            y: qm31(992285211, 649143431, 740191619, 1186584352),
        };

This will be needed outside of tests.
Can define below M31_CIRCLE_GEN

Suggestion:

/// A generator for the circle group over [`SecureField`].
pub const QM31_CIRCLE_GEN: CirclePointQM31 =
    CirclePointQM31 {
        x: QM31 {
            a: CM31 { a: M31 { inner: 1 }, b: M31 { inner: 0 }, },
            b: CM31 { a: M31 { inner: 478637715 }, b: M31 { inner: 513582971 } }
        },
        y: QM31 {
            a: CM31 { a: M31 { inner: 992285211 }, b: M31 { inner: 649143431 } },
            b: CM31 { a: M31 { inner: 740191619 }, b: M31 { inner: 1186584352 } }
        },
    };