Function endo_coefficient 

pub fn endo_coefficient<F: PrimeField>() -> F

Computes a primitive cube root of unity ξ in the field F.

For a prime field F_p where 3 divides p-1, this returns:

ξ = g^((p-1)/3)

where g is a generator of the multiplicative group F_p*.

Properties

• ξ³ = g^(p-1) = 1 (by Fermat’s Little Theorem)
• ξ ≠ 1 (since (p-1)/3 is not a multiple of p-1)
• The three cube roots of unity are: {1, ξ, ξ²}

Usage

This is used in two contexts:

1. Base field (ξ): For the curve endomorphism φ(x,y) = (ξ·x, y)
2. Scalar field (λ): As the scalar such that φ(P) = [λ]P

Both fields (Fp and Fq for Pasta curves) have cube roots of unity, and they correspond to each other via the endomorphism relationship.

References

• Halo paper, Section 6.2: https://eprint.iacr.org/2019/1021