Native field curve arithmetics

This repo implements native field arithmetics for short Weierstrass curves, using a nice trick from Tianyi Liu. It is efficient and is almost generic for both short Weierstrass curves and twisted Edward curves.

Performance

• A group mul takes 1221 rows, 2 witness columns and 3 selector columns. Custom gate has a degree of 5 (coset FFT domain = 4N).
• In comparison, Jellyfish uses 1865 rows, 5 witness columns and 13 selector columns. Also use degree 5 gates.

Gate config

The gate configuration is:

op codes	cost	q_ec	q1	q2	q3	statement
cond ec add	4	1	1	0	0	(x1, y1), (x2, y2) and (x3, -y3) are on a same line
ec double	2	1	0	1	0	(x1, y1) and (x3, -y3) are on a tangential line of the curve
is on curve	1	1	0	1	1	y1^2 = x1^3 - C::b()
partial decompose	3	0	1	0	0	y3 = x1 + y1 + x2 + y2 + x3 and x1, y1, x2, y2 are all binary
add	2	0	0	1	0	a1 = a0 + b0
mul	2	0	0	0	1	a1 = a0 * b0

EC ops

Conditional Addition

index	a	b	q_ec	q1	q2	q3
	p1.x	p1.y	1	1	0	0
	p2.x	p2.y
	cond
offset	p3.x	p3.y

An addition is correct if

• p3 is on curve
• p3 satisfies (x2-x1)/(y2-y1) = (x3-x1)/(-y3-y1)

If cond == 1 return p3; else return p1

Doubling

index	a	b	q_ec	q1	q2	q3
	p1.x	p1.y	1	0	1	0
offset	p3.x	p3.y

A doubling is correct if

• p3 is on curve
• p3 satisfies 2y1 * (y3 + y1) + 3x1^2 * (x3 - x1) = 0

On Curve

index	a	b	q_ec	q1	q2	q3
offset	p1.x	p1.y	1	0	0	1

Field ops

partial_bit_decomp

index	a	b	q_ec	q1	q2	q3
	x1	y1	0	1	0	0
	x2	y2
offset	x3	y3

Assertions:

• x3 = x1 + 2y1 + 4x2 + 8y2 + 16y3
• x1, y1, x2, y2 are all binary