Zellic - Audit Reports

Function: `_modifiedJacobianDouble(uint256 x, uint256 y, uint256 z)`

This function takes as parameters Jacobian coordinates (x:y:z) of a point on the elliptic curve secp256r1 and returns Jacobian coordinates representing double that point. Additionally, when passed (0,0,0), the function should return (0,0,0). (See Finding ref↗.)

Inputs

(x, y, z):

Validation: No checks in this function.

Impact: Caller should ensure that (x:y:z) are valid Jacobian coordinates for a point on the secp256r1 curve, or (x,y,z)=(0,0,0).

Correctness:

pd agrees with the constant pp defined in the library.

The inline constant used for a agree with the constant a defined in the library.

Assuming (x:y:z) are valid Jacobian coordinates for a point on the secp256r1 curve, (x3:y3:z3) = 2(x:y:z).

If (x,y,z)=(0,0,0), then the function returns (0,0,0).

The return values satisfy 0 <= x3, y3, z3 < pd (for this check, we refer also to section ref↗).

Detailed steps taken to check correctness of the result: Assuming (x,y,z) are valid Jacobian coordinates for a point on the secp256r1 curve, we compare each of the steps of the computation done in the function with the reference book. Equality is here to be taken modulo pd.

z2 is what is called

z_{1}^{2}

in the reference book.

az4 is what is called

a z_{1}^{4}

in the reference book.

y2 is what is called

y_{1}^{2}

in the reference book.

s is what is called

v

in the reference book.

u is what is called

8 y_{1}^{4}

in the reference book.

m is what is called

w

in the reference book.

twos is what is called

2 v

in the reference book.

m2 is what is called

w^{2}

in the reference book.

x3 is what is called

x_{3}

in the reference book.

After the first assignment, y3 is what is called

(v - x_{3}) w

in the reference book.

After the last assignment, y3 is what is called

y_{3}

in the reference book.

z3 is what is called

z_{3}

in the reference book.

Function: _modifiedJacobianDouble(uint256 x, uint256 y, uint256 z)

Inputs

Function: `_modifiedJacobianDouble(uint256 x, uint256 y, uint256 z)`