/**
* Invert this field element and return the result. The inverse is found via Fermat's little
* theorem: a^p congruent a mod p and therefore a^(p-2) congruent a^-1 mod p
*
* @return The inverse of this field element.
*/
public Ed25519FieldElement invert() {
Ed25519FieldElement f0, f1;
// comments describe how exponent is created
// 2 == 2 * 1
f0 = this.square();
// 9 == 9
f1 = this.pow2to9();
// 11 == 9 + 2
f0 = f0.multiply(f1);
// 2^252 - 2^2
f1 = this.pow2to252sub4();
// 2^255 - 2^5
for (int i = 1; i < 4; ++i) {
f1 = f1.square();
}
// 2^255 - 21
return f1.multiply(f0);
}
/**
* Calculates and returns one of the square roots of u / v.
*
* <pre>{@code
* x = (u * v^3) * (u * v^7)^((p - 5) / 8) ==> x^2 = +-(u / v).
* }</pre>
*
* Note that this means x can be sqrt(u / v), -sqrt(u / v), +i * sqrt(u / v), -i * sqrt(u / v).
*
* @param u The nominator of the fraction.
* @param v The denominator of the fraction.
* @return The square root of u / v.
*/
public static Ed25519FieldElement sqrt(final Ed25519FieldElement u, final Ed25519FieldElement v) {
Ed25519FieldElement x;
final Ed25519FieldElement v3;
// v3 = v^3
v3 = v.square().multiply(v);
// x = (v3^2) * v * u = u * v^7
x = v3.square().multiply(v).multiply(u);
// x = (u * v^7)^((q - 5) / 8)
x = x.pow2to252sub4().multiply(x); // 2^252 - 3
// x = u * v^3 * (u * v^7)^((q - 5) / 8)
x = v3.multiply(u).multiply(x);
return x;
}
/**
* Computes this field element to the power of (2^252 - 4) and returns the result. This is a
* helper function for calculating the square root.
*
* @return This field element to the power of (2^252 - 4).
*/
private Ed25519FieldElement pow2to252sub4() {
Ed25519FieldElement f0, f1, f2;
// 2 == 2 * 1
f0 = this.square();
// 9
f1 = this.pow2to9();
// 11 == 9 + 2
f0 = f0.multiply(f1);
// 22 == 2 * 11
f0 = f0.square();
// 31 == 22 + 9
f0 = f1.multiply(f0);
// 2^6 - 2^1
f1 = f0.square();
// 2^10 - 2^5
for (int i = 1; i < 5; ++i) {
f1 = f1.square();
}
// 2^10 - 2^0
f0 = f1.multiply(f0);
// 2^11 - 2^1
f1 = f0.square();
// 2^20 - 2^10
for (int i = 1; i < 10; ++i) {
f1 = f1.square();
}
// 2^20 - 2^0
f1 = f1.multiply(f0);
// 2^21 - 2^1
f2 = f1.square();
// 2^40 - 2^20
for (int i = 1; i < 20; ++i) {
f2 = f2.square();
}
// 2^40 - 2^0
f1 = f2.multiply(f1);
// 2^41 - 2^1
f1 = f1.square();
// 2^50 - 2^10
for (int i = 1; i < 10; ++i) {
f1 = f1.square();
}
// 2^50 - 2^0
f0 = f1.multiply(f0);
// 2^51 - 2^1
f1 = f0.square();
// 2^100 - 2^50
for (int i = 1; i < 50; ++i) {
f1 = f1.square();
}
// 2^100 - 2^0
f1 = f1.multiply(f0);
// 2^101 - 2^1
f2 = f1.square();
// 2^200 - 2^100
for (int i = 1; i < 100; ++i) {
f2 = f2.square();
}
// 2^200 - 2^0
f1 = f2.multiply(f1);
// 2^201 - 2^1
f1 = f1.square();
// 2^250 - 2^50
for (int i = 1; i < 50; ++i) {
f1 = f1.square();
}
// 2^250 - 2^0
f0 = f1.multiply(f0);
// 2^251 - 2^1
f0 = f0.square();
// 2^252 - 2^2
return f0.square();
}