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Fix and finite field operations (isSquare, sqrt) #287

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merged 7 commits into from
Jul 31, 2024
Merged

Fix and finite field operations (isSquare, sqrt) #287

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4e5d15e
fix(finite-field): ensure isSquare and sqrt methods take input modulo p
MartinMinkov Jul 23, 2024
901bdde
test(finite-field): add tests for non-canonical zero handling in sqrt…
MartinMinkov Jul 23, 2024
261522b
refactor(finite-field): simplify negate function to use mod(-x, p) in…
MartinMinkov Jul 30, 2024
cc3f632
refactor(finite-field): improve sqrt and isSquare functions by moddin…
MartinMinkov Jul 30, 2024
3674c62
test(finite-field): add test cases for non-canonical zero
MartinMinkov Jul 30, 2024
ba836e3
fix(finite-field): check for even number correctly by first reducing …
MartinMinkov Jul 30, 2024
71f2e69
docs(finite-field): add comment explaining the logic behind the equal…
MartinMinkov Jul 30, 2024
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15 changes: 10 additions & 5 deletions crypto/finite-field.ts
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Original file line number Diff line number Diff line change
Expand Up @@ -184,12 +184,13 @@ function fastInverse(
return s;
}

function sqrt(n: bigint, p: bigint, Q: bigint, c: bigint, M: bigint) {
function sqrt(n_: bigint, p: bigint, Q: bigint, c: bigint, M: bigint) {
// https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm#The_algorithm
// variable naming is the same as in that link ^
// Q is what we call `t` elsewhere - the odd factor in p - 1
// c is a known primitive root of unity
// M is the twoadicity = exponent of 2 in factorization of p - 1
let n = mod(n_, p);
if (n === 0n) return 0n;
let t = power(n, (Q - 1n) >> 1n, p); // n^(Q - 1)/2
let R = mod(t * n, p); // n^((Q - 1)/2 + 1) = n^((Q + 1)/2)
Expand All @@ -212,7 +213,8 @@ function sqrt(n: bigint, p: bigint, Q: bigint, c: bigint, M: bigint) {
}
}

function isSquare(x: bigint, p: bigint) {
function isSquare(x_: bigint, p: bigint) {
let x = mod(x_, p);
if (x === 0n) return true;
let sqrt1 = power(x, (p - 1n) / 2n, p);
return sqrt1 === 1n;
Expand Down Expand Up @@ -282,7 +284,7 @@ function createField(
return mod(2n ** BigInt(bits) - (x + 1n), p);
},
negate(x: bigint) {
return x === 0n ? 0n : p - x;
return x === 0n ? 0n : mod(-x, p);
},
sub(x: bigint, y: bigint) {
return mod(x - y, p);
Expand Down Expand Up @@ -317,10 +319,13 @@ function createField(
return mod(z, p);
},
equal(x: bigint, y: bigint) {
return mod(x - y, p) === 0n;
// We check if x and y are both in the range [0, p). If they are, can do a simple comparison. Otherwise, we need to reduce them to the proper canonical field range.
let x_ = x >= 0n && x < p ? x : mod(x, p);
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I think I am okay with this, but let's add a comment here explaining what and why we do it this particular way, i can imagine it might seem a little weird to the naked eye

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fixed here! 71f2e69

let y_ = y >= 0n && y < p ? y : mod(y, p);
return x_ === y_;
},
isEven(x: bigint) {
return !(x & 1n);
return !(mod(x, p) & 1n);
},
random() {
return randomField(p, sizeInBytes, hiBitMask);
Expand Down
8 changes: 8 additions & 0 deletions crypto/finite-field.unit-test.ts
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Original file line number Diff line number Diff line change
Expand Up @@ -28,9 +28,11 @@ for (let F of fields) {
assert.equal(F.sub(3n, 3n), 0n, 'sub');
assert.equal(F.sub(3n, 8n), p - 5n, 'sub');
assert.equal(F.negate(5n), p - 5n, 'negate');
assert.equal(F.negate(p), 0n, 'non-canonical 0 is negated');
assert.equal(F.add(x, F.negate(x)), 0n, 'add & negate');
assert.equal(F.sub(F.add(x, y), x), y, 'add & sub');
assert.equal(F.isEven(17n), false, 'isEven');
assert.equal(F.isEven(p), true, 'non-canonical 0 is even');
assert.equal(F.isEven(p - 1n), true, 'isEven');

assert.equal(F.mul(p - 1n, 2n), p - 2n, 'mul');
Expand Down Expand Up @@ -70,9 +72,15 @@ for (let F of fields) {
let squareX = F.square(x);
assert(F.isSquare(squareX), 'square + isSquare');
assert([x, F.negate(x)].includes(F.sqrt(squareX)!), 'square + sqrt');
assert.equal(F.sqrt(0n), F.sqrt(p), 'sqrt handles non-canonical 0');

if (F.M >= 2n) {
assert(F.isSquare(p - 1n), 'isSquare -1');
assert.equal(
F.isSquare(0n),
F.isSquare(p),
'isSquare handles non-canonical 0'
);
let i = F.power(F.twoadicRoot, 1n << (F.M - 2n));
assert([i, F.negate(i)].includes(F.sqrt(p - 1n)!), 'sqrt -1');
}
Expand Down
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