Polynomial division done

aux/calculate2roots.js

@@ -0,0 +1,66 @@
+
+const bigInt = require("../src/bigint.js");
+const ZqField = require("../src/zqfield.js");
+
+
+const r = bigInt("21888242871839275222246405745257275088548364400416034343698204186575808495617");
+const s = 28;
+const nqr_to_t = bigInt("19103219067921713944291392827692070036145651957329286315305642004821462161904");
+const t_minus_1_over_2 = bigInt("40770029410420498293352137776570907027550720424234931066070132305055");
+const root_unity = bigInt("19103219067921713944291392827692070036145651957329286315305642004821462161904");
+const t = bigInt("81540058820840996586704275553141814055101440848469862132140264610111");
+
+const F = new ZqField(r);
+
+function sqrt(a) {
+
+    let v = s;
+    let z = nqr_to_t;
+    let w = F.exp(a, t_minus_1_over_2);
+    let x = F.mul(a, w);
+    let b = F.mul(x, w);
+
+
+    // compute square root with Tonelli--Shanks
+    // (does not terminate if not a square!)
+
+    while (!F.equals(b, F.one))
+    {
+        let m = 0;
+        let b2m = b;
+        while (!F.equals(b2m, F.one))
+        {
+            /* invariant: b2m = b^(2^m) after entering this loop */
+            b2m = F.square(b2m);
+            m += 1;
+        }
+
+        let j = v-m-1;
+        w = z;
+        while (j > 0)
+        {
+            w = F.square(w);
+            --j;
+        } // w = z^2^(v-m-1)
+
+        z = F.square(w);
+        b = F.mul(b, z);
+        x = F.mul(x, w);
+        v = m;
+    }
+
+    return x;
+}
+
+const p_minus1= F.sub(r,bigInt(1));
+const gen = bigInt(bigInt(5));
+const twoto28= F.exp(bigInt(2), bigInt(28));
+const rem = F.div(p_minus1, twoto28);
+const w28 = F.exp(gen, rem);
+
+const one = F.exp(w28, twoto28);
+
+
+console.log(F.toString(w28));
+console.log(w28.toString(10));
+console.log(F.toString(one));

file%3a/Users/jbaylina/git/personal/zksnark/src/polfield.js

@@ -0,0 +1,260 @@
+/*
+    This library do operations on polinomials where their coefficients are in field F
+
+    The polynomial P(x) = p0 + p1 * x + p2 * x^2 + p3 * x^3, ...
+    is represented by the array [ p0, p1, p2, p3, ...  ]
+ */
+
+const bigInt = require("./bigInt");
+const ZqField = require("./zqfield");
+
+class PolFieldZq {
+    constructor (q) {
+        this.F = new ZqField(q);
+
+        let rem = q.sub(bigInt(1));
+        let s = 0;
+        while (!rem.isOdd()) {
+            s ++;
+            rem = rem.shiftRight(1);
+        }
+
+        this.w = new Array(s+1);
+        this.wi = new Array(s+1);
+        this.w[s] = this.F.exp(bigInt(5), rem);
+        this.wi[s] = this.F.inverse(this.w[s]);
+
+        let n=s-1;
+        while (n>=0) {
+            this.w[n] = this.F.square(this.w[n+1]);
+            this.wi[n] = this.F.square(this.wi[n+1]);
+            n--;
+        }
+
+    }
+
+    add(a, b) {
+        const m = Math.max(a.length, b.length);
+        const res = new Array(m);
+        for (let i=0; i<m; i++) {
+            res[i] = this.F.add(a[i] || this.F.zero, b[i] || this.F.zero);
+        }
+        return this.reduce(res);
+    }
+
+    double(a) {
+        return this.add(a,a);
+    }
+
+    sub(a, b) {
+        const m = Math.max(a.length, b.length);
+        const res = new Array(m);
+        for (let i=0; i<m; i++) {
+            res[i] = this.F.sub(a[i] || this.F.zero, b[i] || this.F.zero);
+        }
+        return this.reduce(res);
+    }
+
+    mulEscalar(a, b) {
+        if (this.F.isZero(b)) return [];
+        const res = new Array(a.length);
+        for (let i=0; i<a.length; i++) {
+            res[i] = this.F.mul(a[i], b);
+        }
+        return res;
+    }
+
+    mul(a, b) {
+        if (a.length == 0) return [];
+        if (b.length == 0) return [];
+        if (a.length == 1) return this.mulEscalar(b, a[0]);
+        if (b.length == 1) return this.mulEscalar(a, b[0]);
+
+        const longestN = Math.max(a.length, b.length);
+        const bitsResult = log2(longestN-1)+2;
+        const m = 1 << bitsResult;
+        const ea = this.extend(a,m);
+        const eb = this.extend(b,m);
+
+        const ta = this._fft(ea, bitsResult, 0, 1, false);
+        const tb = this._fft(eb, bitsResult, 0, 1, false);
+
+        const tres = new Array(m);
+
+        for (let i=0; i<m; i++) {
+            tres[i] = this.F.mul(ta[i], tb[i]);
+        }
+
+        const res = this._fft(tres, bitsResult, 0, 1, true);
+
+        const twoinvm = this.F.inverse(bigInt(m));
+        const resn = new Array(m);
+        for (let i=0; i<m; i++) {
+            resn[i] = this.F.mul(res[(m-i)%m], twoinvm);
+        }
+
+        return this.reduce(this.affine(resn));
+    }
+
+    square(a) {
+        return this.mul(a,a);
+    }
+
+    scaleX(p, n) {
+        if (n==0) {
+            return p;
+        } else if (n>0) {
+            const z = new Array(n).fill(this.F.zero);
+            return z.concat(p);
+        } else {
+            return p.slice(-n);
+        }
+    }
+
+    div(a, b) {
+        throw new Error("Not Implementted");
+    }
+
+    eval(p, x) {
+        let v = this.F.zero;
+        let ix = this.F.one;
+        for (let i=0; i<p.length; i++) {
+            v = this.F.add(v, this.F.mul(p[i], ix));
+            ix = this.F.mul(ix, x);
+        }
+        return v;
+    }
+
+    lagrange(points) {
+        throw new Error("Not Implementted");
+    }
+
+    _fft(pall, bits, offset, step) {
+
+        const n = 1 << bits;
+        if (n==1) {
+            return [ pall[offset] ];
+        }
+
+        const ndiv2 = n >> 1;
+        const p1 = this._fft(pall, bits-1, offset, step*2);
+        const p2 = this._fft(pall, bits-1, offset+step, step*2);
+
+        const out = new Array(n);
+
+        let m= bigInt(1);
+        for (let i=0; i<ndiv2; i++) {
+            out[i] = this.F.add(p1[i], this.F.mul(m, p2[i]));
+            out[i+ndiv2] = this.F.sub(p1[i], this.F.mul(m, p2[i]));
+            m = this.F.mul(m, this.w[bits]);
+        }
+
+        return out;
+    }
+
+    extend(p, e) {
+        if (e == p.length) return p;
+        const z = new Array(e-p.length).fill(this.F.zero);
+
+        return p.concat(z);
+    }
+
+    reduce(p) {
+        if (p.length == 0) return p;
+        if (! this.F.isZero(p[p.length-1]) ) return p;
+        let i=p.length-1;
+        while( i>0 && this.F.isZero(p[i]) ) i--;
+        return p.slice(0, i+1);
+    }
+
+    affine(p) {
+        for (let i=0; i<p.length; i++) {
+            p[i] = this.F.affine(p[i]);
+        }
+        return p;
+    }
+
+    equals(a, b) {
+        const pa = this.reduce(this.affine(a));
+        const pb = this.reduce(this.affine(b));
+
+        if (pa.length != pb.length) return false;
+        for (let i=0; i<pb.length; i++) {
+            if (!this.F.equals(pa[i], pb[i])) return false;
+        }
+
+        return true;
+    }
+
+    _next2Power(v) {
+        v--;
+        v |= v >> 1;
+        v |= v >> 2;
+        v |= v >> 4;
+        v |= v >> 8;
+        v |= v >> 16;
+        v++;
+        return v;
+    }
+
+    toString(p) {
+        const ap = this.affine(p);
+        let S = "";
+        for (let i=ap.length-1; i>=0; i--) {
+            if (!this.F.isZero(p[i])) {
+                if (S!="") S += " + ";
+                S = S + p[i].toString(10);
+                if (i>0) {
+                    S = S + "x";
+                    if (i>1) {
+                        S = S + "^" +i;
+                    }
+                }
+            }
+        }
+        return S;
+    }
+
+
+    _reciprocal(p, bits) {
+        const k = 1 << bits;
+        if (k==1) {
+            return [ this.F.inverse(p[0]) ];
+        }
+        const np = this.scaleX(p, -k/2);
+        const q = this._reciprocal(np, bits-1);
+        const a = this.scaleX(this.double(q), 3*k/2-2);
+        const b = this.mul( this.square(q), p);
+
+        return this.scaleX(this.sub(a,b),   -(k-2));
+    }
+
+    // divides x^m / v
+    _div2(m, v) {
+        const kbits = log2(v.length-1)+1;
+        const k = 1 << kbits;
+
+        const scaleV = k - v.length;
+
+        // rec = x^(k - 2) / v* x^scaleV =>
+        // rec = x^(k-2-scaleV)/ v
+        //
+        // res = x^m/v = x^(m +(2k-2-scaleV) -(2k-2-scaleV)) /v =>
+        // res = rec * x^(m - (2k-2-scaleV)) =>
+        // res = rec * x^(m - 2k +2 + scaleV)
+
+        const rec = this._reciprocal(this.scaleX(v, scaleV), kbits);
+        const res = this.scaleX(rec, m - k*2 +2+scaleV);
+
+        return res;
+
+    }
+
+}
+
+function log2( V )
+{
+    return( ( ( V & 0xFFFF0000 ) !== 0 ? ( V &= 0xFFFF0000, 16 ) : 0 ) | ( ( V & 0xFF00FF00 ) !== 0 ? ( V &= 0xFF00FF00, 8 ) : 0 ) | ( ( V & 0xF0F0F0F0 ) !== 0 ? ( V &= 0xF0F0F0F0, 4 ) : 0 ) | ( ( V & 0xCCCCCCCC ) !== 0 ? ( V &= 0xCCCCCCCC, 2 ) : 0 ) | ( ( V & 0xAAAAAAAA ) !== 0 ) );
+}
+
+module.exports = PolFieldZq;

src/bigint.js

@@ -3,8 +3,6 @@ const bigInt = require("big-integer");
 
 let wBigInt;
 
-console.log("XXX");
-
 if (typeof(BigInt) != "undefined") {
     wBigInt  = BigInt;
     wBigInt.one = wBigInt(1);

src/polfield.js

@@ -5,46 +5,291 @@
     is represented by the array [ p0, p1, p2, p3, ...  ]
  */
 
-class PolField {
+const bigInt = require("./bigInt");
-    constructor (F) {
+const ZqField = require("./zqfield");
-        this.F = F;
+
+class PolFieldZq {
+    constructor (q) {
+        this.F = new ZqField(q);
+
+        let rem = q.sub(bigInt(1));
+        let s = 0;
+        while (!rem.isOdd()) {
+            s ++;
+            rem = rem.shiftRight(1);
+        }
+
+        this.w = new Array(s+1);
+        this.wi = new Array(s+1);
+        this.w[s] = this.F.exp(bigInt(5), rem);
+        this.wi[s] = this.F.inverse(this.w[s]);
+
+        let n=s-1;
+        while (n>=0) {
+            this.w[n] = this.F.square(this.w[n+1]);
+            this.wi[n] = this.F.square(this.wi[n+1]);
+            n--;
         }
 
-    _reduce(a) {
-        let i = a.length-1;
-        while ((i>=0) && (this.F.isZero(a[i]))  ) i--;
-        return (i < a.length-1) ? a.slice(0, i+1) : a;
     }
 
     add(a, b) {
-        const maxGrade = Math.max(a.length, b.length);
+        const m = Math.max(a.length, b.length);
-        const res = new Array(maxGrade);
+        const res = new Array(m);
-        for (let i=0; i<maxGrade; i++) {
+        for (let i=0; i<m; i++) {
-            res[i] = this.F.add(a[i], b[i]);
+            res[i] = this.F.add(a[i] || this.F.zero, b[i] || this.F.zero);
         }
-        return this._reduce(res);
+        return this.reduce(res);
+    }
+
+    double(a) {
+        return this.add(a,a);
     }
 
     sub(a, b) {
-        throw new Error("Not Implementted");
+        const m = Math.max(a.length, b.length);
+        const res = new Array(m);
+        for (let i=0; i<m; i++) {
+            res[i] = this.F.sub(a[i] || this.F.zero, b[i] || this.F.zero);
+        }
+        return this.reduce(res);
+    }
+
+    mulEscalar(a, b) {
+        if (this.F.isZero(b)) return [];
+        const res = new Array(a.length);
+        for (let i=0; i<a.length; i++) {
+            res[i] = this.F.mul(a[i], b);
+        }
+        return res;
     }
 
     mul(a, b) {
-        throw new Error("Not Implementted");
+        if (a.length == 0) return [];
+        if (b.length == 0) return [];
+        if (a.length == 1) return this.mulEscalar(b, a[0]);
+        if (b.length == 1) return this.mulEscalar(a, b[0]);
+
+        const longestN = Math.max(a.length, b.length);
+        const bitsResult = log2(longestN-1)+2;
+        const m = 1 << bitsResult;
+        const ea = this.extend(a,m);
+        const eb = this.extend(b,m);
+
+        const ta = this._fft(ea, bitsResult, 0, 1, false);
+        const tb = this._fft(eb, bitsResult, 0, 1, false);
+
+        const tres = new Array(m);
+
+        for (let i=0; i<m; i++) {
+            tres[i] = this.F.mul(ta[i], tb[i]);
         }
 
-    div(a, b) {
+        const res = this._fft(tres, bitsResult, 0, 1, true);
-        throw new Error("Not Implementted");
+
+        const twoinvm = this.F.inverse(bigInt(m));
+        const resn = new Array(m);
+        for (let i=0; i<m; i++) {
+            resn[i] = this.F.mul(res[(m-i)%m], twoinvm);
+        }
+
+        return this.reduce(this.affine(resn));
+    }
+
+
+
+    square(a) {
+        return this.mul(a,a);
+    }
+
+    scaleX(p, n) {
+        if (n==0) {
+            return p;
+        } else if (n>0) {
+            const z = new Array(n).fill(this.F.zero);
+            return z.concat(p);
+        } else {
+            if (-n >= p.length) return [];
+            return p.slice(-n);
+        }
     }
 
     eval(p, x) {
-        throw new Error("Not Implementted");
+        let v = this.F.zero;
+        let ix = this.F.one;
+        for (let i=0; i<p.length; i++) {
+            v = this.F.add(v, this.F.mul(p[i], ix));
+            ix = this.F.mul(ix, x);
+        }
+        return v;
     }
 
     lagrange(points) {
         throw new Error("Not Implementted");
     }
+
+    _fft(pall, bits, offset, step) {
+
+        const n = 1 << bits;
+        if (n==1) {
+            return [ pall[offset] ];
+        }
+
+        const ndiv2 = n >> 1;
+        const p1 = this._fft(pall, bits-1, offset, step*2);
+        const p2 = this._fft(pall, bits-1, offset+step, step*2);
+
+        const out = new Array(n);
+
+        let m= bigInt(1);
+        for (let i=0; i<ndiv2; i++) {
+            out[i] = this.F.add(p1[i], this.F.mul(m, p2[i]));
+            out[i+ndiv2] = this.F.sub(p1[i], this.F.mul(m, p2[i]));
+            m = this.F.mul(m, this.w[bits]);
+        }
+
+        return out;
+    }
+
+    extend(p, e) {
+        if (e == p.length) return p;
+        const z = new Array(e-p.length).fill(this.F.zero);
+
+        return p.concat(z);
+    }
+
+    reduce(p) {
+        if (p.length == 0) return p;
+        if (! this.F.isZero(p[p.length-1]) ) return p;
+        let i=p.length-1;
+        while( i>0 && this.F.isZero(p[i]) ) i--;
+        return p.slice(0, i+1);
+    }
+
+    affine(p) {
+        for (let i=0; i<p.length; i++) {
+            p[i] = this.F.affine(p[i]);
+        }
+        return p;
+    }
+
+    equals(a, b) {
+        const pa = this.reduce(this.affine(a));
+        const pb = this.reduce(this.affine(b));
+
+        if (pa.length != pb.length) return false;
+        for (let i=0; i<pb.length; i++) {
+            if (!this.F.equals(pa[i], pb[i])) return false;
+        }
+
+        return true;
+    }
+
+    _next2Power(v) {
+        v--;
+        v |= v >> 1;
+        v |= v >> 2;
+        v |= v >> 4;
+        v |= v >> 8;
+        v |= v >> 16;
+        v++;
+        return v;
+    }
+
+    toString(p) {
+        const ap = this.affine(p);
+        let S = "";
+        for (let i=ap.length-1; i>=0; i--) {
+            if (!this.F.isZero(p[i])) {
+                if (S!="") S += " + ";
+                S = S + p[i].toString(10);
+                if (i>0) {
+                    S = S + "x";
+                    if (i>1) {
+                        S = S + "^" +i;
+                    }
+                }
+            }
+        }
+        return S;
     }
 
 
-module.exports = PolField;
+    _reciprocal(p, bits) {
+        const k = 1 << bits;
+        if (k==1) {
+            return [ this.F.inverse(p[0]) ];
+        }
+        const np = this.scaleX(p, -k/2);
+        const q = this._reciprocal(np, bits-1);
+        const a = this.scaleX(this.double(q), 3*k/2-2);
+        const b = this.mul( this.square(q), p);
+
+        return this.scaleX(this.sub(a,b),   -(k-2));
+    }
+
+    // divides x^m / v
+    _div2(m, v) {
+        const kbits = log2(v.length-1)+1;
+        const k = 1 << kbits;
+
+        const scaleV = k - v.length;
+
+        // rec = x^(k - 2) / v* x^scaleV =>
+        // rec = x^(k-2-scaleV)/ v
+        //
+        // res = x^m/v = x^(m + (2*k-2 - scaleV) - (2*k-2 - scaleV)) /v =>
+        // res = rec * x^(m - (2*k-2 - scaleV)) =>
+        // res = rec * x^(m - 2*k + 2 + scaleV)
+
+        const rec = this._reciprocal(this.scaleX(v, scaleV), kbits);
+        const res = this.scaleX(rec, m - 2*k + 2 + scaleV);
+
+        return res;
+    }
+
+    div(_u, _v) {
+        if (_u.length < _v.length) return [];
+        const kbits = log2(_v.length-1)+1;
+        const k = 1 << kbits;
+
+        const u = this.scaleX(_u, k-_v.length);
+        const v = this.scaleX(_v, k-_v.length);
+
+        const n = v.length-1;
+        let m = u.length-1;
+
+        const s = this._reciprocal(v, kbits);
+        let t;
+        if (m>2*n) {
+            t = this.sub(this.scaleX([bigInt(1)], 2*n), this.mul(s, v));
+        }
+
+        let q = [];
+        let rem = u;
+        let us, ut;
+        let finish = false;
+
+        while (!finish) {
+            us = this.mul(rem, s);
+            q = this.add(q, this.scaleX(us, -2*n));
+
+            if ( m > 2*n ) {
+                ut = this.mul(rem, t);
+                rem = this.scaleX(ut, -2*n);
+                m = rem.length-1;
+            } else {
+                finish = true;
+            }
+        }
+
+        return q;
+    }
+}
+
+function log2( V )
+{
+    return( ( ( V & 0xFFFF0000 ) !== 0 ? ( V &= 0xFFFF0000, 16 ) : 0 ) | ( ( V & 0xFF00FF00 ) !== 0 ? ( V &= 0xFF00FF00, 8 ) : 0 ) | ( ( V & 0xF0F0F0F0 ) !== 0 ? ( V &= 0xF0F0F0F0, 4 ) : 0 ) | ( ( V & 0xCCCCCCCC ) !== 0 ? ( V &= 0xCCCCCCCC, 2 ) : 0 ) | ( ( V & 0xAAAAAAAA ) !== 0 ) );
+}
+
+module.exports = PolFieldZq;

src/zqfield.js

@@ -16,6 +16,8 @@ class ZqField {
         this.equals = bigInt.genEquals(q);
         this.affine = bigInt.genAffine(q);
         this.isZero = bigInt.genIsZero(q);
+        this.two = this.add(this.one, this.one);
+        this.twoinv = this.inverse(this.two);
     }
 
     copy(a) {

test/algebra.js

@@ -154,7 +154,6 @@ describe("Pairing", () => {
             const g1b = bn128.G1.mulEscalar(bn128.G1.g, 30);
             const g2b = bn128.G2.mulEscalar(bn128.G2.g, 25);
 
-
             const pre1a = bn128.precomputeG1(g1a);
             const pre2a = bn128.precomputeG2(g2a);
             const pre1b = bn128.precomputeG1(g1b);

test/pols.js

@@ -0,0 +1,107 @@
+const chai = require("chai");
+
+const bigInt = require("../src/bigint.js");
+const PolField = require("../src/polfield.js");
+
+const assert = chai.assert;
+
+const r  = bigInt("21888242871839275222246405745257275088548364400416034343698204186575808495617");
+
+describe("Polinomial field", () => {
+    it("Should compute a multiplication", () => {
+        const PF = new PolField(r);
+
+        const a = [bigInt(1), bigInt(2), bigInt(3)];
+        const b = [bigInt(1), bigInt(2), bigInt(3)];
+        const res = PF.mul(a,b);
+
+        assert(PF.equals(res, [bigInt(1), bigInt(4), bigInt(10), bigInt(12), bigInt(9)]));
+    });
+    it("Should compute a multiplication 2", () => {
+        const PF = new PolField(r);
+
+        const a = [bigInt(5), bigInt(1)];
+        const b = [bigInt(-5), bigInt(1)];
+        const res = PF.mul(a,b);
+
+        assert(PF.equals(res, [bigInt(-25), bigInt(0), bigInt(1)]));
+    });
+    it("Should compute an addition", () => {
+        const PF = new PolField(r);
+
+        const a = [bigInt(5), bigInt(1)];
+        const b = [bigInt(-5), bigInt(1)];
+        const res = PF.add(a,b);
+
+        assert(PF.equals(res, [bigInt(0), bigInt(2)]));
+    });
+    it("Should compute a substraction", () => {
+        const PF = new PolField(r);
+
+        const a = [bigInt(5), bigInt(3), bigInt(4)];
+        const b = [bigInt(5), bigInt(1)];
+        const res = PF.sub(a,b);
+
+        assert(PF.equals(res, [bigInt(0), bigInt(2), bigInt(4)]));
+    });
+    it("Should compute reciprocal", () => {
+        const PF = new PolField(r);
+
+        const a = [bigInt(4), bigInt(1), bigInt(-3), bigInt(-1), bigInt(2),bigInt(1), bigInt(-1), bigInt(1)];
+        const res = PF._reciprocal(a, 3, 0);
+
+        assert(PF.equals(res, [bigInt(12), bigInt(15), bigInt(3), bigInt(-4), bigInt(-3), bigInt(0), bigInt(1), bigInt(1)]));
+    });
+    it("Should div2", () => {
+        const PF = new PolField(r);
+
+        // x^6
+        const a = [bigInt(0), bigInt(0), bigInt(0), bigInt(0), bigInt(0),bigInt(0), bigInt(1)];
+        // x^5
+        const b = [bigInt(0), bigInt(0), bigInt(0), bigInt(0), bigInt(0), bigInt(1)];
+
+        const res = PF._div2(6, b);
+        assert(PF.equals(res, [bigInt(0), bigInt(1)]));
+
+        const res2 = PF.div(a,b);
+        assert(PF.equals(res2, [bigInt(0), bigInt(1)]));
+    });
+    it("Should div", () => {
+        const PF = new PolField(r);
+
+        const a = [bigInt(1), bigInt(2), bigInt(3), bigInt(4), bigInt(5),bigInt(6), bigInt(7)];
+        const b = [bigInt(8), bigInt(9), bigInt(10), bigInt(11), bigInt(12), bigInt(13)];
+
+        const c = PF.mul(a,b);
+        const d = PF.div(c,b);
+
+        assert(PF.equals(a, d));
+    });
+
+    it("Should div big/small", () => {
+        const PF = new PolField(r);
+
+        const a = [bigInt(1), bigInt(2), bigInt(3), bigInt(4), bigInt(5),bigInt(6), bigInt(7)];
+        const b = [bigInt(8), bigInt(9)];
+
+        const c = PF.mul(a,b);
+        const d = PF.div(c,b);
+
+        assert(PF.equals(a, d));
+    });
+    it("Should div random big", () => {
+        const PF = new PolField(r);
+
+        const a = [];
+        const b = [];
+        for (let i=0; i<1000; i++) a.push(bigInt(Math.floor(Math.random()*100000) -500000));
+        for (let i=0; i<300; i++) b.push(bigInt(Math.floor(Math.random()*100000) -500000));
+
+        const c = PF.mul(a,b);
+
+        const d = PF.div(c,b);
+
+        assert(PF.equals(a, d));
+    }).timeout(10000000);
+
+});