M README.md → README.md

@@ -1,5 +1,36 @@
 # kzg-commitments-study [![GoDoc](https://godoc.org/github.com/arnaucube/kzg-commitments-study?status.svg)](https://godoc.org/github.com/arnaucube/kzg-commitments-study) [![Go Report Card](https://goreportcard.com/badge/github.com/arnaucube/kzg-commitments-study)](https://goreportcard.com/report/github.com/arnaucube/kzg-commitments-study) [![Test](https://github.com/arnaucube/kzg-commitments-study/workflows/Test/badge.svg)](https://github.com/arnaucube/kzg-commitments-study/actions?query=workflow%3ATest)
 
-Doing this to study and learn [KZG commitments](http://cacr.uwaterloo.ca/techreports/2010/cacr2010-10.pdf), do not use in production.
+Doing this to study and learn [KZG commitments](http://cacr.uwaterloo.ca/techreports/2010/cacr2010-10.pdf), do not use in production. More details at https://arnaucube.com/blog/kzg-commitments.html .
 
 Thanks to [Dankrad Feist](https://dankradfeist.de/ethereum/2020/06/16/kate-polynomial-commitments.html), [Alin Tomescu](https://alinush.github.io/2020/05/06/kzg-polynomial-commitments.html), [Tom Walton-Pocock](https://hackmd.io/@tompocock/Hk2A7BD6U) for their articles, which helped me understand a bit the KZG Commitments.
+
+### Usage
+```go
+// p(x) = x^3 + x + 5
+p := []*big.Int{
+	big.NewInt(5),
+	big.NewInt(1), // x^1
+	big.NewInt(0), // x^2
+	big.NewInt(1), // x^3
+}
+assert.Equal(t, "1x³ + 1x¹ + 5", PolynomialToString(p))
+
+// TrustedSetup
+ts, err := NewTrustedSetup(p)
+assert.Nil(t, err)
+
+// Commit
+c := Commit(ts, p)
+
+// p(z)=y --> p(3)=35
+z := big.NewInt(3)
+y := big.NewInt(35)
+
+// z & y: to prove an evaluation p(z)=y
+proof, err := EvaluationProof(ts, p, z, y)
+assert.Nil(t, err)
+
+// verification
+v := Verify(ts, c, proof, z, y)
+assert.True(t, v)
+```

M arithmetic.go → arithmetic.go

@@ -224,7 +224,9 @@ }
 	return sN
 }
 
-func polynomialToString(p []*big.Int) string {
+// PolynomialToString converts a polynomial represented by a *big.Int array,
+// into its string human readable representation
+func PolynomialToString(p []*big.Int) string {
 	s := ""
 	for i := len(p) - 1; i >= 1; i-- {
 		if !bytes.Equal(p[i].Bytes(), big.NewInt(0).Bytes()) {

M kzg.go → kzg.go

@@ -65,14 +65,15 @@ }
 
 // EvaluationProof generates the evaluation proof
 func EvaluationProof(ts *TrustedSetup, p []*big.Int, z, y *big.Int) (*bn256.G1, error) {
-	n := polynomialSub(p, []*big.Int{y})    // p-y
+	n := polynomialSub(p, []*big.Int{y}) // p-y
+	// n := p // we can omit y (p(z))
 	d := []*big.Int{fNeg(z), big.NewInt(1)} // x-z
 	q, rem := polynomialDiv(n, d)
 	if compareBigIntArray(rem, arrayOfZeroes(len(rem))) {
 		return nil,
 			fmt.Errorf("remainder should be 0, instead is %d", rem)
 	}
-	fmt.Println("q(x):", polynomialToString(q)) // TMP DBG
+	fmt.Println("q(x):", PolynomialToString(q)) // TMP DBG
 
 	// proof: e = [q(s)]₁
 	e := evaluateG1(ts, q)

M kzg_test.go → kzg_test.go

@@ -15,8 +15,7 @@ big.NewInt(1), // x^1
 		big.NewInt(0), // x^2
 		big.NewInt(1), // x^3
 	}
-	// fmt.Println("p(x):", polynomialToString(p))
-	assert.Equal(t, "1x³ + 1x¹ + 5", polynomialToString(p))
+	assert.Equal(t, "1x³ + 1x¹ + 5", PolynomialToString(p))
 
 	// TrustedSetup
 	ts, err := NewTrustedSetup(p)