docs updated

2026-02-28 05:16:46 +01:00 · 2018-10-27 03:02:02 +02:00
parent 3d67f7912e
commit d4433a1008
4 changed files with 494 additions and 13 deletions
--- a/.gitignore
+++ b/.gitignore
@@ -1,2 +1,3 @@
 fmt
 schnorr.goBACKUP
+notes.md
--- a/README.md
+++ b/README.md
@@ -3,6 +3,19 @@
 Crypto algorithms from scratch. Academic purposes only.


+
+- [RSA cryptosystem & Blind signature & Homomorphic Multiplication](#rsa-cryptosystem--blind-signature--homomorphic-multiplication)
+- [Paillier cryptosystem & Homomorphic Addition](#paillier-cryptosystem--homomorphic-addition)
+- [Shamir Secret Sharing](#shamir-secret-sharing)
+- [Diffie-Hellman](#diffie-hellman)
+- [ECC](#ecc)
+- [ECC ElGamal](#ecc-elgamal)
+- [ECC ECDSA](#ecc-ecdsa)
+- [Schnorr signature](#schnorr-signature)
+- [Bn128](#bn128)
+
+---
+
 ## RSA cryptosystem & Blind signature & Homomorphic Multiplication
 - https://en.wikipedia.org/wiki/RSA_(cryptosystem)#
 - https://en.wikipedia.org/wiki/Blind_signature
@@ -13,10 +26,91 @@ Crypto algorithms from scratch. Academic purposes only.
 - [x] Decrypt
 - [x] Blind
 - [x] Blind Signature
- [x] Unblind Signature- RSA- RSA  
+- [x] Unblind Signature
 - [x] Verify Signature
 - [x] Homomorphic Multiplication

+
+#### Usage
+- Key generation, Encryption, Decryption
+```go
+// generate key pair
+key, err := GenerateKeyPair()
+if err!=nil {
+	fmt.Println(err)
+}
+mBytes := []byte("Hi")
+m := new(big.Int).SetBytes(mBytes)
+
+// encrypt message
+c := Encrypt(m, key.PubK)
+
+// decrypt ciphertext
+d := Decrypt(c, key.PrivK)
+if m == d {
+	fmt.Println("correctly decrypted")
+}
+```
+
+- Blind signatures
+```go
+// key generation [Alice]
+key, err := GenerateKeyPair()
+if err!=nil {
+	fmt.Println(err)
+}
+
+// create new message [Alice]
+mBytes := []byte("Hi")
+m := new(big.Int).SetBytes(mBytes)
+
+// define r value [Alice]
+rVal := big.NewInt(int64(101))
+
+// blind message [Alice]
+mBlinded := Blind(m, rVal, key.PubK)
+
+// Blind Sign the blinded message [Bob]
+sigma := BlindSign(mBlinded, key.PrivK)
+
+// unblind the blinded signed message, and get the signature of the message [Alice]
+mSigned := Unblind(sigma, rVal, key.PubK)
+
+// verify the signature [Alice/Bob/Trudy]
+verified := Verify(m, mSigned, key.PubK)
+if !verified {
+	fmt.Println("signature could not be verified")
+}
+```
+
+- Homomorphic Multiplication
+```go
+// key generation [Alice]
+key, err := GenerateKeyPair()
+if err!=nil {
+	fmt.Println(err)
+}
+
+// define values [Alice]
+n1 := big.NewInt(int64(11))
+n2 := big.NewInt(int64(15))
+
+// encrypt the values [Alice]
+c1 := Encrypt(n1, key.PubK)
+c2 := Encrypt(n2, key.PubK)
+
+// compute homomorphic multiplication with the encrypted values [Bob]
+c3c4 := HomomorphicMul(c1, c2, key.PubK)
+
+// decrypt the result [Alice]
+d := Decrypt(c3c4, key.PrivK)
+
+// check that the result is the expected
+if !bytes.Equal(new(big.Int).Mul(n1, n2).Bytes(), d.Bytes()) {
+	fmt.Println("decrypted result not equal to expected result")
+}
+```
+
 ## Paillier cryptosystem & Homomorphic Addition
 - https://en.wikipedia.org/wiki/Paillier_cryptosystem
 - https://en.wikipedia.org/wiki/Homomorphic_encryption
@@ -26,12 +120,104 @@ Crypto algorithms from scratch. Academic purposes only.
 - [x] Decrypt
 - [x] Homomorphic Addition

+#### Usage
+- Encrypt, Decrypt
+```go
+// key generation
+key, err := GenerateKeyPair()
+if err!=nil {
+	fmt.Println(err)
+}
+
+mBytes := []byte("Hi")
+m := new(big.Int).SetBytes(mBytes)
+
+// encryption
+c := Encrypt(m, key.PubK)
+
+// decryption
+d := Decrypt(c, key.PubK, key.PrivK)
+if m == d {
+	fmt.Println("ciphertext decrypted correctly")
+}
+```
+
+- Homomorphic Addition
+```go
+// key generation [Alice]
+key, err := GenerateKeyPair()
+if err!=nil {
+	fmt.Println(err)
+}
+
+// define values [Alice]
+n1 := big.NewInt(int64(110))
+n2 := big.NewInt(int64(150))
+
+// encrypt values [Alice]
+c1 := Encrypt(n1, key.PubK)
+c2 := Encrypt(n2, key.PubK)
+
+// compute homomorphic addition [Bob]
+c3c4 := HomomorphicAddition(c1, c2, key.PubK)
+
+// decrypt the result [Alice]
+d := Decrypt(c3c4, key.PubK, key.PrivK)
+if !bytes.Equal(new(big.Int).Add(n1, n2).Bytes(), d.Bytes()) {
+	fmt.Println("decrypted result not equal to expected result")
+}
+```
+
+
 ## Shamir Secret Sharing
 - https://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing

 - [x] create secret sharing from number of secrets needed, number of shares, random point p, secret to share
 - [x] Lagrange Interpolation to restore the secret from the shares

+#### Usage
+```go
+// define secret to share
+k := 123456789
+
+// define random prime
+p, err := rand.Prime(rand.Reader, bits/2)
+if err!=nil {
+	fmt.Println(err)
+}
+
+// define how many secrets are needed to recover the secret
+nNeededSecrets := big.NewInt(int64(3))
+
+// define how many shares want to generate
+nShares := big.NewInt(int64(6))
+
+// create the shares
+shares, err := Create(
+	nNeededSecrets,
+	nShares,
+	p,
+	big.NewInt(int64(k)))
+assert.Nil(t, err)
+if err!=nil {
+	fmt.Println(err)
+}
+
+// select shares to use
+var sharesToUse [][]*big.Int
+sharesToUse = append(sharesToUse, shares[2])
+sharesToUse = append(sharesToUse, shares[1])
+sharesToUse = append(sharesToUse, shares[0])
+
+// recover the secret using Lagrange Interpolation
+secr := LagrangeInterpolation(sharesToUse, p)
+
+// check that the restored secret matches the original secret
+if !bytes.Equal(k.Bytes(), secr.Bytes()) {
+	fmt.Println("reconstructed secret not correspond to original secret")
+}
+```
+
 ## Diffie-Hellman
 - https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange

@@ -46,6 +232,39 @@ Crypto algorithms from scratch. Academic purposes only.
 - [x] Add two points on the elliptic curve
 - [x] Multiply a point n times on the elliptic curve

+#### Usage
+- ECC basic operations
+```go
+// define new ec
+ec := NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11)))
+
+// define two points over the curve
+p1 := Point{big.NewInt(int64(4)), big.NewInt(int64(7))}
+p2 := Point{big.NewInt(int64(2)), big.NewInt(int64(2))}
+
+// add the two points
+q, err := ec.Add(p1, p2)
+if err!=nil {
+	fmt.Println(err)
+}
+
+// multiply the two points
+q, err := ec.Mul(p, big.NewInt(int64(1)))
+if err!=nil {
+	fmt.Println(err)
+}
+
+// get order of a generator point over the elliptic curve
+g := Point{big.NewInt(int64(7)), big.NewInt(int64(8))}
+order, err := ec.Order(g)
+if err!=nil {
+	fmt.Println(err)
+}
+```
+
+
+
+
 ## ECC ElGamal
 - https://en.wikipedia.org/wiki/ElGamal_encryption

@@ -53,6 +272,52 @@ Crypto algorithms from scratch. Academic purposes only.
 - [x] ECC ElGamal Encrypton
 - [x] ECC ElGamal Decryption

+
+#### Usage
+- NewEG, Encryption, Decryption
+```go
+// define new elliptic curve
+ec := ecc.NewEC(big.NewInt(int64(1)), big.NewInt(int64(18)), big.NewInt(int64(19)))
+
+// define new point
+g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(11))}
+
+// define new ElGamal crypto system with the elliptic curve and the point
+eg, err := NewEG(ec, g)
+if err!=nil {
+	fmt.Println(err)
+}
+
+// define privK&pubK over the elliptic curve
+privK := big.NewInt(int64(5))
+pubK, err := eg.PubK(privK)
+if err!=nil {
+	fmt.Println(err)
+}
+
+// define point to encrypt
+m := ecc.Point{big.NewInt(int64(11)), big.NewInt(int64(12))}
+
+// encrypt
+c, err := eg.Encrypt(m, pubK, big.NewInt(int64(15)))
+if err!=nil {
+	fmt.Println(err)
+}
+
+// decrypt
+d, err := eg.Decrypt(c, privK)
+if err!=nil {
+	fmt.Println(err)
+}
+
+// check that decryption is correct
+if !m.Equal(d) {
+	fmt.Println("decrypted not equal to original")
+}
+```
+
+
+
 ## ECC ECDSA
 - https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm

@@ -61,6 +326,48 @@ Crypto algorithms from scratch. Academic purposes only.
 - [x] ECDSA Verify signature


+#### Usage
+```go
+// define new elliptic curve
+ec := ecc.NewEC(big.NewInt(int64(1)), big.NewInt(int64(18)), big.NewInt(int64(19)))
+// define new point
+g := ecc.Point{big.NewInt(int64(7)), big.NewInt(int64(11))}
+
+// define new ECDSA system
+dsa, err := NewDSA(ec, g)
+if err!=nil {
+	fmt.Println(err)
+}
+
+// define privK&pubK over the elliptic curve
+privK := big.NewInt(int64(5))
+pubK, err := dsa.PubK(privK)
+if err!=nil {
+	fmt.Println(err)
+}
+
+// hash value to sign
+hashval := big.NewInt(int64(40))
+
+// define r
+r := big.NewInt(int64(11))
+
+// sign hashed value
+sig, err := dsa.Sign(hashval, privK, r)
+if err!=nil {
+	fmt.Println(err)
+}
+
+// verify signature
+verified, err := dsa.Verify(hashval, sig, pubK)
+if err!=nil {
+	fmt.Println(err)
+}
+if verified {
+	fmt.Println("signature correctly verified")
+}
+```
+
 ## Schnorr signature
 - https://en.wikipedia.org/wiki/Schnorr_signature

@@ -70,12 +377,54 @@ Crypto algorithms from scratch. Academic purposes only.
 - [x] Verify signature


+#### Usage
+```go
+// define new elliptic curve
+ec := ecc.NewEC(big.NewInt(int64(0)), big.NewInt(int64(7)), big.NewInt(int64(11)))
+// define new point
+g := ecc.Point{big.NewInt(int64(11)), big.NewInt(int64(27))} // Generator
+// define new random r
+r := big.NewInt(int64(23))                                   // random r
+
+// define new Schnorr crypto system using the values
+schnorr, sk, err := Gen(ec, g, r)
+if err!=nil {
+	fmt.println(err)
+}
+
+// define message to sign
+m := []byte("hola")
+
+// also we can hash the message, but it's not mandatory, as it will be done inside the schnorr.Sign, but we can perform it now, just to check the function
+h := Hash([]byte("hola"), c)
+if h.String() != "34719153732582497359642109898768696927847420320548121616059449972754491425079") {
+	fmt.Println("not correctly hashed")
+}
+
+s, rPoint, err := schnorr.Sign(sk, m)
+if err!=nil {
+	fmt.println(err)
+}
+
+// verify Schnorr signature
+verified, err := Verify(schnorr.EC, sk.PubK, m, s, rPoint)
+if err!=nil {
+	fmt.println(err)
+}
+if verified {
+	fmt.Println("Schnorr signature correctly verified")
+}
+```
+
+
+
 ## Bn128
 **[not finished]**

 This is implemented followng the implementations and info from:
 - https://github.com/iden3/zksnark
 - https://github.com/zcash/zcash/tree/master/src/snark
+- https://github.com/ethereum/py_ecc/tree/master/py_ecc/bn128
 - `Multiplication and Squaring on Pairing-Friendly
 Fields`, Augusto Jun Devegili, Colm Ó hÉigeartaigh, Michael Scott, and Ricardo Dahab https://pdfs.semanticscholar.org/3e01/de88d7428076b2547b60072088507d881bf1.pdf
 - `Optimal Pairings`, Frederik Vercauteren https://www.cosic.esat.kuleuven.be/bcrypt/optimal.pdf
@@ -87,6 +436,134 @@ over Elliptic Curves`, Matthieu Rivain https://eprint.iacr.org/2011/338.pdf
 - [x] Fq, Fq2, Fq6, Fq12 operations
 - [x] G1, G2 operations

+
+#### Usage
+First let's define three basic functions to convert integer compositions to big integer compositions:
+```go
+func iToBig(a int) *big.Int {
+	return big.NewInt(int64(a))
+}
+
+func iiToBig(a, b int) [2]*big.Int {
+	return [2]*big.Int{iToBig(a), iToBig(b)}
+}
+
+func iiiToBig(a, b int) [2]*big.Int {
+	return [2]*big.Int{iToBig(a), iToBig(b)}
+}
+```
+
+- Finite Fields (1, 2, 6, 12) operations
+```go
+// new finite field of order 1
+fq1 := NewFq(iToBig(7))
+
+// basic operations of finite field 1
+res := fq1.Add(iToBig(4), iToBig(4))
+res = fq1.Double(iToBig(5))
+res = fq1.Sub(iToBig(5), iToBig(7))
+res = fq1.Neg(iToBig(5))
+res = fq1.Mul(iToBig(5), iToBig(11))
+res = fq1.Inverse(iToBig(4))
+res = fq1.Square(iToBig(5))
+
+// new finite field of order 2
+nonResidueFq2str := "-1" // i / Beta
+nonResidueFq2, ok := new(big.Int).SetString(nonResidueFq2str, 10)
+fq2 := Fq2{fq1, nonResidueFq2}
+nonResidueFq6 := iiToBig(9, 1)
+
+// basic operations of finite field of order 2
+res := fq2.Add(iiToBig(4, 4), iiToBig(3, 4))
+res = fq2.Double(iiToBig(5, 3))
+res = fq2.Sub(iiToBig(5, 3), iiToBig(7, 2))
+res = fq2.Neg(iiToBig(4, 4))
+res = fq2.Mul(iiToBig(4, 4), iiToBig(3, 4))
+res = fq2.Inverse(iiToBig(4, 4))
+res = fq2.Div(iiToBig(4, 4), iiToBig(3, 4))
+res = fq2.Square(iiToBig(4, 4))
+
+
+// new finite field of order 6
+nonResidueFq6 := iiToBig(9, 1) // TODO
+fq6 := Fq6{fq2, nonResidueFq6}
+
+// define two new values of Finite Field 6, in order to be able to perform the operations
+a := [3][2]*big.Int{
+	iiToBig(1, 2),
+	iiToBig(3, 4),
+	iiToBig(5, 6)}
+b := [3][2]*big.Int{
+	iiToBig(12, 11),
+	iiToBig(10, 9),
+	iiToBig(8, 7)}
+
+// basic operations of finite field order 6
+res := fq6.Add(a, b)
+res = fq6.Sub(a, b)
+res = fq6.Mul(a, b)
+divRes := fq6.Div(mulRes, b)
+
+
+// new finite field of order 12
+q, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208583", 10) // i
+if !ok {
+	fmt.Println("error parsing string to big integer")
+}
+
+fq1 := NewFq(q)
+nonResidueFq2, ok := new(big.Int).SetString("21888242871839275222246405745257275088696311157297823662689037894645226208582", 10) // i
+assert.True(t, ok)
+nonResidueFq6 := iiToBig(9, 1)
+
+fq2 := Fq2{fq1, nonResidueFq2}
+fq6 := Fq6{fq2, nonResidueFq6}
+fq12 := Fq12{fq6, fq2, nonResidueFq6}
+
+```
+
+- G1 operations
+```go
+bn128, err := NewBn128()
+assert.Nil(t, err)
+
+r1 := big.NewInt(int64(33))
+r2 := big.NewInt(int64(44))
+
+gr1 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r1))
+gr2 := bn128.G1.MulScalar(bn128.G1.G, bn128.Fq1.Copy(r2))
+
+grsum1 := bn128.G1.Add(gr1, gr2)
+r1r2 := bn128.Fq1.Add(r1, r2)
+grsum2 := bn128.G1.MulScalar(bn128.G1.G, r1r2)
+
+a := bn128.G1.Affine(grsum1)
+b := bn128.G1.Affine(grsum2)
+assert.Equal(t, a, b)
+assert.Equal(t, "0x2f978c0ab89ebaa576866706b14787f360c4d6c3869efe5a72f7c3651a72ff00", utils.BytesToHex(a[0].Bytes()))
+assert.Equal(t, "0x12e4ba7f0edca8b4fa668fe153aebd908d322dc26ad964d4cd314795844b62b2", utils.BytesToHex(a[1].Bytes()))
+```
+
+- G2 operations
+```go
+bn128, err := NewBn128()
+assert.Nil(t, err)
+
+r1 := big.NewInt(int64(33))
+r2 := big.NewInt(int64(44))
+
+gr1 := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(r1))
+gr2 := bn128.G2.MulScalar(bn128.G2.G, bn128.Fq1.Copy(r2))
+
+grsum1 := bn128.G2.Add(gr1, gr2)
+r1r2 := bn128.Fq1.Add(r1, r2)
+grsum2 := bn128.G2.MulScalar(bn128.G2.G, r1r2)
+
+a := bn128.G2.Affine(grsum1)
+b := bn128.G2.Affine(grsum2)
+assert.Equal(t, a, b)
+```
+
 ---

 To run all tests:
--- a/shamirsecretsharing/shamirsecretsharing.go
+++ b/shamirsecretsharing/shamirsecretsharing.go
@@ -7,7 +7,8 @@ import (
 )

 const (
-	bits = 1024
+	// bits = 1024
+	bits = 2048
 )

 // Create calculates the secrets to share from given parameters
--- a/shamirsecretsharing/shamirsecretsharing_test.go
+++ b/shamirsecretsharing/shamirsecretsharing_test.go
@@ -1,7 +1,9 @@
 package shamirsecretsharing

 import (
+	"bytes"
 	"crypto/rand"
+	"fmt"
 	"math/big"
 	"testing"

@@ -9,7 +11,7 @@ import (
 )

 func TestCreate(t *testing.T) {
-	k := 123456789
+	k := big.NewInt(int64(123456789))
 	p, err := rand.Prime(rand.Reader, bits/2)
 	assert.Nil(t, err)

@@ -19,7 +21,7 @@ func TestCreate(t *testing.T) {
 		nNeededSecrets,
 		nShares,
 		p,
-		big.NewInt(int64(k)))
+		k)
 	assert.Nil(t, err)

 	//generate sharesToUse
@@ -29,15 +31,15 @@ func TestCreate(t *testing.T) {
 	sharesToUse = append(sharesToUse, shares[0])
 	secr := LagrangeInterpolation(sharesToUse, p)

-	// fmt.Print("original secret: ")
-	// fmt.Println(k)
-	// fmt.Print("p: ")
-	// fmt.Println(p)
-	// fmt.Print("shares: ")
-	// fmt.Println(shares)
-	// fmt.Print("secret result: ")
-	// fmt.Println(secr)
-	if int64(k) != secr.Int64() {
+	fmt.Print("original secret: ")
+	fmt.Println(k)
+	fmt.Print("p: ")
+	fmt.Println(p)
+	fmt.Print("shares: ")
+	fmt.Println(shares)
+	fmt.Print("secret result: ")
+	fmt.Println(secr)
+	if !bytes.Equal(k.Bytes(), secr.Bytes()) {
 		t.Errorf("reconstructed secret not correspond to original secret")
 	}
 }