zk/dleq: Adding DLEQ proofs for Qn, the subgroup of squares in (Z/nZ)*

README.md

@@ -94,7 +94,7 @@ Alternatively, look at the [Cloudflare Go](https://github.com/cloudflare/go/tree
 
  - [Schnorr](./zk/dl): Prove knowledge of the Discrete Logarithm. ([RFC-8235])
  - [DLEQ](./zk/dleq): Prove knowledge of the Discrete Logarithm Equality. ([RFC-9497])
-
+ - [DLEQ in Qn](./zk/qndleq): Prove knowledge of the Discrete Logarithm Equality for subgroup of squares in (Z/nZ)*.
 
 ### Symmetric Cryptography
 

zk/qndleq/qndleq.go

@@ -0,0 +1,133 @@
+// Package qndleq provides zero-knowledge proofs of Discrete-Logarithm Equivalence (DLEQ) on Qn.
+//
+// This package implements proofs on the group Qn (the subgroup of squares in (Z/nZ)*).
+//
+// # Notation
+//
+//	Z/nZ is the ring of integers modulo N.
+//	(Z/nZ)* is the multiplicative group of Z/nZ, a.k.a. the units of Z/nZ, the elements with inverse mod N.
+//	Qn is the subgroup of squares in (Z/nZ)*.
+//
+// A number x belongs to Qn if
+//
+//	gcd(x, N) = 1, and
+//	exists y such that x = y^2 mod N.
+//
+// # References
+//
+// [DLEQ Proof] "Wallet databases with observers" by Chaum-Pedersen.
+// https://doi.org/10.1007/3-540-48071-4_7
+//
+// [Qn] "Practical Threshold Signatures" by Shoup.
+// https://www.iacr.org/archive/eurocrypt2000/1807/18070209-new.pdf
+package qndleq
+
+import (
+	"crypto/rand"
+	"io"
+	"math/big"
+
+	"github.com/cloudflare/circl/internal/sha3"
+)
+
+type Proof struct {
+	Z, C     *big.Int
+	SecParam uint
+}
+
+// SampleQn returns an element of Qn (the subgroup of squares in (Z/nZ)*).
+// SampleQn will return error for any error returned by crypto/rand.Int.
+func SampleQn(random io.Reader, N *big.Int) (*big.Int, error) {
+	one := big.NewInt(1)
+	gcd := new(big.Int)
+	x := new(big.Int)
+
+	for {
+		y, err := rand.Int(random, N)
+		if err != nil {
+			return nil, err
+		}
+		// x is a square by construction.
+		x.Mul(y, y).Mod(x, N)
+		gcd.GCD(nil, nil, x, N)
+		// now check whether h is coprime to N.
+		if gcd.Cmp(one) == 0 {
+			return x, nil
+		}
+	}
+}
+
+// Prove creates a DLEQ Proof that attests that the pairs (g,gx)
+// and (h,hx) have the same discrete logarithm equal to x.
+//
+// Given g, h in Qn (the subgroup of squares in (Z/nZ)*), it holds
+//
+//	gx = g^x mod N
+//	hx = h^x mod N
+//	x  = Log_g(g^x) = Log_h(h^x)
+//
+// Note: this function does not run in constant time because it uses
+// big.Int arithmetic.
+func Prove(random io.Reader, x, g, gx, h, hx, N *big.Int, secParam uint) (*Proof, error) {
+	rSizeBits := uint(N.BitLen()) + 2*secParam
+	rSizeBytes := (rSizeBits + 7) / 8
+
+	rBytes := make([]byte, rSizeBytes)
+	_, err := io.ReadFull(random, rBytes)
+	if err != nil {
+		return nil, err
+	}
+	r := new(big.Int).SetBytes(rBytes)
+
+	gP := new(big.Int).Exp(g, r, N)
+	hP := new(big.Int).Exp(h, r, N)
+
+	c := doChallenge(g, gx, h, hx, gP, hP, N, secParam)
+	z := new(big.Int)
+	z.Mul(c, x).Add(z, r)
+
+	return &Proof{Z: z, C: c, SecParam: secParam}, nil
+}
+
+// Verify checks whether x = Log_g(g^x) = Log_h(h^x).
+func (p Proof) Verify(g, gx, h, hx, N *big.Int) bool {
+	gPNum := new(big.Int).Exp(g, p.Z, N)
+	gPDen := new(big.Int).Exp(gx, p.C, N)
+	ok := gPDen.ModInverse(gPDen, N)
+	if ok == nil {
+		return false
+	}
+	gP := gPNum.Mul(gPNum, gPDen)
+	gP.Mod(gP, N)
+
+	hPNum := new(big.Int).Exp(h, p.Z, N)
+	hPDen := new(big.Int).Exp(hx, p.C, N)
+	ok = hPDen.ModInverse(hPDen, N)
+	if ok == nil {
+		return false
+	}
+	hP := hPNum.Mul(hPNum, hPDen)
+	hP.Mod(hP, N)
+
+	c := doChallenge(g, gx, h, hx, gP, hP, N, p.SecParam)
+
+	return p.C.Cmp(c) == 0
+}
+
+func doChallenge(g, gx, h, hx, gP, hP, N *big.Int, secParam uint) *big.Int {
+	modulusLenBytes := (N.BitLen() + 7) / 8
+	nBytes := make([]byte, modulusLenBytes)
+	cByteLen := (secParam + 7) / 8
+	cBytes := make([]byte, cByteLen)
+
+	H := sha3.NewShake256()
+	_, _ = H.Write(g.FillBytes(nBytes))
+	_, _ = H.Write(h.FillBytes(nBytes))
+	_, _ = H.Write(gx.FillBytes(nBytes))
+	_, _ = H.Write(hx.FillBytes(nBytes))
+	_, _ = H.Write(gP.FillBytes(nBytes))
+	_, _ = H.Write(hP.FillBytes(nBytes))
+	_, _ = H.Read(cBytes)
+
+	return new(big.Int).SetBytes(cBytes)
+}

zk/qndleq/qndleq_test.go

@@ -0,0 +1,84 @@
+package qndleq_test
+
+import (
+	"crypto/rand"
+	"math/big"
+	"testing"
+
+	"github.com/cloudflare/circl/internal/test"
+	"github.com/cloudflare/circl/zk/qndleq"
+)
+
+func TestProve(t *testing.T) {
+	const testTimes = 1 << 8
+	const SecParam = 128
+	one := big.NewInt(1)
+	max := new(big.Int).Lsh(one, 256)
+
+	for i := 0; i < testTimes; i++ {
+		N, _ := rand.Int(rand.Reader, max)
+		if N.Bit(0) == 0 {
+			N.Add(N, one)
+		}
+		x, _ := rand.Int(rand.Reader, N)
+		g, err := qndleq.SampleQn(rand.Reader, N)
+		test.CheckNoErr(t, err, "failed to sampleQn")
+		h, err := qndleq.SampleQn(rand.Reader, N)
+		test.CheckNoErr(t, err, "failed to sampleQn")
+		gx := new(big.Int).Exp(g, x, N)
+		hx := new(big.Int).Exp(h, x, N)
+
+		proof, err := qndleq.Prove(rand.Reader, x, g, gx, h, hx, N, SecParam)
+		test.CheckNoErr(t, err, "failed to generate proof")
+		test.CheckOk(proof.Verify(g, gx, h, hx, N), "failed to verify", t)
+	}
+}
+
+func TestSampleQn(t *testing.T) {
+	const testTimes = 1 << 7
+	one := big.NewInt(1)
+	max := new(big.Int).Lsh(one, 256)
+
+	for i := 0; i < testTimes; i++ {
+		N, _ := rand.Int(rand.Reader, max)
+		if N.Bit(0) == 0 {
+			N.Add(N, one)
+		}
+		a, err := qndleq.SampleQn(rand.Reader, N)
+		test.CheckNoErr(t, err, "failed to sampleQn")
+		jac := big.Jacobi(a, N)
+		test.CheckOk(jac == 1, "Jacoby symbol should be one", t)
+		gcd := new(big.Int).GCD(nil, nil, a, N)
+		test.CheckOk(gcd.Cmp(one) == 0, "should be coprime to N", t)
+	}
+}
+
+func Benchmark_qndleq(b *testing.B) {
+	const SecParam = 128
+	one := big.NewInt(1)
+	max := new(big.Int).Lsh(one, 256)
+
+	N, _ := rand.Int(rand.Reader, max)
+	if N.Bit(0) == 0 {
+		N.Add(N, one)
+	}
+	x, _ := rand.Int(rand.Reader, N)
+	g, _ := qndleq.SampleQn(rand.Reader, N)
+	h, _ := qndleq.SampleQn(rand.Reader, N)
+	gx := new(big.Int).Exp(g, x, N)
+	hx := new(big.Int).Exp(h, x, N)
+
+	proof, _ := qndleq.Prove(rand.Reader, x, g, gx, h, hx, N, SecParam)
+
+	b.Run("Prove", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			_, _ = qndleq.Prove(rand.Reader, x, g, gx, h, hx, N, SecParam)
+		}
+	})
+
+	b.Run("Verify", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			_ = proof.Verify(g, gx, h, hx, N)
+		}
+	})
+}