diff --git a/sm9/bn_pair_test.go b/sm9/bn_pair_test.go
index c2eea4e..967dfdd 100644
--- a/sm9/bn_pair_test.go
+++ b/sm9/bn_pair_test.go
@@ -1,6 +1,7 @@
 package sm9
 
 import (
+	"encoding/hex"
 	"math/big"
 	"testing"
 )
@@ -80,6 +81,53 @@ func Test_gfpBasicOperations(t *testing.T) {
 	}
 }
 
+func TestGfpSqrt(t *testing.T) {
+	tests := []string{
+		"9093a2b979e6186f43a9b28d41ba644d533377f2ede8c66b19774bf4a9c7a596",
+		"92fe90b700fbd4d8cc177d300ed16e4e15471a681b2c9e3728c1b82c885e49c2",
+	}
+	for i, test := range tests {
+		y2 := bigFromHex(test)
+		y21 := new(big.Int).ModSqrt(y2, p)
+
+		y3 := new(big.Int).Mul(y21, y21)
+		y3.Mod(y3, p)
+		if y2.Cmp(y3) != 0 {
+			t.Error("Invalid sqrt")
+		}
+
+		tmp := fromBigInt(y2)
+		tmp.Sqrt(tmp)
+		montDecode(tmp, tmp)
+		var res [32]byte
+		tmp.Marshal(res[:])
+		if hex.EncodeToString(res[:]) != hex.EncodeToString(y21.Bytes()) {
+			t.Errorf("case %v, got %v, expected %v\n", i, hex.EncodeToString(res[:]), hex.EncodeToString(y21.Bytes()))
+		}
+	}
+}
+
+func TestGfpInvert(t *testing.T) {
+	x := fromBigInt(bigFromHex("9093a2b979e6186f43a9b28d41ba644d533377f2ede8c66b19774bf4a9c7a596"))
+	xInv := &gfP{}
+	xInv.Invert(x)
+	y := &gfP{}
+	gfpMul(y, x, xInv)
+	if *y != *one {
+		t.Errorf("got %v, expected %v", y, one)
+	}
+}
+
+func TestGfpDiv(t *testing.T) {
+	x := fromBigInt(bigFromHex("9093a2b979e6186f43a9b28d41ba644d533377f2ede8c66b19774bf4a9c7a596"))
+	ret := &gfP{}
+	ret.Div2(x)
+	gfpAdd(ret, ret, ret)
+	if *ret != *x {
+		t.Errorf("got %v, expected %v", ret, x)
+	}
+}
+
 func Test_gfp12Gen(t *testing.T) {
 	ret := pairing(twistGen, curveGen)
 	if ret.x != gfP12Gen.x || ret.y != gfP12Gen.y || ret.z != gfP12Gen.z {
@@ -87,6 +135,120 @@ func Test_gfp12Gen(t *testing.T) {
 	}
 }
 
+func Test_gfP2Square(t *testing.T) {
+	x := &gfP2{
+		*fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")),
+		*fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")),
+	}
+
+	xmulx := &gfP2{}
+	xmulx.Mul(x, x)
+	xmulx = gfP2Decode(xmulx)
+
+	x2 := &gfP2{}
+	x2.Square(x)
+	x2 = gfP2Decode(x2)
+
+	if xmulx.x != x2.x || xmulx.y != x2.y {
+		t.Errorf("xmulx=%v, x2=%v", xmulx, x2)
+	}
+}
+
+func Test_gfP2Invert(t *testing.T) {
+	x := &gfP2{
+		*fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")),
+		*fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")),
+	}
+
+	xInv := &gfP2{}
+	xInv.Invert(x)
+
+	y := &gfP2{}
+	y.Mul(x, xInv)
+	expected := (&gfP2{}).SetOne()
+
+	if y.x != expected.x || y.y != expected.y {
+		t.Errorf("got %v, expected %v", y, expected)
+	}
+
+	x = &gfP2{
+		*fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")),
+		*zero,
+	}
+
+	xInv.Invert(x)
+
+	y.Mul(x, xInv)
+
+	if y.x != expected.x || y.y != expected.y {
+		t.Errorf("got %v, expected %v", y, expected)
+	}
+
+	x = &gfP2{
+		*zero,
+		*fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")),
+	}
+
+	xInv.Invert(x)
+
+	y.Mul(x, xInv)
+
+	if y.x != expected.x || y.y != expected.y {
+		t.Errorf("got %v, expected %v", y, expected)
+	}
+}
+
+func Test_gfP2Exp(t *testing.T) {
+	x := &gfP2{
+		*fromBigInt(bigFromHex("17509B092E845C1266BA0D262CBEE6ED0736A96FA347C8BD856DC76B84EBEB96")),
+		*fromBigInt(bigFromHex("A7CF28D519BE3DA65F3170153D278FF247EFBA98A71A08116215BBA5C999A7C7")),
+	}
+	got := &gfP2{}
+	got.Exp(x, big.NewInt(1))
+	if x.x != got.x || x.y != got.y {
+		t.Errorf("got %v, expected %v", got, x)
+	}
+}
+
+func Test_gfP2Frobenius(t *testing.T) {
+	x := &gfP2{
+		*fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")),
+		*fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")),
+	}
+	expected := &gfP2{}
+	expected.Exp(x, p)
+	got := &gfP2{}
+	got.Frobenius(x)
+	if expected.x != got.x || expected.y != got.y {
+		t.Errorf("got %v, expected %v", got, x)
+	}
+
+	// make sure i^(p-1) = -1
+	i := &gfP2{}
+	i.SetU()
+	i.Exp(i, bigFromHex("b640000002a3a6f1d603ab4ff58ec74521f2934b1a7aeedbe56f9b27e351457c"))
+	i = gfP2Decode(i)
+	expected.y.Set(newGFp(-1))
+	expected.x.Set(zero)
+	expected = gfP2Decode(expected)
+	if expected.x != i.x || expected.y != i.y {
+		t.Errorf("got %v, expected %v", i, expected)
+	}
+}
+
+func Test_gfP2Div2(t *testing.T) {
+	x := &gfP2{
+		*fromBigInt(bigFromHex("85AEF3D078640C98597B6027B441A01FF1DD2C190F5E93C454806C11D8806141")),
+		*fromBigInt(bigFromHex("3722755292130B08D2AAB97FD34EC120EE265948D19C17ABF9B7213BAF82D65B")),
+	}
+	ret := &gfP2{}
+	ret.Div2(x)
+	ret.Add(ret, ret)
+	if *ret != *x {
+		t.Errorf("got %v, expected %v", ret, x)
+	}
+}
+
 func Test_Pairing_A2(t *testing.T) {
 	pk := bigFromHex("0130E78459D78545CB54C587E02CF480CE0B66340F319F348A1D5B1F2DC5F4")
 	g2 := &G2{}
diff --git a/sm9/gfp2_test.go b/sm9/gfp2_test.go
index 6f08a10..ba67f64 100644
--- a/sm9/gfp2_test.go
+++ b/sm9/gfp2_test.go
@@ -1,5 +1,6 @@
 package sm9
 
+/*
 import (
 	"math/big"
 	"testing"
@@ -118,3 +119,4 @@ func Test_gfP2Div2(t *testing.T) {
 		t.Errorf("got %v, expected %v", ret, x)
 	}
 }
+*/