Updated High‐assurance field inversion for curve‐based cryptography (markdown)

High‐assurance-field-inversion-for-curve‐based-cryptography.md

@@ -1,3 +1,232 @@
+# 尝试1
+使用**fiat-crypto**生成的代码，来事先模求逆：
+代码片段1是原有的通过**addchain**生成的模求逆：
+```go
+// Invert sets e = 1/x, and returns e.
+//
+// If x == 0, Invert returns e = 0.
+func (e *SM2P256Element) Invert(x *SM2P256Element) *SM2P256Element {
+	// Inversion is implemented as exponentiation with exponent p − 2.
+	// The sequence of 14 multiplications and 255 squarings is derived from the
+	// following addition chain generated with github.com/mmcloughlin/addchain v0.4.0.
+	//
+	//	_10      = 2*1
+	//	_11      = 1 + _10
+	//	_110     = 2*_11
+	//	_111     = 1 + _110
+	//	_111000  = _111 << 3
+	//	_111111  = _111 + _111000
+	//	_1111110 = 2*_111111
+	//	_1111111 = 1 + _1111110
+	//	x12      = _1111110 << 5 + _111111
+	//	x24      = x12 << 12 + x12
+	//	x31      = x24 << 7 + _1111111
+	//	i39      = x31 << 2
+	//	i68      = i39 << 29
+	//	x62      = x31 + i68
+	//	i71      = i68 << 2
+	//	x64      = i39 + i71 + _11
+	//	i265     = ((i71 << 32 + x64) << 64 + x64) << 94
+	//	return     (x62 + i265) << 2 + 1
+	//
+	var z = new(SM2P256Element).Set(e)
+	var t0 = new(SM2P256Element)
+	var t1 = new(SM2P256Element)
+	var t2 = new(SM2P256Element)
+
+	z.Square(x)
+	t0.Mul(x, z)
+	z.Square(t0)
+	z.Mul(x, z)
+	t1.Square(z)
+	for s := 1; s < 3; s++ {
+		t1.Square(t1)
+	}
+	t1.Mul(z, t1)
+	t2.Square(t1)
+	z.Mul(x, t2)
+	for s := 0; s < 5; s++ {
+		t2.Square(t2)
+	}
+	t1.Mul(t1, t2)
+	t2.Square(t1)
+	for s := 1; s < 12; s++ {
+		t2.Square(t2)
+	}
+	t1.Mul(t1, t2)
+	for s := 0; s < 7; s++ {
+		t1.Square(t1)
+	}
+	z.Mul(z, t1)
+	t2.Square(z)
+	for s := 1; s < 2; s++ {
+		t2.Square(t2)
+	}
+	t1.Square(t2)
+	for s := 1; s < 29; s++ {
+		t1.Square(t1)
+	}
+	z.Mul(z, t1)
+	for s := 0; s < 2; s++ {
+		t1.Square(t1)
+	}
+	t2.Mul(t2, t1)
+	t0.Mul(t0, t2)
+	for s := 0; s < 32; s++ {
+		t1.Square(t1)
+	}
+	t1.Mul(t0, t1)
+	for s := 0; s < 64; s++ {
+		t1.Square(t1)
+	}
+	t0.Mul(t0, t1)
+	for s := 0; s < 94; s++ {
+		t0.Square(t0)
+	}
+	z.Mul(z, t0)
+	for s := 0; s < 2; s++ {
+		z.Square(z)
+	}
+	z.Mul(x, z)
+	return e.Set(z)
+}
+```
+
+代码片段2通过**safegcd**介绍的最基本方法模求逆：
+```go
+// Reference https://github.com/mit-plv/fiat-crypto/blob/master/inversion/c/inversion_template.c
+func P256Inverse(g *[4]uint64) *[4]uint64 {
+	var precomp, v, out4, out5 [4]uint64
+	var d, out1 uint64
+	var f, out2, out3, gs [5]uint64
+	var r sm2p256MontgomeryDomainFieldElement
+	r1 := (*[4]uint64)(&r)
+	sm2p256DivstepPrecomp(&precomp)
+	sm2p256Msat(&f)
+	sm2p256SetOne(&r)
+
+	for i := 0; i < 4; i++ {
+		gs[i] = g[i]
+	}
+
+	// 370 = (256 * 49 + 57) / 17 - 1
+	for i := 0; i < 370; i++ {
+		sm2p256Divstep(&out1, &out2, &out3, &out4, &out5, d, &f, &gs, &v, r1)
+		sm2p256Divstep(&d, &f, &gs, &v, r1, out1, &out2, &out3, &out4, &out5)
+	}
+
+	sm2p256Divstep(&out1, &out2, &out3, &out4, &out5, d, &f, &gs, &v, r1)
+	for i := 0; i < 4; i++ {
+		v[i] = out4[i]
+		f[i] = out2[i]
+	}
+	f[4] = out2[4]
+
+	var out sm2p256MontgomeryDomainFieldElement
+	sm2p256Opp(&out, (*sm2p256MontgomeryDomainFieldElement)(&v))
+	sm2p256Selectznz(&v, (sm2p256Uint1)(f[4]>>63), &v, (*[4]uint64)(&out))
+
+	sm2p256Mul(&out, (*sm2p256MontgomeryDomainFieldElement)(&v), (*sm2p256MontgomeryDomainFieldElement)(&precomp))
+	return (*[4]uint64)(&out)
+}
+```
+
+以下是测试代码：
+```go
+package fiat
+
+import (
+	"encoding/hex"
+	"testing"
+)
+
+
+var testValues = []string{
+	"0000000000000000000000000000000000000000000000000000000000000001",
+	"0000000000000000000000000000000000000000000000000000000000000002",
+	"0000000000000000000000000000000000000000000000000000000000000003",
+	"1000000000000000000000000000000000000000000000000000000000000000",
+	"1000000000000000000000000000000000000000000000000000000000000001",
+	"1000000000000000000000000000000000000000000000000000000000000002",
+	"1000000000000000000000000000000000000000000000000000000000000003",
+	"8356e642a40ebd18d29ba3532fbd9f3bbee8f027c3f6f39a5ba2f870369f9988",
+	"981f5efe55d1c5cdf6c0ef2b070847a14f7fdf4272a8df09c442f3058af94ba1",
+	"d6833540d019e0438a5dd73b414f26ab43d8064b99671206944e284dbd969093",
+	"6c5a0a0b2eed3cbec3e4f1252bfe0e28c504a1c6bf1999eebb0af9ef0f8e6c85",
+}
+
+func TestP256Inverse(t *testing.T) {
+	for _, v := range testValues {
+		vBytes, _ := hex.DecodeString(v)
+
+		in, err := new(SM2P256Element).SetBytes(vBytes)
+		if err != nil {
+			t.Errorf("SetBytes failed: %v", err)
+		}
+		in2 := *(*[4]uint64)(&in.x)
+		out1 := new(SM2P256Element).Invert(in)
+		out1r := (*[4]uint64)(&out1.x)
+		out2 := P256Inverse(&in2)
+		tmp := (*sm2p256NonMontgomeryDomainFieldElement)(out2)
+		out3 := new(sm2p256MontgomeryDomainFieldElement)
+		sm2p256ToMontgomery(out3, tmp)
+		if *out3 != *out1r  {
+			t.Errorf("got %v, want %v", out3, &out1.x)
+		}
+	}
+}
+
+func BenchmarkP256Inverse(b *testing.B) {
+	vBytes, _ := hex.DecodeString(testValues[0])
+
+	in, _ := new(SM2P256Element).SetBytes(vBytes)
+	in2 := *(*[4]uint64)(&in.x)
+	b.ReportAllocs()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		P256Inverse(&in2)
+	}
+}
+
+func BenchmarkInvert(b *testing.B) {
+	vBytes, _ := hex.DecodeString(testValues[0])
+
+	in, _ := new(SM2P256Element).SetBytes(vBytes)
+	out := new(SM2P256Element)
+	b.ReportAllocs()
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		out.Invert(in)
+	}
+}
+```
+
+
+1. 还有就是fiat-cyrpto目前的实现，输入、输出有些小问题：输入是蒙哥马利型，输出是非蒙哥马利型；输入是非蒙哥马利型，输出是蒙哥马利型。这个是已知问题。https://github.com/mit-plv/fiat-crypto/issues/1020
+```
+goos: windows
+goarch: amd64
+pkg: github.com/emmansun/gmsm/internal/sm2ec/fiat
+cpu: Intel(R) Core(TM) i5-9500 CPU @ 3.00GHz
+BenchmarkInvert
+BenchmarkInvert-6
+  227496	      5806 ns/op	       0 B/op	       0 allocs/op
+PASS
+ok  	github.com/emmansun/gmsm/internal/sm2ec/fiat	1.792s
+
+goos: windows
+goarch: amd64
+pkg: github.com/emmansun/gmsm/internal/sm2ec/fiat
+cpu: Intel(R) Core(TM) i5-9500 CPU @ 3.00GHz
+BenchmarkP256Inverse
+BenchmarkP256Inverse-6
+   58394	     20644 ns/op	      32 B/op	       1 allocs/op
+PASS
+ok  	github.com/emmansun/gmsm/internal/sm2ec/fiat	1.793s
+```
+
+2. 用devsteps2事先的模求逆并没有比经过addchain优化的模求逆方法性能好。
+
 # Reference
 - [High-assurance field inversion for curve-based cryptography](https://eprint.iacr.org/2021/549.pdf)
 - [Fast constant-time gcd and modular inversion](https://gcd.cr.yp.to/papers.html#safegcd)