kyber: remove division by q in ciphertext compression

On some platforms, division by q leaks some information on the
ciphertext by its timing. If a keypair is reused, and an attacker has access to
a decapsulation oracle, this reveals information on the private key.
This is known as "kyberslash2".

Note that this does not affect to the typical ephemeral usage in TLS.

Author: bwesterb

pke/kyber/internal/common/poly.go

@@ -166,7 +166,7 @@ func (p *Poly) CompressMessageTo(m []byte) {
 
 // Set p to Decompress_q(m, 1).
 //
-// Assumes d is in {3, 4, 5, 10, 11}.  p will be normalized.
+// Assumes d is in {4, 5, 10, 11}.  p will be normalized.
 func (p *Poly) Decompress(m []byte, d int) {
 	// Decompress_q(x, d) = ⌈(q/2ᵈ)x⌋
 	//                    = ⌊(q/2ᵈ)x+½⌋
@@ -244,20 +244,28 @@ func (p *Poly) Decompress(m []byte, d int) {
 
 // Writes Compress_q(p, d) to m.
 //
-// Assumes p is normalized and d is in {3, 4, 5, 10, 11}.
+// Assumes p is normalized and d is in {4, 5, 10, 11}.
 func (p *Poly) CompressTo(m []byte, d int) {
 	// Compress_q(x, d) = ⌈(2ᵈ/q)x⌋ mod⁺ 2ᵈ
 	//                  = ⌊(2ᵈ/q)x+½⌋ mod⁺ 2ᵈ
 	//					= ⌊((x << d) + q/2) / q⌋ mod⁺ 2ᵈ
 	//					= DIV((x << d) + q/2, q) & ((1<<d) - 1)
+	//
+	// We approximate DIV(x, q) by computing (x*a)>>e, where a/(2^e) ≈ 1/q.
+	// For d in {10,11} we use 20,642,679/2^36, which computes division by x/q
+	// correctly for 0 ≤ x < 41,522,616, which fits (q << 11) + q/2 comfortably.
+	// For d in {4,5} we use 315/2^20, which doesn't compute division by x/q
+	// correctly for all inputs, but it's close enough that the end result
+	// of the compression is correct. The advantage is that we do not need
+	// to use a 64-bit intermediate value.
 	switch d {
 	case 4:
 		var t [8]uint16
 		idx := 0
 		for i := 0; i < N/8; i++ {
 			for j := 0; j < 8; j++ {
-				t[j] = uint16(((uint32(p[8*i+j])<<4)+uint32(Q)/2)/
-					uint32(Q)) & ((1 << 4) - 1)
+				t[j] = uint16((((uint32(p[8*i+j])<<4)+uint32(Q)/2)*315)>>
+					20) & ((1 << 4) - 1)
 			}
 			m[idx] = byte(t[0]) | byte(t[1]<<4)
 			m[idx+1] = byte(t[2]) | byte(t[3]<<4)
@@ -271,8 +279,8 @@ func (p *Poly) CompressTo(m []byte, d int) {
 		idx := 0
 		for i := 0; i < N/8; i++ {
 			for j := 0; j < 8; j++ {
-				t[j] = uint16(((uint32(p[8*i+j])<<5)+uint32(Q)/2)/
-					uint32(Q)) & ((1 << 5) - 1)
+				t[j] = uint16((((uint32(p[8*i+j])<<5)+uint32(Q)/2)*315)>>
+					20) & ((1 << 5) - 1)
 			}
 			m[idx] = byte(t[0]) | byte(t[1]<<5)
 			m[idx+1] = byte(t[1]>>3) | byte(t[2]<<2) | byte(t[3]<<7)
@@ -287,8 +295,8 @@ func (p *Poly) CompressTo(m []byte, d int) {
 		idx := 0
 		for i := 0; i < N/4; i++ {
 			for j := 0; j < 4; j++ {
-				t[j] = uint16(((uint32(p[4*i+j])<<10)+uint32(Q)/2)/
-					uint32(Q)) & ((1 << 10) - 1)
+				t[j] = uint16((uint64((uint32(p[4*i+j])<<10)+uint32(Q)/2)*
+					20642679)>>36) & ((1 << 10) - 1)
 			}
 			m[idx] = byte(t[0])
 			m[idx+1] = byte(t[0]>>8) | byte(t[1]<<2)
@@ -302,8 +310,8 @@ func (p *Poly) CompressTo(m []byte, d int) {
 		idx := 0
 		for i := 0; i < N/8; i++ {
 			for j := 0; j < 8; j++ {
-				t[j] = uint16(((uint32(p[8*i+j])<<11)+uint32(Q)/2)/
-					uint32(Q)) & ((1 << 11) - 1)
+				t[j] = uint16((uint64((uint32(p[8*i+j])<<11)+uint32(Q)/2)*
+					20642679)>>36) & ((1 << 11) - 1)
 			}
 			m[idx] = byte(t[0])
 			m[idx+1] = byte(t[0]>>8) | byte(t[1]<<3)

pke/kyber/internal/common/poly_test.go

@@ -1,6 +1,7 @@
 package common
 
 import (
+	"bytes"
 	"crypto/rand"
 	"fmt"
 	"testing"
@@ -273,3 +274,107 @@ func TestNormalizeAgainstGeneric(t *testing.T) {
 		}
 	}
 }
+
+func (p *Poly) OldCompressTo(m []byte, d int) {
+	switch d {
+	case 4:
+		var t [8]uint16
+		idx := 0
+		for i := 0; i < N/8; i++ {
+			for j := 0; j < 8; j++ {
+				t[j] = uint16(((uint32(p[8*i+j])<<4)+uint32(Q)/2)/
+					uint32(Q)) & ((1 << 4) - 1)
+			}
+			m[idx] = byte(t[0]) | byte(t[1]<<4)
+			m[idx+1] = byte(t[2]) | byte(t[3]<<4)
+			m[idx+2] = byte(t[4]) | byte(t[5]<<4)
+			m[idx+3] = byte(t[6]) | byte(t[7]<<4)
+			idx += 4
+		}
+
+	case 5:
+		var t [8]uint16
+		idx := 0
+		for i := 0; i < N/8; i++ {
+			for j := 0; j < 8; j++ {
+				t[j] = uint16(((uint32(p[8*i+j])<<5)+uint32(Q)/2)/
+					uint32(Q)) & ((1 << 5) - 1)
+			}
+			m[idx] = byte(t[0]) | byte(t[1]<<5)
+			m[idx+1] = byte(t[1]>>3) | byte(t[2]<<2) | byte(t[3]<<7)
+			m[idx+2] = byte(t[3]>>1) | byte(t[4]<<4)
+			m[idx+3] = byte(t[4]>>4) | byte(t[5]<<1) | byte(t[6]<<6)
+			m[idx+4] = byte(t[6]>>2) | byte(t[7]<<3)
+			idx += 5
+		}
+
+	case 10:
+		var t [4]uint16
+		idx := 0
+		for i := 0; i < N/4; i++ {
+			for j := 0; j < 4; j++ {
+				t[j] = uint16(((uint32(p[4*i+j])<<10)+uint32(Q)/2)/
+					uint32(Q)) & ((1 << 10) - 1)
+			}
+			m[idx] = byte(t[0])
+			m[idx+1] = byte(t[0]>>8) | byte(t[1]<<2)
+			m[idx+2] = byte(t[1]>>6) | byte(t[2]<<4)
+			m[idx+3] = byte(t[2]>>4) | byte(t[3]<<6)
+			m[idx+4] = byte(t[3] >> 2)
+			idx += 5
+		}
+	case 11:
+		var t [8]uint16
+		idx := 0
+		for i := 0; i < N/8; i++ {
+			for j := 0; j < 8; j++ {
+				t[j] = uint16(((uint32(p[8*i+j])<<11)+uint32(Q)/2)/
+					uint32(Q)) & ((1 << 11) - 1)
+			}
+			m[idx] = byte(t[0])
+			m[idx+1] = byte(t[0]>>8) | byte(t[1]<<3)
+			m[idx+2] = byte(t[1]>>5) | byte(t[2]<<6)
+			m[idx+3] = byte(t[2] >> 2)
+			m[idx+4] = byte(t[2]>>10) | byte(t[3]<<1)
+			m[idx+5] = byte(t[3]>>7) | byte(t[4]<<4)
+			m[idx+6] = byte(t[4]>>4) | byte(t[5]<<7)
+			m[idx+7] = byte(t[5] >> 1)
+			m[idx+8] = byte(t[5]>>9) | byte(t[6]<<2)
+			m[idx+9] = byte(t[6]>>6) | byte(t[7]<<5)
+			m[idx+10] = byte(t[7] >> 3)
+			idx += 11
+		}
+	default:
+		panic("unsupported d")
+	}
+}
+
+func TestCompressFullInputFirstCoeff(t *testing.T) {
+	for _, d := range []int{4, 5, 10, 11} {
+		d := d
+		t.Run(fmt.Sprintf("d=%d", d), func(t *testing.T) {
+			var p, q Poly
+			bound := (Q + (1 << uint(d))) >> uint(d+1)
+			buf := make([]byte, (N*d-1)/8+1)
+			buf2 := make([]byte, len(buf))
+			for i := int16(0); i < Q; i++ {
+				p[0] = i
+				p.CompressTo(buf, d)
+				p.OldCompressTo(buf2, d)
+				if !bytes.Equal(buf, buf2) {
+					t.Fatalf("%d", i)
+				}
+				q.Decompress(buf, d)
+				diff := sModQ(p[0] - q[0])
+				if diff < 0 {
+					diff = -diff
+				}
+				if diff > bound {
+					t.Logf("%v\n", buf)
+					t.Fatalf("|%d - %d mod^± q| = %d > %d",
+						p[0], q[0], diff, bound)
+				}
+			}
+		})
+	}
+}