Improve Shamir Secret Sharing code

shamir/shamir.go

@@ -12,9 +12,7 @@ package shamir
 
 import (
 	"crypto/rand"
-	"crypto/subtle"
 	"fmt"
-	mathrand "math/rand"
 )
 
 const (
@@ -101,63 +99,90 @@ func div(a, b uint8) uint8 {
 		panic("divide by zero")
 	}
 
-	var goodVal, zero uint8
-	logA := logTable[a]
-	logB := logTable[b]
-	diff := (int(logA) - int(logB)) % 255
-	if diff < 0 {
-		diff += 255
-	}
-
-	ret := expTable[diff]
-
-	// Ensure we return zero if a is zero but aren't subject to timing attacks
-	goodVal = ret
-
-	if subtle.ConstantTimeByteEq(a, 0) == 1 {
-		ret = zero
-	} else {
-		ret = goodVal
-	}
+	// a divided by b is the same as a multiplied by the inverse of b:
+	return mult(a, inverse(b))
+}
 
-	return ret
+// inverse calculates the inverse of a number in GF(2^8)
+// Note that a must be non-zero; otherwise 0 is returned
+func inverse(a uint8) uint8 {
+	// This makes use of Fermat's Little Theorem for finite groups:
+	// If G is a finite group with n elements, and a any element of G,
+	// then a raised to the power of n equals the neutral element of G.
+	// (See https://en.wikipedia.org/wiki/Fermat%27s_little_theorem;
+	// the generalization to finite groups follows from Lagrange's theorem:
+	// https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory))
+	//
+	// Here we use the multiplicative group of GF(2^8), which has
+	// n = 2^8 - 1 elements (every element but zero). Thus raising a to
+	// the (n - 1)th = 254th power gives a number x so that a*x = 1.
+	//
+	// If a happens to be 0, which is not part of the multiplicative group,
+	// then a raised to the power of 254 is still 0.
+
+	// (See also https://github.com/openbao/openbao/commit/a209a052024b70bc563d9674cde21a20b5106570)
+
+	// In the comments, we use ^ to denote raising to the power:
+	b := mult(a, a)  // b is now a^2
+	c := mult(a, b)  // c is now a^3
+	b = mult(c, c)  // b is now a^6
+	b = mult(b, b)  // b is now a^12
+	c = mult(b, c)  // c is now a^15
+	b = mult(b, b)  // b is now a^24
+	b = mult(b, b)  // b is now a^48
+	b = mult(b, c)  // b is now a^63
+	b = mult(b, b)  // b is now a^126
+	b = mult(a, b)  // b is now a^127
+	return mult(b, b)  // result is a^254
 }
 
 // mult multiplies two numbers in GF(2^8)
 // GF(2^8) multiplication using log/exp tables
 func mult(a, b uint8) (out uint8) {
-	var goodVal, zero uint8
-	log_a := logTable[a]
-	log_b := logTable[b]
-	sum := (int(log_a) + int(log_b)) % 255
-
-	ret := expTable[sum]
-
-	// Ensure we return zero if either a or b are zero but aren't subject to
-	// timing attacks
-	goodVal = ret
-
-	if subtle.ConstantTimeByteEq(a, 0) == 1 {
-		ret = zero
-	} else {
-		ret = goodVal
-	}
-
-	if subtle.ConstantTimeByteEq(b, 0) == 1 {
-		ret = zero
-	} else {
-		// This operation does not do anything logically useful. It
-		// only ensures a constant number of assignments to thwart
-		// timing attacks.
-		goodVal = zero
+	// This computes a * b in GF(2^8), which is defined as GF(2)[X] / <X^8 + X^4 + X^3 + X + 1>.
+	// This finite field is known as Rijndael's finite field. (Rijndael is the algorithm that
+	// was standardized as AES.)
+	// (See https://en.wikipedia.org/wiki/Finite_field_arithmetic#Rijndael's_(AES)_finite_field)
+	//
+	// We identify elements in GF(2^8) with polynomials of degree < 8. The i-th bit of a field
+	// element is the coefficient of X^i in that polynomial.
+	//
+	// To multiply a and b in this finite field, we use something similar to Russian peasant
+	// multiplication. We iterate over b's bits, starting from the highest to the lowest.
+	// i denotes the bit we're currently processing (7, 6, 5, 4, 3, 2, 1, 0).
+	// The accumulator is set to 0; every iteration, we multiply the accumulator
+	// by X modulo X^8+X^4+X^3+X+1, and then add a to the accumulator in case b's i-th bit is 1.
+	var accumulator uint8 = 0
+	var i uint8 = 8
+
+	for i > 0 {
+		i--
+		// Get the i-th bit of b; bitOfB is either 0 or 1.
+		bitOfB := b >> i & 1
+		// aOrZero is 0 if the i-th bit of b is 0, and a if the i-th bit of b is 1. This is
+		// what we later add to the accumulator.
+		aOrZero := -bitOfB & a
+		// zeroOr1B is 0 if the 7th bit of the accumulator is 0, and 0x1B = 11011_2 if the
+		// 7th bit of accumulator is 1
+		zeroOr1B := -(accumulator >> 7) & 0x1B
+		// accumulatorMultipliedByX equals accumulator multiplied by X modulo X^8+X^4+X^3+X+1
+		// In the expression, accumulator + accumulator equals accumulator << 1, which would be
+		// the accumulator multiplied by X modulo X^8.
+		// By XORing (addition and subtraction in GF(2^8)) with zeroOr1B, we turn this into
+		// accumulator multiplied by X modulo X^8 + X^4 + X^3 + X + 1.
+		accumulatorMultipliedByX := zeroOr1B ^ (accumulator + accumulator)
+		// We can now compute the next value of the accumulator as the sum (in GF(2^8)) of aOrZero
+		// and accumulatorMultipliedByX.
+		accumulator = aOrZero ^ accumulatorMultipliedByX
 	}
 
-	return ret
+	return accumulator
 }
 
 // add combines two numbers in GF(2^8)
 // This can also be used for subtraction since it is symmetric.
 func add(a, b uint8) uint8 {
+	// Addition in GF(2^8) equals XOR:
 	return a ^ b
 }
 
@@ -184,13 +209,6 @@ func Split(secret []byte, parts, threshold int) ([][]byte, error) {
 		return nil, fmt.Errorf("cannot split an empty secret")
 	}
 
-	// Generate random x coordinates for computing points. I don't know
-	// why random x coordinates are used, and I also don't know why
-	// a non-cryptographically secure source of randomness is used.
-	// As far as I know the x coordinates do not need to be random.
-
-	xCoordinates := mathrand.Perm(255)
-
 	// Allocate the output array, initialize the final byte
 	// of the output with the offset. The representation of each
 	// output is {y1, y2, .., yN, x}.
@@ -201,7 +219,7 @@ func Split(secret []byte, parts, threshold int) ([][]byte, error) {
 		// then the result of evaluating the polynomial at that point
 		// will be our secret
 		out[idx] = make([]byte, len(secret)+1)
-		out[idx][len(secret)] = uint8(xCoordinates[idx]) + 1
+		out[idx][len(secret)] = uint8(idx) + 1
 	}
 
 	// Construct a random polynomial for each byte of the secret.
@@ -222,7 +240,7 @@ func Split(secret []byte, parts, threshold int) ([][]byte, error) {
 		for i := 0; i < parts; i++ {
 			// Add 1 to the xCoordinate because if it's 0,
 			// then the result of p.evaluate(x) will be our secret
-			x := uint8(xCoordinates[i]) + 1
+			x := uint8(i) + 1
 			// Evaluate the polynomial at x
 			y := p.evaluate(x)
 			out[i][idx] = y

shamir/shamir_test.go

@@ -115,6 +115,22 @@ func TestCombine(t *testing.T) {
 	}
 }
 
+func TestField_MulDivSmoke(t *testing.T) {
+	for a := range 256 {
+		for b := range 256 {
+			if b == 0 {
+				if out := mult(uint8(a), uint8(b)); out != 0 {
+					t.Fatalf("Bad: %v * %v = %v 0", a, b, out)
+				}
+			} else {
+				if out := div(mult(uint8(a), uint8(b)), uint8(b)); out != uint8(a) {
+					t.Fatalf("Bad: (%v * %v) / %v = %v %v", a, b, b, out, a)
+				}
+			}
+		}
+	}
+}
+
 func TestField_Add(t *testing.T) {
 	if out := add(16, 16); out != 0 {
 		t.Fatalf("Bad: %v 16", out)