Implement generalised RSA public exponents for verified boot

Remove the verified boot limitation that only allows a single
RSA public exponent of 65537 (F4).  This change allows use with
existing PKI infrastructure and has been tested with HSM-based
PKI.

Change the configuration OF tree format to store the RSA public
exponent as a 64 bit integer and implement backward compatibility
for verified boot configuration trees without this extra field.

Parameterise vboot_test.sh to test different public exponents.

Mathematics and other hard work by Andrew Bott.

Tested with the following public exponents: 3, 5, 17, 257, 39981,
50457, 65537 and 4294967297.

Signed-off-by: Andrew Bott <Andrew.Bott@ipaccess.com>
Signed-off-by: Andrew Wishart <Andrew.Wishart@ipaccess.com>
Signed-off-by: Neil Piercy <Neil.Piercy@ipaccess.com>
Signed-off-by: Michael van der Westhuizen <michael@smart-africa.com>
Cc: Simon Glass <sjg@chromium.org>

doc/uImage.FIT/signature.txt

@@ -66,7 +66,8 @@ Creating an RSA key and certificate
 -----------------------------------
 To create a new public key, size 2048 bits:
 
-$ openssl genrsa -F4 -out keys/dev.key 2048
+$ openssl genpkey -algorithm RSA -out keys/dev.key \
+    -pkeyopt rsa_keygen_bits:2048 -pkeyopt rsa_keygen_pubexp:65537
 
 To create a certificate for this:
 
@@ -159,6 +160,7 @@ For RSA the following are mandatory:
 
 - rsa,num-bits: Number of key bits (e.g. 2048)
 - rsa,modulus: Modulus (N) as a big-endian multi-word integer
+- rsa,exponent: Public exponent (E) as a 64 bit unsigned integer
 - rsa,r-squared: (2^num-bits)^2 as a big-endian multi-word integer
 - rsa,n0-inverse: -1 / modulus[0] mod 2^32
 

include/u-boot/rsa.h

@@ -27,6 +27,7 @@ struct rsa_public_key {
 	uint32_t n0inv;		/* -1 / modulus[0] mod 2^32 */
 	uint32_t *modulus;	/* modulus as little endian array */
 	uint32_t *rr;		/* R^2 as little endian array */
+	uint64_t exponent;	/* public exponent */
 };
 
 #if IMAGE_ENABLE_SIGN

lib/rsa/rsa-sign.c

@@ -260,11 +260,58 @@ err_priv:
 	return ret;
 }
 
+/*
+ * rsa_get_exponent(): - Get the public exponent from an RSA key
+ */
+static int rsa_get_exponent(RSA *key, uint64_t *e)
+{
+	int ret;
+	BIGNUM *bn_te;
+	uint64_t te;
+
+	ret = -EINVAL;
+	bn_te = NULL;
+
+	if (!e)
+		goto cleanup;
+
+	if (BN_num_bits(key->e) > 64)
+		goto cleanup;
+
+	*e = BN_get_word(key->e);
+
+	if (BN_num_bits(key->e) < 33) {
+		ret = 0;
+		goto cleanup;
+	}
+
+	bn_te = BN_dup(key->e);
+	if (!bn_te)
+		goto cleanup;
+
+	if (!BN_rshift(bn_te, bn_te, 32))
+		goto cleanup;
+
+	if (!BN_mask_bits(bn_te, 32))
+		goto cleanup;
+
+	te = BN_get_word(bn_te);
+	te <<= 32;
+	*e |= te;
+	ret = 0;
+
+cleanup:
+	if (bn_te)
+		BN_free(bn_te);
+
+	return ret;
+}
+
 /*
  * rsa_get_params(): - Get the important parameters of an RSA public key
  */
-int rsa_get_params(RSA *key, uint32_t *n0_invp, BIGNUM **modulusp,
+int rsa_get_params(RSA *key, uint64_t *exponent, uint32_t *n0_invp,
-		   BIGNUM **r_squaredp)
+		   BIGNUM **modulusp, BIGNUM **r_squaredp)
 {
 	BIGNUM *big1, *big2, *big32, *big2_32;
 	BIGNUM *n, *r, *r_squared, *tmp;
@@ -286,6 +333,9 @@ int rsa_get_params(RSA *key, uint32_t *n0_invp, BIGNUM **modulusp,
 		return -ENOMEM;
 	}
 
+	if (0 != rsa_get_exponent(key, exponent))
+		ret = -1;
+
 	if (!BN_copy(n, key->n) || !BN_set_word(big1, 1L) ||
 	    !BN_set_word(big2, 2L) || !BN_set_word(big32, 32L))
 		ret = -1;
@@ -386,6 +436,7 @@ static int fdt_add_bignum(void *blob, int noffset, const char *prop_name,
 int rsa_add_verify_data(struct image_sign_info *info, void *keydest)
 {
 	BIGNUM *modulus, *r_squared;
+	uint64_t exponent;
 	uint32_t n0_inv;
 	int parent, node;
 	char name[100];
@@ -397,7 +448,7 @@ int rsa_add_verify_data(struct image_sign_info *info, void *keydest)
 	ret = rsa_get_pub_key(info->keydir, info->keyname, &rsa);
 	if (ret)
 		return ret;
-	ret = rsa_get_params(rsa, &n0_inv, &modulus, &r_squared);
+	ret = rsa_get_params(rsa, &exponent, &n0_inv, &modulus, &r_squared);
 	if (ret)
 		return ret;
 	bits = BN_num_bits(modulus);
@@ -441,6 +492,9 @@ int rsa_add_verify_data(struct image_sign_info *info, void *keydest)
 		ret = fdt_setprop_u32(keydest, node, "rsa,num-bits", bits);
 	if (!ret)
 		ret = fdt_setprop_u32(keydest, node, "rsa,n0-inverse", n0_inv);
+	if (!ret) {
+		ret = fdt_setprop_u64(keydest, node, "rsa,exponent", exponent);
+	}
 	if (!ret) {
 		ret = fdt_add_bignum(keydest, node, "rsa,modulus", modulus,
 				     bits);

lib/rsa/rsa-verify.c

@@ -26,6 +26,9 @@
 #define get_unaligned_be32(a) fdt32_to_cpu(*(uint32_t *)a)
 #define put_unaligned_be32(a, b) (*(uint32_t *)(b) = cpu_to_fdt32(a))
 
+/* Default public exponent for backward compatibility */
+#define RSA_DEFAULT_PUBEXP	65537
+
 /**
  * subtract_modulus() - subtract modulus from the given value
  *
@@ -122,6 +125,48 @@ static void montgomery_mul(const struct rsa_public_key *key,
 		montgomery_mul_add_step(key, result, a[i], b);
 }
 
+/**
+ * num_pub_exponent_bits() - Number of bits in the public exponent
+ *
+ * @key:	RSA key
+ * @num_bits:	Storage for the number of public exponent bits
+ */
+static int num_public_exponent_bits(const struct rsa_public_key *key,
+		int *num_bits)
+{
+	uint64_t exponent;
+	int exponent_bits;
+	const uint max_bits = (sizeof(exponent) * 8);
+
+	exponent = key->exponent;
+	exponent_bits = 0;
+
+	if (!exponent) {
+		*num_bits = exponent_bits;
+		return 0;
+	}
+
+	for (exponent_bits = 1; exponent_bits < max_bits + 1; ++exponent_bits)
+		if (!(exponent >>= 1)) {
+			*num_bits = exponent_bits;
+			return 0;
+		}
+
+	return -EINVAL;
+}
+
+/**
+ * is_public_exponent_bit_set() - Check if a bit in the public exponent is set
+ *
+ * @key:	RSA key
+ * @pos:	The bit position to check
+ */
+static int is_public_exponent_bit_set(const struct rsa_public_key *key,
+		int pos)
+{
+	return key->exponent & (1ULL << pos);
+}
+
 /**
  * pow_mod() - in-place public exponentiation
  *
@@ -132,6 +177,7 @@ static int pow_mod(const struct rsa_public_key *key, uint32_t *inout)
 {
 	uint32_t *result, *ptr;
 	uint i;
+	int j, k;
 
 	/* Sanity check for stack size - key->len is in 32-bit words */
 	if (key->len > RSA_MAX_KEY_BITS / 32) {
@@ -141,18 +187,48 @@ static int pow_mod(const struct rsa_public_key *key, uint32_t *inout)
 	}
 
 	uint32_t val[key->len], acc[key->len], tmp[key->len];
+	uint32_t a_scaled[key->len];
 	result = tmp;  /* Re-use location. */
 
 	/* Convert from big endian byte array to little endian word array. */
 	for (i = 0, ptr = inout + key->len - 1; i < key->len; i++, ptr--)
 		val[i] = get_unaligned_be32(ptr);
 
-	montgomery_mul(key, acc, val, key->rr);  /* axx = a * RR / R mod M */
+	if (0 != num_public_exponent_bits(key, &k))
-	for (i = 0; i < 16; i += 2) {
+		return -EINVAL;
-		montgomery_mul(key, tmp, acc, acc); /* tmp = acc^2 / R mod M */
+
-		montgomery_mul(key, acc, tmp, tmp); /* acc = tmp^2 / R mod M */
+	if (k < 2) {
+		debug("Public exponent is too short (%d bits, minimum 2)\n",
+		      k);
+		return -EINVAL;
 	}
-	montgomery_mul(key, result, acc, val);  /* result = XX * a / R mod M */
+
+	if (!is_public_exponent_bit_set(key, 0)) {
+		debug("LSB of RSA public exponent must be set.\n");
+		return -EINVAL;
+	}
+
+	/* the bit at e[k-1] is 1 by definition, so start with: C := M */
+	montgomery_mul(key, acc, val, key->rr); /* acc = a * RR / R mod n */
+	/* retain scaled version for intermediate use */
+	memcpy(a_scaled, acc, key->len * sizeof(a_scaled[0]));
+
+	for (j = k - 2; j > 0; --j) {
+		montgomery_mul(key, tmp, acc, acc); /* tmp = acc^2 / R mod n */
+
+		if (is_public_exponent_bit_set(key, j)) {
+			/* acc = tmp * val / R mod n */
+			montgomery_mul(key, acc, tmp, a_scaled);
+		} else {
+			/* e[j] == 0, copy tmp back to acc for next operation */
+			memcpy(acc, tmp, key->len * sizeof(acc[0]));
+		}
+	}
+
+	/* the bit at e[0] is always 1 */
+	montgomery_mul(key, tmp, acc, acc); /* tmp = acc^2 / R mod n */
+	montgomery_mul(key, acc, tmp, val); /* acc = tmp * a / R mod M */
+	memcpy(result, acc, key->len * sizeof(result[0]));
 
 	/* Make sure result < mod; result is at most 1x mod too large. */
 	if (greater_equal_modulus(key, result))
@@ -229,6 +305,8 @@ static int rsa_verify_with_keynode(struct image_sign_info *info,
 	const void *blob = info->fdt_blob;
 	struct rsa_public_key key;
 	const void *modulus, *rr;
+	const uint64_t *public_exponent;
+	int length;
 	int ret;
 
 	if (node < 0) {
@@ -241,6 +319,11 @@ static int rsa_verify_with_keynode(struct image_sign_info *info,
 	}
 	key.len = fdtdec_get_int(blob, node, "rsa,num-bits", 0);
 	key.n0inv = fdtdec_get_int(blob, node, "rsa,n0-inverse", 0);
+	public_exponent = fdt_getprop(blob, node, "rsa,exponent", &length);
+	if (!public_exponent || length < sizeof(*public_exponent))
+		key.exponent = RSA_DEFAULT_PUBEXP;
+	else
+		key.exponent = fdt64_to_cpu(*public_exponent);
 	modulus = fdt_getprop(blob, node, "rsa,modulus", NULL);
 	rr = fdt_getprop(blob, node, "rsa,r-squared", NULL);
 	if (!key.len || !modulus || !rr) {

test/vboot/vboot_test.sh

@@ -54,8 +54,16 @@ echo ${mkimage} -D "${dtc}"
 echo "Build keys"
 mkdir -p ${keys}
 
+PUBLIC_EXPONENT=${1}
+
+if [ -z "${PUBLIC_EXPONENT}" ]; then
+	PUBLIC_EXPONENT=65537
+fi
+
 # Create an RSA key pair
-openssl genrsa -F4 -out ${keys}/dev.key 2048 2>/dev/null
+openssl genpkey -algorithm RSA -out ${keys}/dev.key \
+    -pkeyopt rsa_keygen_bits:2048 \
+    -pkeyopt rsa_keygen_pubexp:${PUBLIC_EXPONENT} 2>/dev/null
 
 # Create a certificate containing the public key
 openssl req -batch -new -x509 -key ${keys}/dev.key -out ${keys}/dev.crt