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Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography Specifications Version 2.1

Part 2 of 3, p. 14 to 43

6. Overview of schemes

A scheme combines cryptographic primitives and other techniques to
achieve a particular security goal.  Two types of scheme are
specified in this document: encryption schemes and signature schemes
with appendix.

The schemes specified in this document are limited in scope in that
their operations consist only of steps to process data with an RSA
public or private key, and do not include steps for obtaining or
validating the key.  Thus, in addition to the scheme operations, an
application will typically include key management operations by which

   parties may select RSA public and private keys for a scheme
operation.  The specific additional operations and other details are
outside the scope of this document.

As was the case for the cryptographic primitives (Section 5), the
specifications of scheme operations assume that certain conditions
are met by the inputs, in particular that RSA public and private keys
are valid.  The behavior of an implementation is thus unspecified
when a key is invalid.  The impact of such unspecified behavior
depends on the application.  Possible means of addressing key
validation include explicit key validation by the application; key
validation within the public-key infrastructure; and assignment of
liability for operations performed with an invalid key to the party
who generated the key.

A generally good cryptographic practice is to employ a given RSA key
pair in only one scheme.  This avoids the risk that vulnerability in
one scheme may compromise the security of the other, and may be
essential to maintain provable security.  While RSAES-PKCS1-v1_5
(Section 7.2) and RSASSA-PKCS1-v1_5 (Section 8.2) have traditionally
been employed together without any known bad interactions (indeed,
this is the model introduced by PKCS #1 v1.5), such a combined use of
an RSA key pair is not recommended for new applications.

To illustrate the risks related to the employment of an RSA key pair
in more than one scheme, suppose an RSA key pair is employed in both
RSAES-OAEP (Section 7.1) and RSAES-PKCS1-v1_5.  Although RSAES-OAEP
by itself would resist attack, an opponent might be able to exploit a
weakness in the implementation of RSAES-PKCS1-v1_5 to recover
messages encrypted with either scheme.  As another example, suppose
an RSA key pair is employed in both RSASSA-PSS (Section 8.1) and
RSASSA-PKCS1-v1_5.  Then the security proof for RSASSA-PSS would no
longer be sufficient since the proof does not account for the
possibility that signatures might be generated with a second scheme.
Similar considerations may apply if an RSA key pair is employed in
one of the schemes defined here and in a variant defined elsewhere.

7. Encryption schemes

For the purposes of this document, an encryption scheme consists of
an encryption operation and a decryption operation, where the
encryption operation produces a ciphertext from a message with a
recipient's RSA public key, and the decryption operation recovers the
message from the ciphertext with the recipient's corresponding RSA
private key.

   An encryption scheme can be employed in a variety of applications.  A
typical application is a key establishment protocol, where the
message contains key material to be delivered confidentially from one
party to another.  For instance, PKCS #7 [45] employs such a protocol
to deliver a content-encryption key from a sender to a recipient; the
encryption schemes defined here would be suitable key-encryption
algorithms in that context.

Two encryption schemes are specified in this document: RSAES-OAEP and
RSAES-PKCS1-v1_5.  RSAES-OAEP is recommended for new applications;
RSAES-PKCS1-v1_5 is included only for compatibility with existing
applications, and is not recommended for new applications.

The encryption schemes given here follow a general model similar to
that employed in IEEE Std 1363-2000 [26], combining encryption and
decryption primitives with an encoding method for encryption.  The
encryption operations apply a message encoding operation to a message
to produce an encoded message, which is then converted to an integer
message representative.  An encryption primitive is applied to the
message representative to produce the ciphertext.  Reversing this,
the decryption operations apply a decryption primitive to the
ciphertext to recover a message representative, which is then
converted to an octet string encoded message.  A message decoding
operation is applied to the encoded message to recover the message
and verify the correctness of the decryption.

To avoid implementation weaknesses related to the way errors are
handled within the decoding operation (see [6] and [36]), the
encoding and decoding operations for RSAES-OAEP and RSAES-PKCS1-v1_5
are embedded in the specifications of the respective encryption
schemes rather than defined in separate specifications.  Both
encryption schemes are compatible with the corresponding schemes in
PKCS #1 v2.0.

7.1 RSAES-OAEP

RSAES-OAEP combines the RSAEP and RSADP primitives (Sections 5.1.1
and 5.1.2) with the EME-OAEP encoding method (step 1.b in Section
7.1.1 and step 3 in Section 7.1.2).  EME-OAEP is based on Bellare and
Rogaway's Optimal Asymmetric Encryption scheme [3].  (OAEP stands for
"Optimal Asymmetric Encryption Padding.").  It is compatible with the
IFES scheme defined in IEEE Std 1363-2000 [26], where the encryption
and decryption primitives are IFEP-RSA and IFDP-RSA and the message
encoding method is EME-OAEP.  RSAES-OAEP can operate on messages of
length up to k - 2hLen - 2 octets, where hLen is the length of the
output from the underlying hash function and k is the length in
octets of the recipient's RSA modulus.

   Assuming that computing e-th roots modulo n is infeasible and the
mask generation function in RSAES-OAEP has appropriate properties,
RSAES-OAEP is semantically secure against adaptive chosen-ciphertext
attacks.  This assurance is provable in the sense that the difficulty
of breaking RSAES-OAEP can be directly related to the difficulty of
inverting the RSA function, provided that the mask generation
function is viewed as a black box or random oracle; see [21] and the
note below for further discussion.

Both the encryption and the decryption operations of RSAES-OAEP take
the value of a label L as input.  In this version of PKCS #1, L is
the empty string; other uses of the label are outside the scope of
this document.  See Appendix A.2.1 for the relevant ASN.1 syntax.

RSAES-OAEP is parameterized by the choice of hash function and mask
generation function.  This choice should be fixed for a given RSA
key.  Suggested hash and mask generation functions are given in
Appendix B.

Note.  Recent results have helpfully clarified the security
properties of the OAEP encoding method [3] (roughly the procedure
described in step 1.b in Section 7.1.1).  The background is as
follows.  In 1994, Bellare and Rogaway [3] introduced a security
concept that they denoted plaintext awareness (PA94).  They proved
that if a deterministic public-key encryption primitive (e.g., RSAEP)
is hard to invert without the private key, then the corresponding
OAEP-based encryption scheme is plaintext-aware (in the random oracle
model), meaning roughly that an adversary cannot produce a valid
ciphertext without actually "knowing" the underlying plaintext.
Plaintext awareness of an encryption scheme is closely related to the
resistance of the scheme against chosen-ciphertext attacks.  In such
attacks, an adversary is given the opportunity to send queries to an
oracle simulating the decryption primitive.  Using the results of
these queries, the adversary attempts to decrypt a challenge
ciphertext.

However, there are two flavors of chosen-ciphertext attacks, and PA94
implies security against only one of them.  The difference relies on
what the adversary is allowed to do after she is given the challenge
ciphertext.  The indifferent attack scenario (denoted CCA1) does not
given the challenge ciphertext, whereas the adaptive scenario
(denoted CCA2) does (except that the decryption oracle refuses to
decrypt the challenge ciphertext once it is published).  In 1998,
Bellare and Rogaway, together with Desai and Pointcheval [2], came up
with a new, stronger notion of plaintext awareness (PA98) that does
imply security against CCA2.

   To summarize, there have been two potential sources for
misconception: that PA94 and PA98 are equivalent concepts; or that
CCA1 and CCA2 are equivalent concepts.  Either assumption leads to
the conclusion that the Bellare-Rogaway paper implies security of
OAEP against CCA2, which it does not.

(Footnote: It might be fair to mention that PKCS #1 v2.0 cites [3]
and claims that "a chosen ciphertext attack is ineffective against a
plaintext-aware encryption scheme such as RSAES-OAEP" without
specifying the kind of plaintext awareness or chosen ciphertext
attack considered.)

OAEP has never been proven secure against CCA2; in fact, Victor Shoup
[48] has demonstrated that such a proof does not exist in the general
case.  Put briefly, Shoup showed that an adversary in the CCA2
scenario who knows how to partially invert the encryption primitive
but does not know how to invert it completely may well be able to
break the scheme.  For example, one may imagine an attacker who is
able to break RSAES-OAEP if she knows how to recover all but the
first 20 bytes of a random integer encrypted with RSAEP.  Such an
attacker does not need to be able to fully invert RSAEP, because she
does not use the first 20 octets in her attack.

Still, RSAES-OAEP is secure against CCA2, which was proved by
Fujisaki, Okamoto, Pointcheval, and Stern [21] shortly after the
announcement of Shoup's result.  Using clever lattice reduction
techniques, they managed to show how to invert RSAEP completely given
a sufficiently large part of the pre-image.  This observation,
combined with a proof that OAEP is secure against CCA2 if the
underlying encryption primitive is hard to partially invert, fills
the gap between what Bellare and Rogaway proved about RSAES-OAEP and
what some may have believed that they proved.  Somewhat
paradoxically, we are hence saved by an ostensible weakness in RSAEP
(i.e., the whole inverse can be deduced from parts of it).

Unfortunately however, the security reduction is not efficient for
concrete parameters.  While the proof successfully relates an
Inv inverting RSA, the probability of success for Inv is only
approximately \epsilon^2 / 2^18, where \epsilon is the probability of

(Footnote: In [21] the probability of success for the inverter was
\epsilon^2 / 4.  The additional factor 1 / 2^16 is due to the eight
fixed zero bits at the beginning of the encoded message EM, which are
not present in the variant of OAEP considered in [21] (Inv must apply
Adv twice to invert RSA, and each application corresponds to a factor
1 / 2^8).)

   In addition, the running time for Inv is approximately t^2, where t
is the running time of the adversary.  The consequence is that we
cannot exclude the possibility that attacking RSAES-OAEP is
considerably easier than inverting RSA for concrete parameters.
Still, the existence of a security proof provides some assurance that
the RSAES-OAEP construction is sounder than ad hoc constructions such
as RSAES-PKCS1-v1_5.

Hybrid encryption schemes based on the RSA-KEM key encapsulation
paradigm offer tight proofs of security directly applicable to
concrete parameters; see [30] for discussion.  Future versions of
PKCS #1 may specify schemes based on this paradigm.

7.1.1 Encryption operation

RSAES-OAEP-ENCRYPT ((n, e), M, L)

Options:
Hash     hash function (hLen denotes the length in octets of the hash
function output)

Input:
(n, e)   recipient's RSA public key (k denotes the length in octets
of the RSA modulus n)
M        message to be encrypted, an octet string of length mLen,
where mLen <= k - 2hLen - 2
L        optional label to be associated with the message; the
default value for L, if L is not provided, is the empty
string

Output:
C        ciphertext, an octet string of length k

Errors:  "message too long"; "label too long"

Assumption: RSA public key (n, e) is valid

Steps:

1. Length checking:

a. If the length of L is greater than the input limitation for the
hash function (2^61 - 1 octets for SHA-1), output "label too
long" and stop.

b. If mLen > k - 2hLen - 2, output "message too long" and stop.

   2. EME-OAEP encoding (see Figure 1 below):

a. If the label L is not provided, let L be the empty string. Let
lHash = Hash(L), an octet string of length hLen (see the note
below).

b. Generate an octet string PS consisting of k - mLen - 2hLen - 2
zero octets.  The length of PS may be zero.

c. Concatenate lHash, PS, a single octet with hexadecimal value
0x01, and the message M to form a data block DB of length k -
hLen - 1 octets as

DB = lHash || PS || 0x01 || M.

d. Generate a random octet string seed of length hLen.

e. Let dbMask = MGF(seed, k - hLen - 1).

i. Concatenate a single octet with hexadecimal value 0x00,
length k octets as

3. RSA encryption:

a. Convert the encoded message EM to an integer message
representative m (see Section 4.2):

m = OS2IP (EM).

b. Apply the RSAEP encryption primitive (Section 5.1.1) to the RSA
public key (n, e) and the message representative m to produce
an integer ciphertext representative c:

c = RSAEP ((n, e), m).

c. Convert the ciphertext representative c to a ciphertext C of
length k octets (see Section 4.1):

C = I2OSP (c, k).

   4. Output the ciphertext C.

Note.  If L is the empty string, the corresponding hash value lHash
has the following hexadecimal representation for different choices of
Hash:

SHA-1:   (0x)da39a3ee 5e6b4b0d 3255bfef 95601890 afd80709
SHA-256: (0x)e3b0c442 98fc1c14 9afbf4c8 996fb924 27ae41e4 649b934c
a495991b 7852b855
SHA-384: (0x)38b060a7 51ac9638 4cd9327e b1b1e36a 21fdb711 14be0743
4c0cc7bf 63f6e1da 274edebf e76f65fb d51ad2f1 4898b95b
SHA-512: (0x)cf83e135 7eefb8bd f1542850 d66d8007 d620e405 0b5715dc
83f4a921 d36ce9ce 47d0d13c 5d85f2b0 ff8318d2 877eec2f
63b931bd 47417a81 a538327a f927da3e

__________________________________________________________________

+----------+---------+-------+
DB = |  lHash   |    PS   |   M   |
+----------+---------+-------+
|
+----------+              V
|   seed   |--> MGF ---> xor
+----------+              |
|                   |
+--+     V                   |
|00|    xor <----- MGF <-----|
+--+     |                   |
|      |                   |
V      V                   V
+--+----------+----------------------------+
+--+----------+----------------------------+
__________________________________________________________________

Figure 1: EME-OAEP encoding operation.  lHash is the hash of the
optional label L.  Decoding operation follows reverse steps to
recover M and verify lHash and PS.

7.1.2 Decryption operation

RSAES-OAEP-DECRYPT (K, C, L)

Options:
Hash     hash function (hLen denotes the length in octets of the hash
function output)

   Input:
K        recipient's RSA private key (k denotes the length in octets
of the RSA modulus n)
C        ciphertext to be decrypted, an octet string of length k,
where k = 2hLen + 2
L        optional label whose association with the message is to be
verified; the default value for L, if L is not provided, is
the empty string

Output:
M        message, an octet string of length mLen, where mLen <= k -
2hLen - 2

Error: "decryption error"

Steps:

1. Length checking:

a. If the length of L is greater than the input limitation for the
hash function (2^61 - 1 octets for SHA-1), output "decryption
error" and stop.

b. If the length of the ciphertext C is not k octets, output
"decryption error" and stop.

c. If k < 2hLen + 2, output "decryption error" and stop.

2.    RSA decryption:

a. Convert the ciphertext C to an integer ciphertext
representative c (see Section 4.2):

c = OS2IP (C).

b. Apply the RSADP decryption primitive (Section 5.1.2) to the
RSA private key K and the ciphertext representative c to
produce an integer message representative m:

If RSADP outputs "ciphertext representative out of range"
(meaning that c >= n), output "decryption error" and stop.

c. Convert the message representative m to an encoded message EM
of length k octets (see Section 4.1):

EM = I2OSP (m, k).

   3. EME-OAEP decoding:

a. If the label L is not provided, let L be the empty string. Let
lHash = Hash(L), an octet string of length hLen (see the note
in Section 7.1.1).

b. Separate the encoded message EM into a single octet Y, an octet
of length k - hLen - 1 as

e. Let dbMask = MGF(seed, k - hLen - 1).

g. Separate DB into an octet string lHash' of length hLen, a
(possibly empty) padding string PS consisting of octets with
hexadecimal value 0x00, and a message M as

DB = lHash' || PS || 0x01 || M.

If there is no octet with hexadecimal value 0x01 to separate PS
from M, if lHash does not equal lHash', or if Y is nonzero,
output "decryption error" and stop.  (See the note below.)

4. Output the message M.

Note.  Care must be taken to ensure that an opponent cannot
distinguish the different error conditions in Step 3.g, whether by
error message or timing, or, more generally, learn partial
information about the encoded message EM.  Otherwise an opponent may
be able to obtain useful information about the decryption of the
ciphertext C, leading to a chosen-ciphertext attack such as the one
observed by Manger [36].

7.2 RSAES-PKCS1-v1_5

RSAES-PKCS1-v1_5 combines the RSAEP and RSADP primitives (Sections
5.1.1 and 5.1.2) with the EME-PKCS1-v1_5 encoding method (step 1 in
Section 7.2.1 and step 3 in Section 7.2.2).  It is mathematically
equivalent to the encryption scheme in PKCS #1 v1.5.  RSAES-PKCS1-
v1_5 can operate on messages of length up to k - 11 octets (k is the
octet length of the RSA modulus), although care should be taken to

   avoid certain attacks on low-exponent RSA due to Coppersmith,
Franklin, Patarin, and Reiter when long messages are encrypted (see
the third bullet in the notes below and [10]; [14] contains an
improved attack).  As a general rule, the use of this scheme for
encrypting an arbitrary message, as opposed to a randomly generated
key, is not recommended.

It is possible to generate valid RSAES-PKCS1-v1_5 ciphertexts without
knowing the corresponding plaintexts, with a reasonable probability
of success.  This ability can be exploited in a chosen- ciphertext
attack as shown in [6].  Therefore, if RSAES-PKCS1-v1_5 is to be
used, certain easily implemented countermeasures should be taken to
thwart the attack found in [6].  Typical examples include the
addition of structure to the data to be encoded, rigorous checking of
PKCS #1 v1.5 conformance (and other redundancy) in decrypted
messages, and the consolidation of error messages in a client-server
protocol based on PKCS #1 v1.5.  These can all be effective
countermeasures and do not involve changes to a PKCS #1 v1.5-based
protocol.  See [7] for a further discussion of these and other
countermeasures.  It has recently been shown that the security of the
SSL/TLS handshake protocol [17], which uses RSAES-PKCS1-v1_5 and
certain countermeasures, can be related to a variant of the RSA
problem; see [32] for discussion.

Note.  The following passages describe some security recommendations
pertaining to the use of RSAES-PKCS1-v1_5.  Recommendations from
version 1.5 of this document are included as well as new
intervening years.

* It is recommended that the pseudorandom octets in step 2 in
Section 7.2.1 be generated independently for each encryption
process, especially if the same data is input to more than one
encryption process.  Haastad's results [24] are one motivation for
this recommendation.

* The padding string PS in step 2 in Section 7.2.1 is at least eight
octets long, which is a security condition for public-key
operations that makes it difficult for an attacker to recover data
by trying all possible encryption blocks.

* The pseudorandom octets can also help thwart an attack due to
Coppersmith et al. [10] (see [14] for an improvement of the
attack) when the size of the message to be encrypted is kept
small.  The attack works on low-exponent RSA when similar messages
are encrypted with the same RSA public key.  More specifically, in
one flavor of the attack, when two inputs to RSAEP agree on a
large fraction of bits (8/9) and low-exponent RSA (e = 3) is used

      to encrypt both of them, it may be possible to recover both inputs
with the attack.  Another flavor of the attack is successful in
decrypting a single ciphertext when a large fraction (2/3) of the
input to RSAEP is already known.  For typical applications, the
message to be encrypted is short (e.g., a 128-bit symmetric key)
so not enough information will be known or common between two
messages to enable the attack.  However, if a long message is
encrypted, or if part of a message is known, then the attack may
be a concern.  In any case, the RSAES-OAEP scheme overcomes the
attack.

7.2.1 Encryption operation

RSAES-PKCS1-V1_5-ENCRYPT ((n, e), M)

Input:
(n, e)   recipient's RSA public key (k denotes the length in octets
of the modulus n)
M        message to be encrypted, an octet string of length mLen,
where mLen <= k - 11

Output:
C        ciphertext, an octet string of length k

Error: "message too long"

Steps:

1. Length checking: If mLen > k - 11, output "message too long" and
stop.

2. EME-PKCS1-v1_5 encoding:

a. Generate an octet string PS of length k - mLen - 3 consisting
of pseudo-randomly generated nonzero octets.  The length of PS
will be at least eight octets.

b. Concatenate PS, the message M, and other padding to form an
encoded message EM of length k octets as

EM = 0x00 || 0x02 || PS || 0x00 || M.

   3. RSA encryption:

a. Convert the encoded message EM to an integer message
representative m (see Section 4.2):

m = OS2IP (EM).

b. Apply the RSAEP encryption primitive (Section 5.1.1) to the RSA
public key (n, e) and the message representative m to produce
an integer ciphertext representative c:

c = RSAEP ((n, e), m).

c. Convert the ciphertext representative c to a ciphertext C of
length k octets (see Section 4.1):

C = I2OSP (c, k).

4. Output the ciphertext C.

7.2.2 Decryption operation

RSAES-PKCS1-V1_5-DECRYPT (K, C)

Input:
K        recipient's RSA private key
C        ciphertext to be decrypted, an octet string of length k,
where k is the length in octets of the RSA modulus n

Output:
M        message, an octet string of length at most k - 11

Error: "decryption error"

Steps:

1. Length checking: If the length of the ciphertext C is not k octets
(or if k < 11), output "decryption error" and stop.

2. RSA decryption:

a. Convert the ciphertext C to an integer ciphertext
representative c (see Section 4.2):

c = OS2IP (C).

      b. Apply the RSADP decryption primitive (Section 5.1.2) to the RSA
private key (n, d) and the ciphertext representative c to
produce an integer message representative m:

m = RSADP ((n, d), c).

If RSADP outputs "ciphertext representative out of range"
(meaning that c >= n), output "decryption error" and stop.

c. Convert the message representative m to an encoded message EM
of length k octets (see Section 4.1):

EM = I2OSP (m, k).

3. EME-PKCS1-v1_5 decoding: Separate the encoded message EM into an
octet string PS consisting of nonzero octets and a message M as

EM = 0x00 || 0x02 || PS || 0x00 || M.

If the first octet of EM does not have hexadecimal value 0x00, if
the second octet of EM does not have hexadecimal value 0x02, if
there is no octet with hexadecimal value 0x00 to separate PS from
M, or if the length of PS is less than 8 octets, output
"decryption error" and stop.  (See the note below.)

4. Output M.

Note.  Care shall be taken to ensure that an opponent cannot
distinguish the different error conditions in Step 3, whether by
error message or timing.  Otherwise an opponent may be able to obtain
to a strengthened version of Bleichenbacher's attack [6]; compare to
Manger's attack [36].

8. Signature schemes with appendix

For the purposes of this document, a signature scheme with appendix
consists of a signature generation operation and a signature
verification operation, where the signature generation operation
produces a signature from a message with a signer's RSA private key,
and the signature verification operation verifies the signature on
the message with the signer's corresponding RSA public key.  To
verify a signature constructed with this type of scheme it is
necessary to have the message itself.  In this way, signature schemes
with appendix are distinguished from signature schemes with message
recovery, which are not supported in this document.

   A signature scheme with appendix can be employed in a variety of
applications.  For instance, the signature schemes with appendix
defined here would be suitable signature algorithms for X.509
certificates [28].  Related signature schemes could be employed in
PKCS #7 [45], although for technical reasons the current version of
PKCS #7 separates a hash function from a signature scheme, which is
different than what is done here; see the note in Appendix A.2.3 for
more discussion.

Two signature schemes with appendix are specified in this document:
RSASSA-PSS and RSASSA-PKCS1-v1_5.  Although no attacks are known
against RSASSA-PKCS1-v1_5, in the interest of increased robustness,
RSASSA-PSS is recommended for eventual adoption in new applications.
RSASSA-PKCS1-v1_5 is included for compatibility with existing
applications, and while still appropriate for new applications, a
gradual transition to RSASSA-PSS is encouraged.

The signature schemes with appendix given here follow a general model
similar to that employed in IEEE Std 1363-2000 [26], combining
signature and verification primitives with an encoding method for
signatures.  The signature generation operations apply a message
encoding operation to a message to produce an encoded message, which
is then converted to an integer message representative.  A signature
primitive is applied to the message representative to produce the
signature.  Reversing this, the signature verification operations
apply a signature verification primitive to the signature to recover
a message representative, which is then converted to an octet string
encoded message.  A verification operation is applied to the message
and the encoded message to determine whether they are consistent.

If the encoding method is deterministic (e.g., EMSA-PKCS1-v1_5), the
verification operation may apply the message encoding operation to
the message and compare the resulting encoded message to the
previously derived encoded message.  If there is a match, the
signature is considered valid.  If the method is randomized (e.g.,
EMSA-PSS), the verification operation is typically more complicated.
For example, the verification operation in EMSA-PSS extracts the
random salt and a hash output from the encoded message and checks
whether the hash output, the salt, and the message are consistent;
the hash output is a deterministic function in terms of the message
and the salt.

For both signature schemes with appendix defined in this document,
the signature generation and signature verification operations are
readily implemented as "single-pass" operations if the signature is
placed after the message.  See PKCS #7 [45] for an example format in
the case of RSASSA-PKCS1-v1_5.

8.1 RSASSA-PSS

RSASSA-PSS combines the RSASP1 and RSAVP1 primitives with the EMSA-
PSS encoding method.  It is compatible with the IFSSA scheme as
amended in the IEEE P1363a draft [27], where the signature and
verification primitives are IFSP-RSA1 and IFVP-RSA1 as defined in
IEEE Std 1363-2000 [26] and the message encoding method is EMSA4.
EMSA4 is slightly more general than EMSA-PSS as it acts on bit
strings rather than on octet strings.  EMSA-PSS is equivalent to
EMSA4 restricted to the case that the operands as well as the hash
and salt values are octet strings.

The length of messages on which RSASSA-PSS can operate is either
unrestricted or constrained by a very large number, depending on the
hash function underlying the EMSA-PSS encoding method.

Assuming that computing e-th roots modulo n is infeasible and the
hash and mask generation functions in EMSA-PSS have appropriate
properties, RSASSA-PSS provides secure signatures.  This assurance is
provable in the sense that the difficulty of forging signatures can
be directly related to the difficulty of inverting the RSA function,
provided that the hash and mask generation functions are viewed as
black boxes or random oracles.  The bounds in the security proof are
essentially "tight", meaning that the success probability and running
time for the best forger against RSASSA-PSS are very close to the
corresponding parameters for the best RSA inversion algorithm; see
[4][13][31] for further discussion.

In contrast to the RSASSA-PKCS1-v1_5 signature scheme, a hash
function identifier is not embedded in the EMSA-PSS encoded message,
so in theory it is possible for an adversary to substitute a
different (and potentially weaker) hash function than the one
selected by the signer.  Therefore, it is recommended that the EMSA-
PSS mask generation function be based on the same hash function.  In
this manner the entire encoded message will be dependent on the hash
function and it will be difficult for an opponent to substitute a
different hash function than the one intended by the signer.  This
matching of hash functions is only for the purpose of preventing hash
function substitution, and is not necessary if hash function
substitution is addressed by other means (e.g., the verifier accepts
only a designated hash function).  See [34] for further discussion of
these points.  The provable security of RSASSA-PSS does not rely on
the hash function in the mask generation function being the same as
the hash function applied to the message.

RSASSA-PSS is different from other RSA-based signature schemes in
that it is probabilistic rather than deterministic, incorporating a
randomly generated salt value.  The salt value enhances the security

   of the scheme by affording a "tighter" security proof than
deterministic alternatives such as Full Domain Hashing (FDH); see [4]
for discussion.  However, the randomness is not critical to security.
In situations where random generation is not possible, a fixed value
or a sequence number could be employed instead, with the resulting
provable security similar to that of FDH [12].

8.1.1 Signature generation operation

RSASSA-PSS-SIGN (K, M)

Input:
K        signer's RSA private key
M        message to be signed, an octet string

Output:
S        signature, an octet string of length k, where k is the
length in octets of the RSA modulus n

Errors: "message too long;" "encoding error"

Steps:

1. EMSA-PSS encoding: Apply the EMSA-PSS encoding operation (Section
9.1.1) to the message M to produce an encoded message EM of length
\ceil ((modBits - 1)/8) octets such that the bit length of the
integer OS2IP (EM) (see Section 4.2) is at most modBits - 1, where
modBits is the length in bits of the RSA modulus n:

EM = EMSA-PSS-ENCODE (M, modBits - 1).

Note that the octet length of EM will be one less than k if
modBits - 1 is divisible by 8 and equal to k otherwise.  If the
encoding operation outputs "message too long," output "message too
long" and stop.  If the encoding operation outputs "encoding
error," output "encoding error" and stop.

2. RSA signature:

a. Convert the encoded message EM to an integer message
representative m (see Section 4.2):

m = OS2IP (EM).

      b. Apply the RSASP1 signature primitive (Section 5.2.1) to the RSA
private key K and the message representative m to produce an
integer signature representative s:

s = RSASP1 (K, m).

c. Convert the signature representative s to a signature S of
length k octets (see Section 4.1):

S = I2OSP (s, k).

3. Output the signature S.

8.1.2 Signature verification operation

RSASSA-PSS-VERIFY ((n, e), M, S)

Input:
(n, e)   signer's RSA public key
M        message whose signature is to be verified, an octet string
S        signature to be verified, an octet string of length k, where
k is the length in octets of the RSA modulus n

Output:
"valid signature" or "invalid signature"

Steps:

1. Length checking: If the length of the signature S is not k octets,
output "invalid signature" and stop.

2. RSA verification:

a. Convert the signature S to an integer signature representative
s (see Section 4.2):

s = OS2IP (S).

b. Apply the RSAVP1 verification primitive (Section 5.2.2) to the
RSA public key (n, e) and the signature representative s to
produce an integer message representative m:

m = RSAVP1 ((n, e), s).

If RSAVP1 output "signature representative out of range,"
output "invalid signature" and stop.

      c. Convert the message representative m to an encoded message EM
of length emLen = \ceil ((modBits - 1)/8) octets, where modBits
is the length in bits of the RSA modulus n (see Section 4.1):

EM = I2OSP (m, emLen).

Note that emLen will be one less than k if modBits - 1 is
divisible by 8 and equal to k otherwise.  If I2OSP outputs
"integer too large," output "invalid signature" and stop.

3. EMSA-PSS verification: Apply the EMSA-PSS verification operation
(Section 9.1.2) to the message M and the encoded message EM to
determine whether they are consistent:

Result = EMSA-PSS-VERIFY (M, EM, modBits - 1).

4. If Result = "consistent," output "valid signature." Otherwise,
output "invalid signature."

8.2. RSASSA-PKCS1-v1_5

RSASSA-PKCS1-v1_5 combines the RSASP1 and RSAVP1 primitives with the
EMSA-PKCS1-v1_5 encoding method.  It is compatible with the IFSSA
scheme defined in IEEE Std 1363-2000 [26], where the signature and
verification primitives are IFSP-RSA1 and IFVP-RSA1 and the message
encoding method is EMSA-PKCS1-v1_5 (which is not defined in IEEE Std
1363-2000, but is in the IEEE P1363a draft [27]).

The length of messages on which RSASSA-PKCS1-v1_5 can operate is
either unrestricted or constrained by a very large number, depending
on the hash function underlying the EMSA-PKCS1-v1_5 method.

Assuming that computing e-th roots modulo n is infeasible and the
hash function in EMSA-PKCS1-v1_5 has appropriate properties, RSASSA-
PKCS1-v1_5 is conjectured to provide secure signatures.  More
precisely, forging signatures without knowing the RSA private key is
conjectured to be computationally infeasible.  Also, in the encoding
method EMSA-PKCS1-v1_5, a hash function identifier is embedded in the
encoding.  Because of this feature, an adversary trying to find a
message with the same signature as a previously signed message must
find collisions of the particular hash function being used; attacking
a different hash function than the one selected by the signer is not
useful to the adversary.  See [34] for further discussion.

Note.  As noted in PKCS #1 v1.5, the EMSA-PKCS1-v1_5 encoding method
has the property that the encoded message, converted to an integer
message representative, is guaranteed to be large and at least
somewhat "random".  This prevents attacks of the kind proposed by

   Desmedt and Odlyzko [16] where multiplicative relationships between
message representatives are developed by factoring the message
representatives into a set of small values (e.g., a set of small
primes).  Coron, Naccache, and Stern [15] showed that a stronger form
of this type of attack could be quite effective against some
instances of the ISO/IEC 9796-2 signature scheme.  They also analyzed
the complexity of this type of attack against the EMSA-PKCS1-v1_5
encoding method and concluded that an attack would be impractical,
requiring more operations than a collision search on the underlying
hash function (i.e., more than 2^80 operations).  Coppersmith,
Halevi, and Jutla [11] subsequently extended Coron et al.'s attack to
break the ISO/IEC 9796-1 signature scheme with message recovery.  The
various attacks illustrate the importance of carefully constructing
the input to the RSA signature primitive, particularly in a signature
scheme with message recovery.  Accordingly, the EMSA-PKCS-v1_5
encoding method explicitly includes a hash operation and is not
intended for signature schemes with message recovery.  Moreover,
while no attack is known against the EMSA-PKCS-v1_5 encoding method,
a gradual transition to EMSA-PSS is recommended as a precaution
against future developments.

8.2.1 Signature generation operation

RSASSA-PKCS1-V1_5-SIGN (K, M)

Input:
K        signer's RSA private key
M        message to be signed, an octet string

Output:
S        signature, an octet string of length k, where k is the
length in octets of the RSA modulus n

Errors: "message too long"; "RSA modulus too short"

Steps:

1. EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
operation (Section 9.2) to the message M to produce an encoded
message EM of length k octets:

EM = EMSA-PKCS1-V1_5-ENCODE (M, k).

If the encoding operation outputs "message too long," output
"message too long" and stop.  If the encoding operation outputs
"intended encoded message length too short," output "RSA modulus
too short" and stop.

   2. RSA signature:

a. Convert the encoded message EM to an integer message
representative m (see Section 4.2):

m = OS2IP (EM).

b. Apply the RSASP1 signature primitive (Section 5.2.1) to the RSA
private key K and the message representative m to produce an
integer signature representative s:

s = RSASP1 (K, m).

c. Convert the signature representative s to a signature S of
length k octets (see Section 4.1):

S = I2OSP (s, k).

3. Output the signature S.

8.2.2 Signature verification operation

RSASSA-PKCS1-V1_5-VERIFY ((n, e), M, S)

Input:
(n, e)   signer's RSA public key
M        message whose signature is to be verified, an octet string
S        signature to be verified, an octet string of length k, where
k is the length in octets of the RSA modulus n

Output:
"valid signature" or "invalid signature"

Errors: "message too long"; "RSA modulus too short"

Steps:

1. Length checking: If the length of the signature S is not k octets,
output "invalid signature" and stop.

2. RSA verification:

a. Convert the signature S to an integer signature representative
s (see Section 4.2):

s = OS2IP (S).

      b. Apply the RSAVP1 verification primitive (Section 5.2.2) to the
RSA public key (n, e) and the signature representative s to
produce an integer message representative m:

m = RSAVP1 ((n, e), s).

If RSAVP1 outputs "signature representative out of range,"
output "invalid signature" and stop.

c. Convert the message representative m to an encoded message EM
of length k octets (see Section 4.1):

EM' = I2OSP (m, k).

If I2OSP outputs "integer too large," output "invalid
signature" and stop.

3. EMSA-PKCS1-v1_5 encoding: Apply the EMSA-PKCS1-v1_5 encoding
operation (Section 9.2) to the message M to produce a second
encoded message EM' of length k octets:

EM' = EMSA-PKCS1-V1_5-ENCODE (M, k).

If the encoding operation outputs "message too long," output
"message too long" and stop.  If the encoding operation outputs
"intended encoded message length too short," output "RSA modulus
too short" and stop.

4. Compare the encoded message EM and the second encoded message EM'.
If they are the same, output "valid signature"; otherwise, output
"invalid signature."

Note.  Another way to implement the signature verification operation
is to apply a "decoding" operation (not specified in this document)
to the encoded message to recover the underlying hash value, and then
to compare it to a newly computed hash value.  This has the advantage
that it requires less intermediate storage (two hash values rather
than two encoded messages), but the disadvantage that it requires

9. Encoding methods for signatures with appendix

Encoding methods consist of operations that map between octet string
messages and octet string encoded messages, which are converted to
and from integer message representatives in the schemes.  The integer
message representatives are processed via the primitives.  The
encoding methods thus provide the connection between the schemes,
which process messages, and the primitives.

   An encoding method for signatures with appendix, for the purposes of
this document, consists of an encoding operation and optionally a
verification operation.  An encoding operation maps a message M to an
encoded message EM of a specified length.  A verification operation
determines whether a message M and an encoded message EM are
consistent, i.e., whether the encoded message EM is a valid encoding
of the message M.

The encoding operation may introduce some randomness, so that
different applications of the encoding operation to the same message
will produce different encoded messages, which has benefits for
provable security.  For such an encoding method, both an encoding and
a verification operation are needed unless the verifier can reproduce
the randomness (e.g., by obtaining the salt value from the signer).
For a deterministic encoding method only an encoding operation is
needed.

Two encoding methods for signatures with appendix are employed in the
signature schemes and are specified here: EMSA-PSS and EMSA-PKCS1-
v1_5.

9.1 EMSA-PSS

This encoding method is parameterized by the choice of hash function,
mask generation function, and salt length.  These options should be
fixed for a given RSA key, except that the salt length can be
variable (see [31] for discussion).  Suggested hash and mask
generation functions are given in Appendix B.  The encoding method is
based on Bellare and Rogaway's Probabilistic Signature Scheme (PSS)
[4][5].  It is randomized and has an encoding operation and a
verification operation.

   Figure 2 illustrates the encoding operation.

__________________________________________________________________

+-----------+
|     M     |
+-----------+
|
V
Hash
|
V
+--------+----------+----------+
M' = |Padding1|  mHash   |   salt   |
+--------+----------+----------+
|
+--------+----------+     V
+--------+----------+     |
|               |
V               |    +--+
xor <--- MGF <---|    |bc|
|               |    +--+
|               |      |
V               V      V
+-------------------+----------+--+
+-------------------+----------+--+
__________________________________________________________________

Figure 2: EMSA-PSS encoding operation.  Verification operation
follows reverse steps to recover salt, then forward steps to
recompute and compare H.

Notes.

1. The encoding method defined here differs from the one in Bellare
and Rogaway's submission to IEEE P1363a [5] in three respects:

*  It applies a hash function rather than a mask generation
function to the message.  Even though the mask generation
function is based on a hash function, it seems more natural to
apply a hash function directly.

*  The value that is hashed together with the salt value is the
string (0x)00 00 00 00 00 00 00 00 || mHash rather than the
message M itself.  Here, mHash is the hash of M.  Note that the

         hash function is the same in both steps.  See Note 3 below for
further discussion.  (Also, the name "salt" is used instead of
"seed", as it is more reflective of the value's role.)

*  The encoded message in EMSA-PSS has nine fixed bits; the first
bit is 0 and the last eight bits form a "trailer field", the
octet 0xbc.  In the original scheme, only the first bit is
fixed.  The rationale for the trailer field is for
compatibility with the Rabin-Williams IFSP-RW signature
primitive in IEEE Std 1363-2000 [26] and the corresponding
primitive in the draft ISO/IEC 9796-2 [29].

2. Assuming that the mask generation function is based on a hash
function, it is recommended that the hash function be the same as
the one that is applied to the message; see Section 8.1 for
further discussion.

3. Without compromising the security proof for RSASSA-PSS, one may
perform steps 1 and 2 of EMSA-PSS-ENCODE and EMSA-PSS-VERIFY (the
application of the hash function to the message) outside the
module that computes the rest of the signature operation, so that
mHash rather than the message M itself is input to the module.  In
other words, the security proof for RSASSA-PSS still holds even if
an opponent can control the value of mHash.  This is convenient if
the module has limited I/O bandwidth, e.g., a smart card.  Note
that previous versions of PSS [4][5] did not have this property.
Of course, it may be desirable for other security reasons to have
the module process the full message.  For instance, the module may
need to "see" what it is signing if it does not trust the
component that computes the hash value.

4. Typical salt lengths in octets are hLen (the length of the output
of the hash function Hash) and 0.  In both cases the security of
RSASSA-PSS can be closely related to the hardness of inverting
RSAVP1.  Bellare and Rogaway [4] give a tight lower bound for the
security of the original RSA-PSS scheme, which corresponds roughly
to the former case, while Coron [12] gives a lower bound for the
related Full Domain Hashing scheme, which corresponds roughly to
the latter case.  In [13] Coron provides a general treatment with
various salt lengths ranging from 0 to hLen; see [27] for
[4][13] to address the differences between the original and the
present version of RSA-PSS as listed in Note 1 above.

5. As noted in IEEE P1363a [27], the use of randomization in
signature schemes - such as the salt value in EMSA-PSS - may
provide a "covert channel" for transmitting information other than
the message being signed.  For more on covert channels, see [50].

9.1.1 Encoding operation

EMSA-PSS-ENCODE (M, emBits)

Options:

Hash     hash function (hLen denotes the length in octets of the hash
function output)
sLen     intended length in octets of the salt

Input:
M        message to be encoded, an octet string
emBits   maximal bit length of the integer OS2IP (EM) (see Section
4.2), at least 8hLen + 8sLen + 9

Output:
EM       encoded message, an octet string of length emLen = \ceil
(emBits/8)

Errors:  "encoding error"; "message too long"

Steps:

1.  If the length of M is greater than the input limitation for the
hash function (2^61 - 1 octets for SHA-1), output "message too
long" and stop.

2.  Let mHash = Hash(M), an octet string of length hLen.

3.  If emLen < hLen + sLen + 2, output "encoding error" and stop.

4.  Generate a random octet string salt of length sLen; if sLen = 0,
then salt is the empty string.

5.  Let
M' = (0x)00 00 00 00 00 00 00 00 || mHash || salt;

M' is an octet string of length 8 + hLen + sLen with eight
initial zero octets.

6.  Let H = Hash(M'), an octet string of length hLen.

7.  Generate an octet string PS consisting of emLen - sLen - hLen - 2
zero octets.  The length of PS may be 0.

8.  Let DB = PS || 0x01 || salt; DB is an octet string of length
emLen - hLen - 1.

   9.  Let dbMask = MGF(H, emLen - hLen - 1).

11. Set the leftmost 8emLen - emBits bits of the leftmost octet in

12. Let EM = maskedDB || H || 0xbc.

13. Output EM.

9.1.2 Verification operation

EMSA-PSS-VERIFY (M, EM, emBits)

Options:
Hash     hash function (hLen denotes the length in octets of the hash
function output)
sLen     intended length in octets of the salt

Input:
M        message to be verified, an octet string
EM       encoded message, an octet string of length emLen = \ceil
(emBits/8)
emBits   maximal bit length of the integer OS2IP (EM) (see Section
4.2), at least 8hLen + 8sLen + 9

Output:
"consistent" or "inconsistent"

Steps:

1.  If the length of M is greater than the input limitation for the
hash function (2^61 - 1 octets for SHA-1), output "inconsistent"
and stop.

2.  Let mHash = Hash(M), an octet string of length hLen.

3.  If emLen < hLen + sLen + 2, output "inconsistent" and stop.

4.  If the rightmost octet of EM does not have hexadecimal value
0xbc, output "inconsistent" and stop.

5.  Let maskedDB be the leftmost emLen - hLen - 1 octets of EM, and
let H be the next hLen octets.

   6.  If the leftmost 8emLen - emBits bits of the leftmost octet in
maskedDB are not all equal to zero, output "inconsistent" and
stop.

7.  Let dbMask = MGF(H, emLen - hLen - 1).

9.  Set the leftmost 8emLen - emBits bits of the leftmost octet in DB
to zero.

10. If the emLen - hLen - sLen - 2 leftmost octets of DB are not zero
or if the octet at position emLen - hLen - sLen - 1 (the leftmost
position is "position 1") does not have hexadecimal value 0x01,
output "inconsistent" and stop.

11.  Let salt be the last sLen octets of DB.

12.  Let
M' = (0x)00 00 00 00 00 00 00 00 || mHash || salt ;

M' is an octet string of length 8 + hLen + sLen with eight
initial zero octets.

13. Let H' = Hash(M'), an octet string of length hLen.

14. If H = H', output "consistent." Otherwise, output "inconsistent."

9.2 EMSA-PKCS1-v1_5

This encoding method is deterministic and only has an encoding
operation.

EMSA-PKCS1-v1_5-ENCODE (M, emLen)

Option:
Hash     hash function (hLen denotes the length in octets of the hash
function output)

Input:
M        message to be encoded
emLen    intended length in octets of the encoded message, at least
tLen + 11, where tLen is the octet length of the DER
encoding T of a certain value computed during the encoding
operation

   Output:
EM       encoded message, an octet string of length emLen

Errors:
"message too long"; "intended encoded message length too short"

Steps:

1. Apply the hash function to the message M to produce a hash value
H:

H = Hash(M).

If the hash function outputs "message too long," output "message
too long" and stop.

2. Encode the algorithm ID for the hash function and the hash value
into an ASN.1 value of type DigestInfo (see Appendix A.2.4) with
the Distinguished Encoding Rules (DER), where the type DigestInfo
has the syntax

DigestInfo ::= SEQUENCE {
digestAlgorithm AlgorithmIdentifier,
digest OCTET STRING
}

The first field identifies the hash function and the second
contains the hash value.  Let T be the DER encoding of the
DigestInfo value (see the notes below) and let tLen be the length
in octets of T.

3. If emLen < tLen + 11, output "intended encoded message length too
short" and stop.

4. Generate an octet string PS consisting of emLen - tLen - 3 octets
with hexadecimal value 0xff.  The length of PS will be at least 8
octets.

5. Concatenate PS, the DER encoding T, and other padding to form the
encoded message EM as

EM = 0x00 || 0x01 || PS || 0x00 || T.

6. Output EM.

   Notes.

1. For the six hash functions mentioned in Appendix B.1, the DER
encoding T of the DigestInfo value is equal to the following:

MD2:     (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 02 05 00 04
10 || H.
MD5:     (0x)30 20 30 0c 06 08 2a 86 48 86 f7 0d 02 05 05 00 04
10 || H.
SHA-1:   (0x)30 21 30 09 06 05 2b 0e 03 02 1a 05 00 04 14 || H.
SHA-256: (0x)30 31 30 0d 06 09 60 86 48 01 65 03 04 02 01 05 00
04 20 || H.
SHA-384: (0x)30 41 30 0d 06 09 60 86 48 01 65 03 04 02 02 05 00
04 30 || H.
SHA-512: (0x)30 51 30 0d 06 09 60 86 48 01 65 03 04 02 03 05 00
04 40 || H.

2. In version 1.5 of this document, T was defined as the BER
encoding, rather than the DER encoding, of the DigestInfo value.
In particular, it is possible - at least in theory - that the
verification operation defined in this document (as well as in
version 2.0) rejects a signature that is valid with respect to the
specification given in PKCS #1 v1.5.  This occurs if other rules
than DER are applied to DigestInfo (e.g., an indefinite length
encoding of the underlying SEQUENCE type).  While this is unlikely
to be a concern in practice, a cautious implementer may choose to
employ a verification operation based on a BER decoding operation
as specified in PKCS #1 v1.5.  In this manner, compatibility with
any valid implementation based on PKCS #1 v1.5 is obtained.  Such
a verification operation should indicate whether the underlying
BER encoding is a DER encoding and hence whether the signature is
valid with respect to the specification given in this document.