PKCS #1: RSA Cryptography Specifications Version 2.0RSA Laboratories East20 Crosby DriveBedfordMA 01730(617) 687-7000burt@rsa.comRSA Laboratories West2955 Campus DriveSuite 400San MateoCA 94403(650) 295-7600jstaddon@rsa.com
Security
cryptographicpublic key cryptography standardspublic key encryption
This memo is the successor to RFC 2313. This document provides
recommendations for the implementation of public-key cryptography
based on the RSA algorithm , covering the following aspects:
-cryptographic primitives
-encryption schemes
-signature schemes with appendix
-ASN.1 syntax for representing keys and for identifying the
schemes
The recommendations are intended for general application within
computer and communications systems, and as such include a fair
amount of flexibility. It is expected that application standards
based on these specifications may include additional constraints. The
recommendations are intended to be compatible with draft standards
currently being developed by the ANSI X9F1 and IEEE P1363 working
groups . This document supersedes PKCS #1 version 1.5 .
Editor's note. It is expected that subsequent versions of PKCS #1 may
cover other aspects of the RSA algorithm such as key size, key
generation, key validation, and signature schemes with message
recovery.
The organization of this document is as follows:
-Section 1 is an introduction.
-Section 2 defines some notation used in this document.
-Section 3 defines the RSA public and private key types.
-Sections 4 and 5 define several primitives, or basic mathematical
operations. Data conversion primitives are in Section 4, and
cryptographic primitives (encryption-decryption,
signature-verification) are in Section 5.
-Section 6, 7 and 8 deal with the encryption and signature schemes
in this document. Section 6 gives an overview. Section 7 defines
an OAEP-based encryption scheme along with the method found
in PKCS #1 v1.5. Section 8 defines a signature scheme with
appendix; the method is identical to that of PKCS #1 v1.5.
-Section 9 defines the encoding methods for the encryption and
signature schemes in Sections 7 and 8.
-Section 10 defines the hash functions and the mask generation
function used in this document.
-Section 11 defines the ASN.1 syntax for the keys defined in
Section 3 and the schemes gives in Sections 7 and 8.
-Section 12 outlines the revision history of PKCS #1.
-Section 13 contains references to other publications and
standards.
Two key types are employed in the primitives and schemes defined in
this document: RSA public key and RSA private key. Together, an RSA
public key and an RSA private key form an RSA key pair.
For the purposes of this document, an RSA public key consists of two
components:
n, the modulus, a nonnegative integer
e, the public exponent, a nonnegative integer
In a valid RSA public key, the modulus n is a product of two odd
primes p and q, and the public exponent e is an integer between 3 and
n-1 satisfying gcd (e, \lambda(n)) = 1, where \lambda(n) = lcm (p-
1,q-1). A recommended syntax for interchanging RSA public keys
between implementations is given in Section 11.1.1; an
implementation's internal representation may differ.
For the purposes of this document, an RSA private key may have either
of two representations.
1. The first representation consists of the pair (n, d), where the
components have the following meanings:
n, the modulus, a nonnegative integer
d, the private exponent, a nonnegative integer
2. The second representation consists of a quintuple (p, q, dP, dQ,
qInv), where the components have the following meanings:
p, the first factor, a nonnegative integer
q, the second factor, a nonnegative integer
dP, the first factor's exponent, a nonnegative integer
dQ, the second factor's exponent, a nonnegative integer
qInv, the CRT coefficient, a nonnegative integer
In a valid RSA private key with the first representation, the modulus
n is the same as in the corresponding public key and is the product
of two odd primes p and q, and the private exponent d is a positive
integer less than n satisfying:
where e is the corresponding public exponent and \lambda(n) is as
defined above.
In a valid RSA private key with the second representation, the two
factors p and q are the prime factors of the modulus n, the exponents
dP and dQ are positive integers less than p and q respectively
satisfying
and the CRT coefficient qInv is a positive integer less than p
satisfying:
A recommended syntax for interchanging RSA private keys between
implementations, which includes components from both representations,
is given in Section 11.1.2; an implementation's internal
representation may differ.
Two data conversion primitives are employed in the schemes defined in
this document:
I2OSP: Integer-to-Octet-String primitive
OS2IP: Octet-String-to-Integer primitive
For the purposes of this document, and consistent with ASN.1 syntax, an
octet string is an ordered sequence of octets (eight-bit bytes). The
sequence is indexed from first (conventionally, leftmost) to last
(rightmost). For purposes of conversion to and from integers, the first
octet is considered the most significant in the following conversion
primitives
I2OSP converts a nonnegative integer to an octet string of a specified
length.
Steps:
1. If x>=256^l, output "integer too large" and stop.
2. Write the integer x in its unique l-digit representation base 256:
x = x_{l-1}256^{l-1} + x_{l-2}256^{l-2} +... + x_1 256 + x_0
where 0 <= x_i < 256 (note that one or more leading digits will be
zero if x < 256^{l-1}).
3. Let the octet X_i have the value x_{l-i} for 1 <= i <= l. Output
the octet string:
X = X_1 X_2 ... X_l.
OS2IP converts an octet string to a nonnegative integer.
Steps:
1. Let X_1 X_2 ... X_l be the octets of X from first to last, and
let x{l-i} have value X_i for 1<= i <= l.
2. Let x = x{l-1} 256^{l-1} + x_{l-2} 256^{l-2} +...+ x_1 256 + x_0.
3. Output x.
Cryptographic primitives are basic mathematical operations on which
cryptographic schemes can be built. They are intended for
implementation in hardware or as software modules, and are not
intended to provide security apart from a scheme.
Four types of primitive are specified in this document, organized in
pairs: encryption and decryption; and signature and verification.
The specifications of the primitives assume that certain conditions
are met by the inputs, in particular that public and private keys are
valid.
An encryption primitive produces a ciphertext representative from a
message representative under the control of a public key, and a
decryption primitive recovers the message representative from the
ciphertext representative under the control of the corresponding
private key.
One pair of encryption and decryption primitives is employed in the
encryption schemes defined in this document and is specified here:
RSAEP/RSADP. RSAEP and RSADP involve the same mathematical operation,
with different keys as input.
The primitives defined here are the same as in the draft IEEE P1363
and are compatible with PKCS #1 v1.5.
The main mathematical operation in each primitive is exponentiation.
Assumptions: public key (n, e) is valid
Steps:
1. If the message representative m is not between 0 and n-1, output
message representative out of range and stop.
2. Let c = m^e mod n.
3. Output c.
Assumptions: private key K is valid
Steps:
1. If the ciphertext representative c is not between 0 and n-1,
output "ciphertext representative out of range" and stop.
2. If the first form (n, d) of K is used:
2.1 Let m = c^d mod n. Else, if the second form (p, q, dP,
dQ, qInv) of K is used:
2.2 Let m_1 = c^dP mod p.
2.3 Let m_2 = c^dQ mod q.
2.4 Let h = qInv ( m_1 - m_2 ) mod p.
2.5 Let m = m_2 + hq.
3. Output m.
A signature primitive produces a signature representative from a
message representative under the control of a private key, and a
verification primitive recovers the message representative from the
signature representative under the control of the corresponding
public key. One pair of signature and verification primitives is
employed in the signature schemes defined in this document and is
specified here: RSASP1/RSAVP1.
The primitives defined here are the same as in the draft IEEE P1363
and are compatible with PKCS #1 v1.5.
The main mathematical operation in each primitive is exponentiation,
as in the encryption and decryption primitives of Section 5.1. RSASP1
and RSAVP1 are the same as RSADP and RSAEP except for the names of
their input and output arguments; they are distinguished as they are
intended for different purposes.
Assumptions:
private key K is valid
Steps:
1. If the message representative m is not between 0 and n-1, output
"message representative out of range" and stop.
2. If the first form (n, d) of K is used:
2.1 Let s = m^d mod n. Else, if the second form (p, q, dP,
dQ, qInv) of K is used:
2.2 Let s_1 = m^dP mod p.
2.3 Let s_2 = m^dQ mod q.
2.4 Let h = qInv ( s_1 - s_2 ) mod p.
2.5 Let s = s_2 + hq.
3. Output S.
Assumptions:
public key (n, e) is valid
Steps:
1. If the signature representative s is not between 0 and n-1, output
"invalid" and stop.
2. Let m = s^e mod n.
3. Output m.
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 a 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 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 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.
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 public key, and the
decryption operation recovers the message from the ciphertext with
the recipient's corresponding 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 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 P1363, by 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.
RSAES-OAEP combines the RSAEP and RSADP primitives (Sections 5.1.1
and 5.1.2) with the EME-OAEP encoding method (Section 9.1.1) EME-OAEP
is based on the method found in . It is compatible with the IFES
scheme defined in the draft P1363 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-2-
2hLen octets, where hLen is the length of the hash function output
for EME-OAEP and k is the length in octets of the recipient's RSA
modulus. Assuming that the hash function in EME-OAEP has appropriate
properties, and the key size is sufficiently large, RSAEP-OAEP
provides "plaintext-aware encryption," meaning that it is
computationally infeasible to obtain full or partial information
about a message from a ciphertext, and computationally infeasible to
generate a valid ciphertext without knowing the corresponding
message. Therefore, a chosen-ciphertext attack is ineffective
against a plaintext-aware encryption scheme such as RSAES-OAEP.
Both the encryption and the decryption operations of RSAES-OAEP take
the value of the parameter string P as input. In this version of PKCS
#1, P is an octet string that is specified explicitly. See Section
11.2.1 for the relevant ASN.1 syntax. We briefly note that to receive
the full security benefit of RSAES-OAEP, it should not be used in a
protocol involving RSAES-PKCS1-v1_5. It is possible that in a
protocol on which both encryption schemes are present, an adaptive
chosen ciphertext attack such as would be useful.
Both the encryption and the decryption operations of RSAES-OAEP take
the value of the parameter string P as input. In this version of PKCS
#1, P is an octet string that is specified explicitly. See Section
11.2.1 for the relevant ASN.1 syntax.
Assumptions: public key (n, e) is valid
Steps:
1. Apply the EME-OAEP encoding operation (Section 9.1.1.2) to the
message M and the encoding parameters P to produce an encoded message
EM of length k-1 octets:
EM = EME-OAEP-ENCODE (M, P, k-1)
If the encoding operation outputs "message too long," then output
"message too long" and stop.
2. Convert the encoded message EM to an integer message
representative m: m = OS2IP (EM)
3. Apply the RSAEP encryption primitive (Section 5.1.1) to the public
key (n, e) and the message representative m to produce an integer
ciphertext representative c:
c = RSAEP ((n, e), m)
4. Convert the ciphertext representative c to a ciphertext C of
length k octets: C = I2OSP (c, k)
5. Output the ciphertext C.
Steps:
1. If the length of the ciphertext C is not k octets, output
"decryption error" and stop.
2. Convert the ciphertext C to an integer ciphertext representative
c: c = OS2IP (C).
3. Apply the RSADP decryption primitive (Section 5.1.2) to the
private key K and the ciphertext representative c to produce an
integer message representative m:
m = RSADP (K, c)
If RSADP outputs "ciphertext out of range," then output "decryption
error" and stop.
4. Convert the message representative m to an encoded message EM of
length k-1 octets: EM = I2OSP (m, k-1)
If I2OSP outputs "integer too large," then output "decryption error"
and stop.
5. Apply the EME-OAEP decoding operation to the encoded message EM
and the encoding parameters P to recover a message M:
M = EME-OAEP-DECODE (EM, P)
If the decoding operation outputs "decoding error," then output
"decryption error" and stop.
6. Output the message M.
Note. It is important that the error messages output in steps 4 and 5
be the same, otherwise an adversary may be able to extract useful
information from the type of error message received. Error message
information is used to mount a chosen-ciphertext attack on PKCS #1
v1.5 encrypted messages in .
RSAES-PKCS1-v1_5 combines the RSAEP and RSADP primitives with the
EME-PKCS1-v1_5 encoding method. It is the same as the encryption
scheme in PKCS #1 v1.5. RSAES-PKCS1-v1_5 can operate on messages of
length up to k-11 octets, although care should be taken to avoid
certain attacks on low-exponent RSA due to Coppersmith, et al. when
long messages are encrypted (see the third bullet in the notes below
and ).
RSAES-PKCS1-v1_5 does not provide "plaintext aware" encryption. In
particular, it is possible to generate valid 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 . Therefore, if RSAES-PKCS1-v1_5 is to be used,
certain easily implemented countermeasures should be taken to thwart
the attack found in . 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 can all be
effective countermeasures and don't involve changes to a PKCS #1
v1.5-based protocol. These and other countermeasures are discussed in
.
Notes. 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
recommendations motivated by cryptanalytic advances made in the
intervening years.
-It is recommended that the pseudorandom octets in EME-PKCS1-v1_5 be
generated independently for each encryption process, especially if
the same data is input to more than one encryption process. Hastad's
results are one motivation for this recommendation.
-The padding string PS in EME-PKCS1-v1_5 is at least eight octets
long, which is a security condition for public-key operations that
prevents an attacker from recovering data by trying all possible
encryption blocks.
-The pseudorandom octets can also help thwart an attack due to
Coppersmith et al. 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 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
RSAEP-OAEP scheme overcomes the attack.
Steps:
1. Apply the EME-PKCS1-v1_5 encoding operation (Section 9.1.2.1) to
the message M to produce an encoded message EM of length k-1 octets:
EM = EME-PKCS1-V1_5-ENCODE (M, k-1)
If the encoding operation outputs "message too long," then output
"message too long" and stop.
2. Convert the encoded message EM to an integer message
representative m: m = OS2IP (EM)
3. Apply the RSAEP encryption primitive (Section 5.1.1) to the public
key (n, e) and the message representative m to produce an integer
ciphertext representative c: c = RSAEP ((n, e), m)
4. Convert the ciphertext representative c to a ciphertext C of
length k octets: C = I2OSP (c, k)
5. Output the ciphertext C.
Steps:
1. If the length of the ciphertext C is not k octets, output
"decryption error" and stop.
2. Convert the ciphertext C to an integer ciphertext representative
c: c = OS2IP (C).
3. Apply the RSADP decryption primitive to the 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 out of range," then output "decryption
error" and stop.
4. Convert the message representative m to an encoded message EM of
length k-1 octets: EM = I2OSP (m, k-1)
If I2OSP outputs "integer too large," then output "decryption error"
and stop.
5. Apply the EME-PKCS1-v1_5 decoding operation to the encoded message
EM to recover a message M: M = EME-PKCS1-V1_5-DECODE (EM).
If the decoding operation outputs "decoding error," then output
"decryption error" and stop.
6. Output the message M.
Note. It is important that only one type of error message is output
by EME-PKCS1-v1_5, as ensured by steps 4 and 5. If this is not done,
then an adversary may be able to use information extracted form the
type of error message received to mount a chosen-ciphertext attack
such as the one found in .
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 private key, and the signature verification operation
verifies the signature on the message with the signer's corresponding
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, X.509 employs such a scheme to
authenticate the content of a certificate; the signature scheme with
appendix defined here would be a suitable signature algorithm in that
context. A related signature scheme could be employed in PKCS #7
, 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.
One signature scheme with appendix is specified in this document:
RSASSA-PKCS1-v1_5.
The signature scheme with appendix given here follows a general model
similar to that employed in IEEE P1363, by 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 then
applied to the message representative to produce the signature. 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. The message encoding operation
is again applied to the message, and the result is compared to the
recovered octet string. If there is a match, the signature is
considered valid. (Note that this approach assumes that the signature
and verification primitives have the message-recovery form and the
encoding method is deterministic, as is the case for RSASP1/RSAVP1
and EMSA-PKCS1-v1_5. The signature generation and verification
operations have a different form in P1363 for other primitives and
encoding methods.)
Editor's note. RSA Laboratories is investigating the possibility of
including a scheme based on the PSS encoding methods specified in
, which would be recommended for new applications.
RSASSA-PKCS1-v1_5 combines the RSASP1 and RSAVP1 primitives with the
EME-PKCS1-v1_5 encoding method. It is compatible with the IFSSA
scheme defined in the draft P1363 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 P1363).
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 message encoding method.
Assuming that the hash function in EMSA-PKCS1-v1_5 has appropriate
properties and the key size is sufficiently large, RSASSA-PKCS1-v1_5
provides secure signatures, meaning that it is computationally
infeasible to generate a signature without knowing the private key,
and computationally infeasible to find a message with a given
signature or two messages with the same signature. Also, in the
encoding method EMSA-PKCS1-v1_5, a hash function identifier is
embedded in the encoding. Because of this feature, an adversary must
invert or 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.
Steps:
1. Apply the EMSA-PKCS1-v1_5 encoding operation (Section 9.2.1) to
the message M to produce an encoded message EM of length k-1 octets:
EM = EMSA-PKCS1-V1_5-ENCODE (M, k-1)
If the encoding operation outputs "message too long," then output
"message too long" and stop. If the encoding operation outputs
"intended encoded message length too short" then output "modulus too
short".
2. Convert the encoded message EM to an integer message
representative m: m = OS2IP (EM)
3. Apply the RSASP1 signature primitive (Section 5.2.1) to the
private key K and the message representative m to produce an integer
signature representative s: s = RSASP1 (K, m)
4. Convert the signature representative s to a signature S of length
k octets: S = I2OSP (s, k)
5. Output the signature S.
Steps:
1. If the length of the signature S is not k octets, output "invalid
signature" and stop.
2. Convert the signature S to an integer signature representative s:
s = OS2IP (S)
3. Apply the RSAVP1 verification primitive (Section 5.2.2) to the
public key (n, e) and the signature representative s to produce an
integer message representative m:
m = RSAVP1 ((n, e), s) If RSAVP1 outputs "invalid"
then output "invalid signature" and stop.
4. Convert the message representative m to an encoded message EM of
length k-1 octets: EM = I2OSP (m, k-1)
If I2OSP outputs "integer too large," then output "invalid signature"
and stop.
5. Apply the EMSA-PKCS1-v1_5 encoding operation (Section 9.2.1) to
the message M to produce a second encoded message EM' of length k-1
octets:
EM' = EMSA-PKCS1-V1_5-ENCODE (M, k-1)
If the encoding operation outputs "message too long," then output
"message too long" and stop. If the encoding operation outputs
"intended encoded message length too short" then output "modulus too
short".
6. Compare the encoded message EM and the second encoded message EM'.
If they are the same, output "valid signature"; otherwise, output
"invalid signature."
Encoding methods consist of operations that map between octet string
messages and integer message representatives.
Two types of encoding method are considered in this document:
encoding methods for encryption, encoding methods for signatures with
appendix.
An encoding method for encryption consists of an encoding operation
and a decoding operation. An encoding operation maps a message M to a
message representative EM of a specified length; the decoding
operation maps a message representative EM back to a message. The
encoding and decoding operations are inverses.
The message representative EM will typically have some structure that
can be verified by the decoding operation; the decoding operation
will output "decoding error" if the structure is not present. The
encoding operation may also introduce some randomness, so that
different applications of the encoding operation to the same message
will produce different representatives.
Two encoding methods for encryption are employed in the encryption
schemes and are specified here: EME-OAEP and EME-PKCS1-v1_5.
This encoding method is parameterized by the choice of hash function
and mask generation function. Suggested hash and mask generation
functions are given in Section 10. This encoding method is based on
the method found in .
Steps:
1. If the length of P is greater than the input limitation for the
hash function (2^61-1 octets for SHA-1) then output "parameter string
too long" and stop.
2. If ||M|| > emLen-2hLen-1 then output "message too long" and stop.
3. Generate an octet string PS consisting of emLen-||M||-2hLen-1 zero
octets. The length of PS may be 0.
4. Let pHash = Hash(P), an octet string of length hLen.
5. Concatenate pHash, PS, the message M, and other padding to form a
data block DB as: DB = pHash || PS || 01 || M
6. Generate a random octet string seed of length hLen.
7. Let dbMask = MGF(seed, emLen-hLen).
8. Let maskedDB = DB \xor dbMask.
9. Let seedMask = MGF(maskedDB, hLen).
10. Let maskedSeed = seed \xor seedMask.
11. Let EM = maskedSeed || maskedDB.
12. Output EM.
Steps:
1. If the length of P is greater than the input limitation for the
hash function (2^61-1 octets for SHA-1) then output "parameter string
too long" and stop.
2. If ||EM|| < 2hLen+1, then output "decoding error" and stop.
3. Let maskedSeed be the first hLen octets of EM and let maskedDB be
the remaining ||EM|| - hLen octets.
4. Let seedMask = MGF(maskedDB, hLen).
5. Let seed = maskedSeed \xor seedMask.
6. Let dbMask = MGF(seed, ||EM|| - hLen).
7. Let DB = maskedDB \xor dbMask.
8. Let pHash = Hash(P), an octet string of length hLen.
9. Separate DB into an octet string pHash' consisting of the first
hLen octets of DB, a (possibly empty) octet string PS consisting of
consecutive zero octets following pHash', and a message M as:
DB = pHash' || PS || 01 || M
If there is no 01 octet to separate PS from M, output "decoding
error" and stop.
10. If pHash' does not equal pHash, output "decoding error" and stop.
11. Output M.
This encoding method is the same as in PKCS #1 v1.5, Section 8:
Encryption Process.
Steps:
1. If the length of the message M is greater than emLen - 10 octets,
output "message too long" and stop.
2. Generate an octet string PS of length emLen-||M||-2 consisting of
pseudorandomly generated nonzero octets. The length of PS will be at
least 8 octets.
3. Concatenate PS, the message M, and other padding to form the
encoded message EM as:
EM = 02 || PS || 00 || M
4. Output EM.
Steps:
1. If the length of the encoded message EM is less than 10, output
"decoding error" and stop.
2. Separate the encoded message EM into an octet string PS consisting
of nonzero octets and a message M as: EM = 02 || PS || 00 || M.
If the first octet of EM is not 02, or if there is no 00 octet to
separate PS from M, output "decoding error" and stop.
3. If the length of PS is less than 8 octets, output "decoding error"
and stop.
4. Output M.
An encoding method for signatures with appendix, for the purposes of
this document, consists of an encoding operation. An encoding
operation maps a message M to a message representative EM of a
specified length. (In future versions of this document, encoding
methods may be added that also include a decoding operation.)
One encoding method for signatures with appendix is employed in the
encryption schemes and is specified here: EMSA-PKCS1-v1_5.
This encoding method only has an encoding operation.
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," then output "message
too long".
2. Encode the algorithm ID for the hash function and the hash value
into an ASN.1 value of type DigestInfo (see Section 11) with the
Distinguished Encoding Rules (DER), where the type DigestInfo has the
syntax
The first field identifies the hash function and the second contains
the hash value. Let T be the DER encoding.
3. If emLen is less than ||T|| + 10 then output "intended encoded
message length too short".
4. Generate an octet string PS consisting of emLen-||T||-2 octets
with value FF (hexadecimal). 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 = 01 || PS || 00 || T
6. Output EM.
This section specifies the hash functions and the mask generation
functions that are mentioned in the encoding methods (Section 9).
Hash functions are used in the operations contained in Sections 7, 8
and 9. Hash functions are deterministic, meaning that the output is
completely determined by the input. Hash functions take octet strings
of variable length, and generate fixed length octet strings. The hash
functions used in the operations contained in Sections 7, 8 and 9
should be collision resistant. This means that it is infeasible to
find two distinct inputs to the hash function that produce the same
output. A collision resistant hash function also has the desirable
property of being one-way; this means that given an output, it is
infeasible to find an input whose hash is the specified output. The
property of collision resistance is especially desirable for RSASSA-
PKCS1-v1_5, as it makes it infeasible to forge signatures. In
addition to the requirements, the hash function should yield a mask
generation function (Section 10.2) with pseudorandom output.
Three hash functions are recommended for the encoding methods in this
document: MD2 , MD5 , and SHA-1 . For the EME-OAEP
encoding method, only SHA-1 is recommended. For the EMSA-PKCS1-v1_5
encoding method, SHA-1 is recommended for new applications. MD2 and
MD5 are recommended only for compatibility with existing applications
based on PKCS #1 v1.5.
The hash functions themselves are not defined here; readers are
referred to the appropriate references (, and ).
Note. Version 1.5 of this document also allowed for the use of MD4 in
signature schemes. The cryptanalysis of MD4 has progressed
significantly in the intervening years. For example, Dobbertin
demonstrated how to find collisions for MD4 and that the first two
rounds of MD4 are not one-way . Because of these results and
others (e.g. ), MD4 is no longer recommended. There have also been
advances in the cryptanalysis of MD2 and MD5, although not enough to
warrant removal from existing applications. Rogier and Chauvaud
demonstrated how to find collisions in a modified version of MD2. No
one has demonstrated how to find collisions for the full MD5
algorithm, although partial results have been found (e.g. ). For
new applications, to address these concerns, SHA-1 is preferred.
A mask generation function takes an octet string of variable length
and a desired output length as input, and outputs an octet string of
the desired length. There may be restrictions on the length of the
input and output octet strings, but such bounds are generally very
large. Mask generation functions are deterministic; the octet string
output is completely determined by the input octet string. The output
of a mask generation function should be pseudorandom, that is, if the
seed to the function is unknown, it should be infeasible to
distinguish the output from a truly random string. The plaintext-
awareness of RSAES-OAEP relies on the random nature of the output of
the mask generation function, which in turn relies on the random
nature of the underlying hash.
One mask generation function is recommended for the encoding methods
in this document, and is defined here: MGF1, which is based on a hash
function. Future versions of this document may define other mask
generation functions.
MGF1 is a Mask Generation Function based on a hash function.
Steps:
1.If l > 2^32(hLen), output "mask too long" and stop.
2.Let T be the empty octet string.
3.For counter from 0 to \lceil{l / hLen}\rceil-1, do the following:
a.Convert counter to an octet string C of length 4 with the primitive
I2OSP: C = I2OSP (counter, 4)
b.Concatenate the hash of the seed Z and C to the octet string T: T =
T || Hash (Z || C)
4.Output the leading l octets of T as the octet string mask.
This section defines ASN.1 object identifiers for RSA public and
private keys, and defines the types RSAPublicKey and RSAPrivateKey.
The intended application of these definitions includes X.509
certificates, PKCS #8 , and PKCS #12 .
The object identifier rsaEncryption identifies RSA public and private
keys as defined in Sections 11.1.1 and 11.1.2. The parameters field
associated with this OID in an AlgorithmIdentifier shall have type
NULL.
All of the definitions in this section are the same as in PKCS #1
v1.5.
An RSA public key should be represented with the ASN.1 type
RSAPublicKey:
(This type is specified in X.509 and is retained here for
compatibility.)
The fields of type RSAPublicKey have the following meanings:
-modulus is the modulus n.
-publicExponent is the public exponent e.
An RSA private key should be represented with ASN.1 type
RSAPrivateKey:
The fields of type RSAPrivateKey have the following meanings:
-version is the version number, for compatibility with future
revisions of this document. It shall be 0 for this version of the
document.
-modulus is the modulus n.
-publicExponent is the public exponent e.
-privateExponent is the private exponent d.
-prime1 is the prime factor p of n.
-prime2 is the prime factor q of n.
-exponent1 is d mod (p-1).
-exponent2 is d mod (q-1).
-coefficient is the Chinese Remainder Theorem coefficient q-1 mod p.
This section defines object identifiers for the encryption and
signature schemes. The schemes compatible with PKCS #1 v1.5 have the
same definitions as in PKCS #1 v1.5. The intended application of
these definitions includes X.509 certificates and PKCS #7.
The object identifier id-RSAES-OAEP identifies the RSAES-OAEP
encryption scheme.
The parameters field associated with this OID in an
AlgorithmIdentifier shall have type RSAEP-OAEP-params:
The fields of type RSAES-OAEP-params have the following meanings:
-hashFunc identifies the hash function. It shall be an algorithm ID
with an OID in the set oaepDigestAlgorithms, which for this version
shall consist of id-sha1, identifying the SHA-1 hash function. The
parameters field for id-sha1 shall have type NULL.
The default hash function is SHA-1:
sha1Identifier ::= AlgorithmIdentifier {id-sha1, NULL}
-maskGenFunc identifies the mask generation function. It shall be an
algorithm ID with an OID in the set pkcs1MGFAlgorithms, which for
this version shall consist of id-mgf1, identifying the MGF1 mask
generation function (see Section 10.2.1). The parameters field for
id-mgf1 shall have type AlgorithmIdentifier, identifying the hash
function on which MGF1 is based, where the OID for the hash function
shall be in the set oaepDigestAlgorithms.
The default mask generation function is MGF1 with SHA-1:
-pSourceFunc identifies the source (and possibly the value) of the
encoding parameters P. It shall be an algorithm ID with an OID in the
set pkcs1pSourceAlgorithms, which for this version shall consist of
id-pSpecified, indicating that the encoding parameters are specified
explicitly. The parameters field for id-pSpecified shall have type
OCTET STRING, containing the encoding parameters.
The default encoding parameters is an empty string (so that pHash in
EME-OAEP will contain the hash of the empty string):
If all of the default values of the fields in RSAES-OAEP-params are
used, then the algorithm identifier will have the following value:
The object identifier rsaEncryption (Section 11.1) identifies the
RSAES-PKCS1-v1_5 encryption scheme. The parameters field associated
with this OID in an AlgorithmIdentifier shall have type NULL. This is
the same as in PKCS #1 v1.5.
The object identifier for RSASSA-PKCS1-v1_5 shall be one of the
following. The choice of OID depends on the choice of hash algorithm:
MD2, MD5 or SHA-1. Note that if either MD2 or MD5 is used then the
OID is just as in PKCS #1 v1.5. For each OID, the parameters field
associated with this OID in an AlgorithmIdentifier shall have type
NULL.
If the hash function to be used is MD2, then the OID should be:
If the hash function to be used is MD5, then the OID should be:
If the hash function to be used is SHA-1, then the OID should be:
In the digestInfo type mentioned in Section 9.2.1 the OIDS for the
digest algorithm are the following:
The parameters field of the digest algorithm has ASN.1 type NULL for
these OIDs.
The Internet Standards Process as defined in RFC 1310 requires a
written statement from the Patent holder that a license will be made
available to applicants under reasonable terms and conditions prior
to approving a specification as a Proposed, Draft or Internet
Standard.
The Internet Society, Internet Architecture Board, Internet
Engineering Steering Group and the Corporation for National Research
Initiatives take no position on the validity or scope of the
following patents and patent applications, nor on the appropriateness
of the terms of the assurance. The Internet Society and other groups
mentioned above have not made any determination as to any other
intellectual property rights which may apply to the practice of this
standard. Any further consideration of these matters is the user's
responsibility.
The Massachusetts Institute of Technology has granted RSA Data
Security, Inc., exclusive sub-licensing rights to the following
patent issued in the United States:
Cryptographic Communications System and Method ("RSA"), No. 4,405,829
RSA Data Security, Inc. has provided the following statement with
regard to this patent:
It is RSA's business practice to make licenses to its patents
available on reasonable and nondiscriminatory terms. Accordingly, RSA
is willing, upon request, to grant non-exclusive licenses to such
patent on reasonable and non-discriminatory terms and conditions to
those who respect RSA's intellectual property rights and subject to
RSA's then current royalty rate for the patent licensed. The royalty
rate for the RSA patent is presently set at 2% of the licensee's
selling price for each product covered by the patent. Any requests
for license information may be directed to:
A license under RSA's patent(s) does not include any rights to know-
how or other technical information or license under other
intellectual property rights. Such license does not extend to any
activities which constitute infringement or inducement thereto. A
licensee must make his own determination as to whether a license is
necessary under patents of others.
Versions 1.0-1.3
Versions 1.0-1.3 were distributed to participants in RSA Data
Security, Inc.'s Public-Key Cryptography Standards meetings in
February and March 1991.
Version 1.4
Version 1.4 was part of the June 3, 1991 initial public release of
PKCS. Version 1.4 was published as NIST/OSI Implementors' Workshop
document SEC-SIG-91-18.
Version 1.5
Version 1.5 incorporates several editorial changes, including updates
to the references and the addition of a revision history. The
following substantive changes were made: -Section 10: "MD4 with RSA"
signature and verification processes were added.
-Section 11: md4WithRSAEncryption object identifier was added.
Version 2.0 [DRAFT]
Version 2.0 incorporates major editorial changes in terms of the
document structure, and introduces the RSAEP-OAEP encryption scheme.
This version continues to support the encryption and signature
processes in version 1.5, although the hash algorithm MD4 is no
longer allowed due to cryptanalytic advances in the intervening
years.
This RFC contained boilerplate in this section which has been moved
to the RFC2223-compliant unnumbered section "References."
Security issues are discussed throughout this memo.
This document is based on a contribution of RSA Laboratories, a
division of RSA Data Security, Inc. Any substantial use of the text
from this document must acknowledge RSA Data Security, Inc. RSA Data
Security, Inc. requests that all material mentioning or referencing
this document identify this as "RSA Data Security, Inc. PKCS #1
v2.0".
ANSI, ANSI X9.44: Key Management Using Reversible Public Key Cryptography for the Financial Services Industry. Work in ProgressM. Bellare and P. Rogaway. Optimal Asymmetric Encryption - How to Encrypt with RSA. In Advances in Cryptology-Eurocrypt '94, pp. 92-111, Springer-VerlagM. Bellare and P. Rogaway. The Exact Security of Digital Signatures - How to Sign with RSA and Rabin. In Advances in Cryptology-Eurocrypt '96, pp. 399-416, Springer-VerlagD. Bleichenbacher. Chosen Ciphertext Attacks against Protocols Based on the RSA Encryption Standard PKCS #1. To appear in Advances in Cryptology-Crypto '98B. Kaliski and J. Staddon. Recent Results on PKCS #1: RSA Encryption Standard. RSA Laboratories' Bulletin, Number 7CCITT. Recommendation X.509: The Directory-Authentication FrameworkJ. Patarin and M. Reiter. Low-Exponent RSA with Related Messages. In Advances in Cryptology-Eurocrypt '96, pp. 1-9, Springer-Verlag, 1996B. Den Boer and Bosselaers. Collisions for the Compression Function of MD5. In Advances in Cryptology-Eurocrypt '93, pp 293-304, Springer-VerlagB. den Boer, and A. Bosselaers. An Attack on the Last Two Rounds of MD4. In Advances in Cryptology-Crypto '91, pp.194-203, Springer-VerlagH. Dobbertin. Cryptanalysis of MD4. Fast Software Encryption. Lecture Notes in Computer Science, Springer-Verlag pp. 55-72H. Dobbertin. Cryptanalysis of MD5 Compress. Presented at the rump session of Eurocrypt `96, 1996H. Dobbertin.The First Two Rounds of MD4 are Not One-Way. Fast Software Encryption. Lecture Notes in Computer Science, Springer-Verlag pp. 284-292J. Hastad. Solving Simultaneous Modular Equations of Low Degree. SIAM Journal of Computing, 17, pp. 336-341IEEE. IEEE P1363: Standard Specifications for Public Key Cryptography. Draft Version 4The MD2 Message-Digest AlgorithmRFC 1319National Institute of Standards and Technology (NIST). FIPS Publication 180-1: Secure Hash StandardThe MD5 Message-Digest AlgorithmRFC 1321A. Shamir and L. Adleman. A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM, 21(2), pp. 120-126N. Rogier and P. Chauvaud. The Compression Function of MD2 is not Collision Free. Presented at Selected Areas of Cryptography `95. Carleton University, Ottawa, Canada. -19, 1995RSA Laboratories. PKCS #1: RSA Encryption Standard. Version 1.5RSA Laboratories. PKCS #7: Cryptographic Message Syntax Standard. Version 1.5RSA Laboratories. PKCS #8: Private-Key Information Syntax Standard. Version 1.2RSA Laboratories. PKCS #12: Personal Information Exchange Syntax Standard. Version 1.0, Work in Progress