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Versions: 00 01 RFC 2437

Network Working Group                          B. Kaliski and J. Staddon
Internet-Draft                                          RSA Laboratories
Category: Informational                                   September 1998

         PKCS #1: RSA Cryptography Specifications
                         Version 2.0

Status of this Memo

     This document is an Internet-Draft.  Internet-Drafts are working
     documents of the Internet Engineering Task Force (IETF), its
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This memo provides information for the Internet community. It does not
Specify an Internet standard of any kind. Distribution of this memo
is unlimited.

Copyright Notice

Copyright (C) The Internet Society (1998). All Rights Reserved.


Table of Contents
1.       Introduction.....................................2
1.1      Overview.........................................2
2.       Notation.........................................3
3.       Key types........................................4
3.1      RSA public key...................................4
3.2      RSA private key..................................4
4.       Data conversion primitives.......................5
4.1      I2OSP............................................6
4.2      OS2IP............................................6
5.       Cryptographic primitives.........................7
5.1      Encryption and decryption primitives.............7
5.1.1    RSAEP............................................7
5.1.2    RSADP............................................8
5.2      Signature and verification primitives............8
5.2.1    RSASP1...........................................9
5.2.2    RSAVP1...........................................9
6.       Overview of schemes.............................10
7.       Encryption schemes..............................10
7.1      RSAES-OAEP......................................11
7.1.1    Encryption operation............................11
7.1.2    Decryption operation............................12
7.2      RSAES-PKCS1-v1_5................................13
7.2.1    Encryption operation............................14
7.2.2    Decryption operation............................15
8.       Signature schemes with appendix.................16
8.1      RSASSA-PKCS1-v1_5...............................17
8.1.1    Signature generation operation..................17
8.1.2    Signature verification operation................18
9.       Encoding methods................................19
9.1      Encoding methods for encryption.................19
9.1.1    EME-OAEP........................................19
9.1.2    EME-PKCS1-v1_5..................................21
9.2      Encoding methods for signatures with appendix...22
9.2.1    EMSA-PKCS1-v1_5.................................22
10.      Auxiliary Functions.............................23
10.1     Hash Functions..................................23
10.2     Mask Generation Functions.......................24

Internet-Draft   RSA Encryption and Signature Generation  September 1998

10.2.1   MGF1............................................24
11.      ASN.1 syntax....................................25
11.1     Key representation..............................25
11.1.1   Public-key syntax...............................25
11.1.2   Private-key syntax..............................26
11.2     Scheme identification...........................26
11.2.1   Syntax for RSAES-OAEP...........................26
11.2.2   Syntax for RSAES-PKCS1-v1_5.....................28
11.2.3   Syntax for RSASSA-PKCS1-v1_5....................28
12       Patent Statement................................29
12.1     Patent statement for the RSA algorithm..........30
13.      Revision history................................30
14.      References......................................30



1. Introduction
This Internet-Draft is proposed as a successor to RFC-2313. This
document provides recommendations for the implementation of public-
key cryptography based on the RSA algorithm [18], 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 [1] and IEEE P1363 working groups [14].
This document supersedes PKCS #1 version 1.5 [20].

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.

1.1 Overview

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 [2] encryption scheme along with the method found in PKCS #1 v1.5.

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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.

2. Notation

(n, e)         RSA public key

c              ciphertext representative, an integer between 0 and n-1

C              ciphertext, an octet string

d              private exponent

dP             p's exponent, a positive integer such that:
                e(dP)\equiv 1 (mod(p-1))

dQ             q's exponent, a positive integer such that:
                e(dQ)\equiv 1 (mod(q-1))

e              public exponent

EM             encoded message, an octet string

emLen          intended length in octets of an encoded message

H              hash value, an output of Hash

Hash           hash function

hLen           output length in octets of hash function Hash

K              RSA private key

k              length in octets of the modulus

l              intended length of octet string

lcm(.,.)       least common multiple of two
               nonnegative integers

m              message representative, an integer between
               0 and n-1

M              message, an octet string


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MGF            mask generation function

n              modulus

P              encoding parameters, an octet string

p,q            prime factors of the modulus

qInv           CRT coefficient, a positive integer less
               than p such: q(qInv)\equiv 1 (mod p)

s              signature representative, an integer
               between 0 and n-1

S              signature, an octet string

x              a nonnegative integer

X              an octet string corresponding to x

\xor           bitwise exclusive-or of two octet strings

\lambda(n)     lcm(p-1, q-1), where n = pq

||             concatenation operator

||.||          octet length operator

3. Key types

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.

3.1 RSA public key

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.

3.2 RSA private key

For the purposes of this document, an RSA private key may have either of
two representations.
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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:

ed \equiv 1 (mod \lambda(n))

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

e(dP)\equiv 1(mod(p-1))
e(dQ)\equiv 1(mod(q-1)),

and the CRT coefficient qInv is a positive integer less than p
satisfying:

q(qInv)\equiv 1 (mod p).

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.

4. Data conversion primitives

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


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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

4.1 I2OSP

I2OSP converts a nonnegative integer to an octet string of a specified
length.

I2OSP (x, l)

Input:
x         nonnegative integer to be converted
l         intended length of the resulting octet string

Output:
X         corresponding octet string of length l; or "integer too large"

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.

4.2 OS2IP

OS2IP converts an octet string to a nonnegative integer.

OS2IP (X)

Input:
X         octet string to be converted

Output:
x         corresponding 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.

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3.Output x.

5. Cryptographic primitives

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.

5.1 Encryption and decryption primitives

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.

5.1.1 RSAEP

RSAEP((n, e), m)

Input:
(n, e)    RSA public key
m         message representative, an integer between 0 and n-1

Output:
c         ciphertext representative, an integer between 0 and n-1;
          or "message representative out of range"

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.



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2. Let c = m^e mod n.
3. Output c.

5.1.2 RSADP

RSADP (K, c)

Input:

K         RSA private key, where K has one of the following forms
              -a pair (n, d)
              -a quintuple (p, q, dP, dQ, qInv)
c         ciphertext representative, an integer between 0 and n-1

Output:
m         message representative, an integer between 0 and n-1; or
          "ciphertext representative out of range"

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.

5.2 Signature and verification primitives

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.

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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.

5.2.1 RSASP1

RSASP1 (K, m)

Input:
K             RSA private key, where K has one of the following
              forms:
                 -a pair (n, d)
                 -a quintuple (p, q, dP, dQ, qInv)

m             message representative, an integer between 0 and n-1

Output:
s             signature representative, an integer between  0 and
              n-1, or "message representative out of range"

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.

5.2.2 RSAVP1

RSAVP1 ((n, e), s)

Input:


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(n, e)  RSA public key
s       signature representative, an integer between 0 and n-1

Output:
m       message representative, an integer between 0 and n-1;
        or "invalid"

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.

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 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.

7. Encryption schemes

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


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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 [21] 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.

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 (Section 9.1.1) EME-OAEP is
based on the method found in [2]. 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 [4] would be useful.


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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.

7.1.1 Encryption operation

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

Input:
(n, e)    recipient's RSA public key

M         message to be encrypted, an octet string of length at
          most k-2-2hLen, where k is the length in octets of the
          modulus n and hLen is the length in octets of the hash
          function output for EME-OAEP

P         encoding parameters, an octet string that may be empty

Output:
C         ciphertext, an octet string of length k; or "message too
          long"

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.

7.1.2 Decryption operation

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


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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 modulus n
P          encoding parameters, an octet string that may be empty

Output:
M          message, an octet string of length at most k-2-2hLen,
           where hLen is the length in octets of the hash
           function output for EME-OAEP; or "decryption error"


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 [4].

7.2 RSAES-PKCS1-v1_5


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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 [7]).

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
[4]. Therefore, if RSAES-PKCS1-v1_5 is to be used, certain easily
implemented countermeasures should be taken to thwart the attack found
in [4]. 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 [5].

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
[13] 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. [7] 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.

7.2.1 Encryption operation

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


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

Output:
C       ciphertext, an octet string of length k; or "message too
        long"


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.


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 modulus n

Output:
M       message, an octet string of length at most k-11; or
        "decryption error"

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).
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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 [4].

8. Signature schemes with appendix

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 [6] 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 [21],
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

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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 [3],
which would be recommended for new applications.

8.1 RSASSA-PKCS1-v1_5

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.

8.1.1 Signature generation operation

RSASSA-PKCS1-V1_5-SIGN (K, M)
Input:
K         signer's RSA private ke
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 modulus n; "message too long" or
          "modulus too short"
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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.

8.1.2 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 modulus n

Output: "valid signature," "invalid signature," or "message too long",
or "modulus too short"

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.


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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."


9. Encoding methods

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.

9.1 Encoding methods for encryption

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.

9.1.1 EME-OAEP

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 [2].


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9.1.1.1 Encoding operation

EME-OAEP-ENCODE (M, P, emLen)

Options:
Hash      hash function (hLen denotes the length in octet of the
          hash function output)
MGF       mask generation function

Input:
M         message to be encoded, an octet string of length at most
          emLen- 1-2hLen
P         encoding parameters, an octet string
emLen     intended length in octets of the encoded message, at least
          2hLen+1

Output:
EM        encoded message, an octet string of length emLen;
          "message too long" or "parameter string too long"

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.




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9.1.1.2 Decoding operation
EME-OAEP-DECODE (EM, P)

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

MGF       mask generation function

Input:

EM        encoded message, an octet string of length at least 2hLen+1
P         encoding parameters, an octet string

Output:
M         recovered message, an octet string of length at most ||EM||-1-
          2hLen; or "decoding error"

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.


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9.1.2 EME-PKCS1-v1_5

This encoding method is the same as in PKCS #1 v1.5, Section 8:
Encryption Process.

9.1.2.1 Encoding operation

EME-PKCS1-V1_5-ENCODE (M, emLen)

Input:
M         message to be encoded, an octet string of length at most emLen-10
emLen     intended length in octets of the encoded message

Output:
EM        encoded message, an octet string of length emLen; or "message too
          long"

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.

9.1.2.2 Decoding operation

EME-PKCS1-V1_5-DECODE (EM)

Input:
EM      encoded message, an octet string of length at least 10

Output:
M       recovered message, an octet string of length at most ||EM||-10;
        or "decoding error"

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.


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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.

9.2 Encoding methods for signatures with appendix

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.

9.2.1 EMSA-PKCS1-v1_5

This encoding method only has an encoding operation.

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

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

Input:
M         message to be encoded
emLen     intended length in octets of the encoded message, at least
          ||T|| + 10, where T is the DER encoding of a certain value computed
          during the encoding operation

Output:
EM        encoded message, an octet string of length emLen; or "message
          too long" or "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," 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


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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.

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.


10. Auxiliary Functions

This section specifies the hash functions and the mask generation
functions that are mentioned in the encoding methods (Section 9).

10.1 Hash Functions

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 [15], MD5 [17], and SHA-1 [16]. 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 ([15], [17] and [16]).

Note. Version 1.5 of this document also allowed for the use of MD4 in

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signature schemes. The cryptanalysis of MD4 has progressed significantly
in the intervening years. For example, Dobbertin [10] demonstrated how
to find collisions for MD4 and that the first two rounds of MD4 are not
one-way [11]. Because of these results and others (e.g. [9]), 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 [19] 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. [8]). For new applications, to address these
concerns, SHA-1 is preferred.

10.2 Mask Generation Functions

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.

10.2.1 MGF1

MGF1 is a Mask Generation Function based on a hash function.

MGF1 (Z, l)

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

Input:
Z       seed from which mask is generated, an octet string
l       intended length in octets of the mask, at most 2^32(hLen)

Output:
mask    mask, an octet string of length l; or "mask too long"

Steps:

1.If l > 2^32(hLen), output "mask too long" and stop.


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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.

11. ASN.1 syntax

11.1 Key representation

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
[22], and PKCS #12 [23].

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.

rsaEncryption OBJECT IDENTIFIER ::= {pkcs-1 1}

All of the definitions in this section are the same as in PKCS #1 v1.5.


11.1.1 Public-key syntax

An RSA public key should be represented with the ASN.1 type
RSAPublicKey:

RSAPublicKey::=SEQUENCE{
  modulus INTEGER, -- n
  publicExponent INTEGER -- e }

(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.

11.1.2 Private-key syntax

An RSA private key should be represented with ASN.1 type RSAPrivateKey:




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RSAPrivateKey ::= SEQUENCE {
  version Version,
  modulus INTEGER, -- n
  publicExponent INTEGER, -- e
  privateExponent INTEGER, -- d
  prime1 INTEGER, -- p
  prime2 INTEGER, -- q
  exponent1 INTEGER, -- d mod (p-1)
  exponent2 INTEGER, -- d mod (q-1)
  coefficient INTEGER -- (inverse of q) mod p }

Version ::= INTEGER

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.

11.2 Scheme identification

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.

11.2.1 Syntax for RSAES-OAEP

The object identifier id-RSAES-OAEP identifies the RSAES-OAEP encryption
scheme.

id-RSAES-OAEP OBJECT IDENTIFIER ::= {pkcs-1 7}

The parameters field associated with this OID in an AlgorithmIdentifier
shall have type RSAEP-OAEP-params:

RSAES-OAEP-params ::=  SEQUENCE {
  hashFunc [0] AlgorithmIdentifier {{oaepDigestAlgorithms}}
    DEFAULT sha1Identifier,
  maskGenFunc [1] AlgorithmIdentifier {{pkcs1MGFAlgorithms}}
    DEFAULT mgf1SHA1Identifier,
  pSourceFunc [2] AlgorithmIdentifier
    {{pkcs1pSourceAlgorithms}}
    DEFAULT pSpecifiedEmptyIdentifier }



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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.

oaepDigestAlgorithms ALGORITHM-IDENTIFIER ::= {
  {NULL IDENTIFIED BY id-sha1} }

id-sha1 OBJECT IDENTIFIER ::=
  {iso(1) identified-organization(3) oiw(14) secsig(3)
    algorithms(2) 26}


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.

pkcs1MGFAlgorithms ALGORITHM-IDENTIFIER ::= {
  {AlgorithmIdentifier {{oaepDigestAlgorithms}} IDENTIFIED
    BY id-mgf1} }

id-mgf1 OBJECT IDENTIFIER ::= {pkcs-1 8}

The default mask generation function is MGF1 with SHA-1:

mgf1SHA1Identifier ::= AlgorithmIdentifier {
  id-mgf1, sha1Identifier }

-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.

pkcs1pSourceAlgorithms ALGORITHM-IDENTIFIER ::= {
  {OCTET STRING IDENTIFIED BY id-pSpecified} }

id-pSpecified OBJECT IDENTIFIER ::= {pkcs-1 9}

The default encoding parameters is an empty string (so that pHash in
EME-OAEP will contain the hash of the empty string):


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pSpecifiedEmptyIdentifier ::= AlgorithmIdentifier {
  id-pSpecified, OCTET STRING SIZE (0) }


If all of the default values of the fields in RSAES-OAEP-params are
used, then the algorithm identifier will have the following value:

RSAES-OAEP-Default-Identifier ::= AlgorithmIdentifier {
  id-RSAES-OAEP,
  {sha1Identifier,
   mgf1SHA1Identifier,
   pSpecifiedEmptyIdentifier } }

11.2.2 Syntax for RSAES-PKCS1-v1_5

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.

RsaEncryption   OBJECT IDENTIFIER ::= {PKCS-1 1}

11.2.3 Syntax for RSASSA-PKCS1-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:

md2WithRSAEncryption ::= {PKCS-1 2}

If the hash function to be used is MD5, then the OID should be:

md5WithRSAEncryption ::= {PKCS-1 4}

If the hash function to be used is SHA-1, then the OID should be:

sha1WithRSAEncryption ::= {pkcs-1 5}

In the digestInfo type mentioned in Section 9.2.1 the OIDS for the
digest algorithm are the following:

id-SHA1 OBJECT IDENTIFIER ::=
        {iso(1) identified-organization(3) oiw(14) secsig(3) algorithms(2) 26 }

md2 OBJECT IDENTIFIER ::=
        {iso(1) member-body(2) US(840) rsadsi(113549) digestAlgorithm(2) 2}



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md5 OBJECT IDENTIFIER ::=
        {iso(1) member-body(2) US(840) rsadsi(113549) digestAlgorithm(2) 5}


The parameters field of the digest algorithm has ASN.1 type NULL for
these OIDs.

12. Patent statement

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 Stan-
dard.

The Internet Society, Internet Architecture Board, Internet Engineer-
ing Steering Group and the Corporation for National Research Initia-
tives 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 men-
tioned 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.

12.1 Patent statement for the RSA algorithm

The Massachusetts Institute of Technology has granted RSA Data Secu-
rity, 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 roy-
alty 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:



         Director of Licensing
         RSA Data Security, Inc.
         2955 Campus Drive
         Suite 400
         San Mateo, CA 94403
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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.

13. Revision history

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.

14. References

[1] ANSI, ANSI X9.44: Key Management Using Reversible Public Key
Cryptography for the Financial Services Industry. Working Draft.

[2] M. Bellare and P. Rogaway. Optimal Asymmetric Encryption-
How to Encrypt with RSA. In Advances in Cryptology-Eurocrypt '94,
pp. 92-111, Springer-Verlag, 1994.

[3] M. 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-Verlag, 1996.
Kaliski and Staddon          Informational                Page 31
Internet-Draft  RSA Encryption and Signature Generation  September 1998

[4] D. Bleichenbacher. Chosen Ciphertext Attacks against Protocols Based on
the RSA Encryption Standard PKCS #1. To appear in Advances in
Cryptology-Crypto '98.

[5] D. Bleichenbacher, B. Kaliski and J. Staddon. Recent Results on PKCS #1:
RSA Encryption Standard. RSA Laboratories' Bulletin, Number 7, June 24,
1998.

[6] CCITT. Recommendation X.509: The Directory-Authentication Framework.
1988.

[7] D. Coppersmith, M. Franklin, J. Patarin and M. Reiter. Low-Exponent
RSA with Related Messages. In Advances in Cryptology-Eurocrypt '96,
pp. 1-9, Springer-Verlag, 1996

[8] B. Den Boer and Bosselaers. Collisions for the Compression
Function of MD5. In Advances in Cryptology-Eurocrypt '93, pp 293-304,
Springer-Verlag, 1994.

[9] B. den Boer, and A. Bosselaers. An Attack on the Last Two Rounds of MD4.
In Advances in Cryptology-Crypto '91, pp.194-203, Springer-Verlag, 1992.

[10] H. Dobbertin. Cryptanalysis of MD4. Fast Software Encryption. Lecture
Notes in Computer Science, Springer-Verlag 1996, pp. 55-72.

[11] H. Dobbertin. Cryptanalysis of MD5 Compress. Presented at the rump
session of Eurocrypt `96, May 14, 1996

[12] H. Dobbertin.The First Two Rounds of MD4 are Not One-Way. Fast Software
Encryption. Lecture Notes in Computer Science, Springer-Verlag 1998, pp.
284-292.

[13] J. Hastad. Solving Simultaneous Modular Equations of Low Degree.
SIAM Journal of Computing, 17, 1988, pp. 336-341.

[14] IEEE. IEEE P1363: Standard Specifications for Public Key Cryptography.
Draft Version 4.

[15] B. Kaliski. RFC 1319: The MD2 Message-Digest Algorithm. Internet
Activities Board, April 1992.

[16] National Institute of Standards and Technology (NIST). FIPS Publication
180-1: Secure Hash Standard. April 1994.

[17] R. Rivest. RFC 1321: The MD5 Message-Digest Algorithm. Internet
Activities Board, April 1992.

[18] R. Rivest, A. Shamir and L. Adleman. A Method for Obtaining Digital
Signatures and Public-Key Cryptosystems. Communications of the ACM,
21(2), pp. 120-126, February 1978.

[19] N. Rogier and P. Chauvaud. The Compression Function of MD2 is not
Collision Free. Presented at Selected Areas of Cryptography `95.
Carleton University, Ottawa, Canada. May 18-19, 1995.
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[20] RSA Laboratories. PKCS #1: RSA Encryption Standard. Version 1.5,
November 1993.

[21] RSA Laboratories. PKCS #7: Cryptographic Message Syntax Standard.
Version 1.5, November 1993.

[22] RSA  Laboratories. PKCS #8: Private-Key Information Syntax Standard.
Version 1.2, November 1993.

[23] RSA Laboratories. PKCS #12: Personal Information Exchange Syntax
Standard. Version 1.0, DRAFT, April 1997.


Acknowledgements

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".

Authors' Addresses

Burt Kaliski
RSA Laboratories East
20 Crosby Drive
Bedford, MA  01730
Phone: (617) 687-7000
EMail: burt@rsa.com

Jessica Staddon
RSA Laboratories West
2955 Campus Drive
Suite 400
San Mateo, CA 94403
Phone: (650) 295-7600
Email: jstaddon@rsa.com


Full Copyright Statement

   Copyright (C) The Internet Society (1998).  All Rights Reserved.

   This document and translations of it may be copied and furnished to
   others, and derivative works that comment on or otherwise explain it
   or assist in its implementation may be prepared, copied, published
   and distributed, in whole or in part, without restriction of any
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   the copyright notice or references to the Internet Society or other
   Internet organizations, except as needed for the purpose of

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   developing Internet standards in which case the procedures for
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   The limited permissions granted above are perpetual and will not be
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Kaliski and Staddon          Informational                Page 34


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