Network Working Group                                         J. Jonsson
Request for Comments: 3447                                    B. Kaliski
Obsoletes: 2437                                         RSA Laboratories
Category: Informational                                    February 2003


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

Status of this Memo

   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 (2003).  All Rights Reserved.

Abstract

   This memo represents a republication of PKCS #1 v2.1 from RSA
   Laboratories' Public-Key Cryptography Standards (PKCS) series, and
   change control is retained within the PKCS process.  The body of this
   document is taken directly from the PKCS #1 v2.1 document, with
   certain corrections made during the publication process.

Table of Contents

   1.       Introduction...............................................2
   2.       Notation...................................................3
   3.       Key types..................................................6
      3.1      RSA public key..........................................6
      3.2      RSA private key.........................................7
   4.       Data conversion primitives.................................8
      4.1      I2OSP...................................................9
      4.2      OS2IP...................................................9
   5.       Cryptographic primitives..................................10
      5.1      Encryption and decryption primitives...................10
      5.2      Signature and verification primitives..................12
   6.       Overview of schemes.......................................14
   7.       Encryption schemes........................................15
      7.1      RSAES-OAEP.............................................16
      7.2      RSAES-PKCS1-v1_5.......................................23
   8.       Signature schemes with appendix...........................27
      8.1      RSASSA-PSS.............................................29
      8.2      RSASSA-PKCS1-v1_5......................................32
   9.       Encoding methods for signatures with appendix.............35

      9.1      EMSA-PSS...............................................36
      9.2      EMSA-PKCS1-v1_5........................................41
   Appendix A. ASN.1 syntax...........................................44
      A.1      RSA key representation.................................44
      A.2      Scheme identification..................................46
   Appendix B. Supporting techniques..................................52
      B.1      Hash functions.........................................52
      B.2      Mask generation functions..............................54
   Appendix C. ASN.1 module...........................................56
   Appendix D. Intellectual Property Considerations...................63
   Appendix E. Revision history.......................................64
   Appendix F. References.............................................65
   Appendix G. About PKCS.............................................70
   Appendix H. Corrections Made During RFC Publication Process........70
   Security Considerations............................................70
   Acknowledgements...................................................71
   Authors' Addresses.................................................71
   Full Copyright Statement...........................................72

1. Introduction

   This document provides recommendations for the implementation of
   public-key cryptography based on the RSA algorithm [42], 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 the standard
   IEEE-1363-2000 [26] and draft standards currently being developed by
   the ANSI X9F1 [1] and IEEE P1363 [27] working groups.

   This document supersedes PKCS #1 version 2.0 [35][44] but includes
   compatible techniques.

   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.

    * Sections 6, 7, and 8 deal with the encryption and signature
      schemes in this document.  Section 6 gives an overview.  Along
      with the methods found in PKCS #1 v1.5, Section 7 defines an
      OAEP-based [3] encryption scheme and Section 8 defines a PSS-based
      [4][5] signature scheme with appendix.

    * Section 9 defines the encoding methods for the signature schemes
      in Section 8.

    * Appendix A defines the ASN.1 syntax for the keys defined in
      Section 3 and the schemes in Sections 7 and 8.

    * Appendix B defines the hash functions and the mask generation
      function used in this document, including ASN.1 syntax for the
      techniques.

    * Appendix C gives an ASN.1 module.

    * Appendices D, E, F and G cover intellectual property issues,
      outline the revision history of PKCS #1, give references to other
      publications and standards, and provide general information about
      the Public-Key Cryptography Standards.

2. Notation

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

   C              ciphertext, an octet string

   d              RSA private exponent

   d_i            additional factor r_i's CRT exponent, a positive
                  integer such that

                    e * d_i == 1 (mod (r_i-1)), i = 3, ..., u

   dP             p's CRT exponent, a positive integer such that

                    e * dP == 1 (mod (p-1))

   dQ             q's CRT exponent, a positive integer such that

                    e * dQ == 1 (mod (q-1))

   e              RSA public exponent

   EM             encoded message, an octet string

   emBits         (intended) length in bits of an encoded message EM

   emLen          (intended) length in octets of an encoded message EM

   GCD(. , .)     greatest common divisor of two nonnegative integers

   Hash           hash function

   hLen           output length in octets of hash function Hash

   k              length in octets of the RSA modulus n

   K              RSA private key

   L              optional RSAES-OAEP label, an octet string

   LCM(., ..., .) least common multiple of a list of nonnegative
                  integers

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

   M              message, an octet string

   mask           MGF output, an octet string

   maskLen        (intended) length of the octet string mask

   MGF            mask generation function

   mgfSeed        seed from which mask is generated, an octet string

   mLen           length in octets of a message M

   n              RSA modulus, n = r_1 * r_2 * ... * r_u , u >= 2

   (n, e)         RSA public key

   p, q           first two prime factors of the RSA modulus n

   qInv           CRT coefficient, a positive integer less than p such
                  that

                    q * qInv == 1 (mod p)

   r_i            prime factors of the RSA modulus n, including r_1 = p,
                  r_2 = q, and additional factors if any

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

   S              signature, an octet string

   sLen           length in octets of the EMSA-PSS salt

   t_i            additional prime factor r_i's CRT coefficient, a
                  positive integer less than r_i such that

                    r_1 * r_2 * ... * r_(i-1) * t_i == 1 (mod r_i) ,

                  i = 3, ... , u

   u              number of prime factors of the RSA modulus, u >= 2

   x              a nonnegative integer

   X              an octet string corresponding to x

   xLen           (intended) length of the octet string X

   0x             indicator of hexadecimal representation of an octet or
                  an octet string; "0x48" denotes the octet with
                  hexadecimal value 48; "(0x)48 09 0e" denotes the
                  string of three consecutive octets with hexadecimal
                  value 48, 09, and 0e, respectively

   \lambda(n)     LCM(r_1-1, r_2-1, ... , r_u-1)

   \xor           bit-wise exclusive-or of two octet strings

   \ceil(.)       ceiling function; \ceil(x) is the smallest integer
                  larger than or equal to the real number x

   ||             concatenation operator

   ==             congruence symbol; a == b (mod n) means that the
                  integer n divides the integer a - b

   Note.  The CRT can be applied in a non-recursive as well as a
   recursive way.  In this document a recursive approach following
   Garner's algorithm [22] is used.  See also Note 1 in Section 3.2.

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.

   This specification supports so-called "multi-prime" RSA where the
   modulus may have more than two prime factors.  The benefit of multi-
   prime RSA is lower computational cost for the decryption and
   signature primitives, provided that the CRT (Chinese Remainder
   Theorem) is used.  Better performance can be achieved on single
   processor platforms, but to a greater extent on multiprocessor
   platforms, where the modular exponentiations involved can be done in
   parallel.

   For a discussion on how multi-prime affects the security of the RSA
   cryptosystem, the reader is referred to [49].

3.1 RSA public key

   For the purposes of this document, an RSA public key consists of two
   components:

      n        the RSA modulus, a positive integer
      e        the RSA public exponent, a positive integer

   In a valid RSA public key, the RSA modulus n is a product of u
   distinct odd primes r_i, i = 1, 2, ..., u, where u >= 2, and the RSA
   public exponent e is an integer between 3 and n - 1 satisfying GCD(e,
   \lambda(n)) = 1, where \lambda(n) = LCM(r_1 - 1, ..., r_u - 1).  By
   convention, the first two primes r_1 and r_2 may also be denoted p
   and q respectively.

   A recommended syntax for interchanging RSA public keys between
   implementations is given in Appendix A.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.

   1. The first representation consists of the pair (n, d), where the
      components have the following meanings:

         n        the RSA modulus, a positive integer
         d        the RSA private exponent, a positive integer

   2. The second representation consists of a quintuple (p, q, dP, dQ,
      qInv) and a (possibly empty) sequence of triplets (r_i, d_i, t_i),
      i = 3, ..., u, one for each prime not in the quintuple, where the
      components have the following meanings:

         p        the first factor, a positive integer
         q        the second factor, a positive integer
         dP       the first factor's CRT exponent, a positive integer
         dQ       the second factor's CRT exponent, a positive integer
         qInv     the (first) CRT coefficient, a positive integer
         r_i      the i-th factor, a positive integer
         d_i      the i-th factor's CRT exponent, a positive integer
         t_i      the i-th factor's CRT coefficient, a positive integer

   In a valid RSA private key with the first representation, the RSA
   modulus n is the same as in the corresponding RSA public key and is
   the product of u distinct odd primes r_i, i = 1, 2, ..., u, where u
   >= 2.  The RSA private exponent d is a positive integer less than n
   satisfying

      e * d == 1 (mod \lambda(n)),

   where e is the corresponding RSA public exponent and \lambda(n) is
   defined as in Section 3.1.

   In a valid RSA private key with the second representation, the two
   factors p and q are the first two prime factors of the RSA modulus n
   (i.e., r_1 and r_2), the CRT exponents dP and dQ are positive
   integers less than p and q respectively satisfying

      e * dP == 1 (mod (p-1))
      e * dQ == 1 (mod (q-1)) ,

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

      q * qInv == 1 (mod p).

   If u > 2, the representation will include one or more triplets (r_i,
   d_i, t_i), i = 3, ..., u.  The factors r_i are the additional prime
   factors of the RSA modulus n.  Each CRT exponent d_i (i = 3, ..., u)
   satisfies

      e * d_i == 1 (mod (r_i - 1)).

   Each CRT coefficient t_i (i = 3, ..., u) is a positive integer less
   than r_i satisfying

      R_i * t_i == 1 (mod r_i) ,

   where R_i = r_1 * r_2 * ... * r_(i-1).

   A recommended syntax for interchanging RSA private keys between
   implementations, which includes components from both representations,
   is given in Appendix A.1.2; an implementation's internal
   representation may differ.

   Notes.

   1. The definition of the CRT coefficients here and the formulas that
      use them in the primitives in Section 5 generally follow Garner's
      algorithm [22] (see also Algorithm 14.71 in [37]). However, for
      compatibility with the representations of RSA private keys in PKCS
      #1 v2.0 and previous versions, the roles of p and q are reversed
      compared to the rest of the primes.  Thus, the first CRT
      coefficient, qInv, is defined as the inverse of q mod p, rather
      than as the inverse of R_1 mod r_2, i.e., of p mod q.

   2. Quisquater and Couvreur [40] observed the benefit of applying the
      Chinese Remainder Theorem to RSA operations.

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 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, xLen)

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

   Output:
   X        corresponding octet string of length xLen

   Error: "integer too large"

   Steps:

   1. If x >= 256^xLen, output "integer too large" and stop.

   2. Write the integer x in its unique xLen-digit representation in
      base 256:

         x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) + ...
         + x_1 256 + x_0,

      where 0 <= x_i < 256 (note that one or more leading digits will be
      zero if x is less than 256^(xLen-1)).

   3. Let the octet X_i have the integer value x_(xLen-i) for 1 <= i <=
      xLen.  Output the octet string

         X = X_1 X_2 ... X_xLen.

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_xLen be the octets of X from first to last,
      and let x_(xLen-i) be the integer value of the octet X_i for
      1 <= i <= xLen.

   2. Let x = x_(xLen-1) 256^(xLen-1) + x_(xLen-2) 256^(xLen-2) + ...
      + x_1 256 + x_0.

   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 RSA 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 IFEP-RSA/IFDP-RSA in IEEE
   Std 1363-2000 [26] (except that support for multi-prime RSA has been
   added) 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

   Error: "message representative out of range"

   Assumption: RSA 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.

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) and a possibly empty
              sequence of triplets (r_i, d_i, t_i), i = 3, ..., u
   c        ciphertext representative, an integer between 0 and n - 1

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

   Error: "ciphertext representative out of range"

   Assumption: RSA 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. The message representative m is computed as follows.

      a. If the first form (n, d) of K is used, let m = c^d mod n.

      b. If the second form (p, q, dP, dQ, qInv) and (r_i, d_i, t_i)
         of K is used, proceed as follows:

         i.    Let m_1 = c^dP mod p and m_2 = c^dQ mod q.

         ii.   If u > 2, let m_i = c^(d_i) mod r_i, i = 3, ..., u.

         iii.  Let h = (m_1 - m_2) * qInv mod p.

         iv.   Let m = m_2 + q * h.

         v.    If u > 2, let R = r_1 and for i = 3 to u do

                  1. Let R = R * r_(i-1).

                  2. Let h = (m_i - m) * t_i mod r_i.

                  3. Let m = m + R * h.

   3.   Output m.

   Note.  Step 2.b can be rewritten as a single loop, provided that one
   reverses the order of p and q.  For consistency with PKCS #1 v2.0,
   however, the first two primes p and q are treated separately from
   the additional primes.

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 IFSP-RSA1/IFVP-RSA1 in
   IEEE 1363-2000 [26] (except that support for multi-prime RSA has
   been added) 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.

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) and a (possibly empty)
              sequence of triplets (r_i, d_i, t_i), i = 3, ..., u
   m        message representative, an integer between 0 and n - 1

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

   Error: "message representative out of range"

   Assumption: RSA 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. The signature representative s is computed as follows.

      a. If the first form (n, d) of K is used, let s = m^d mod n.

         b. If the second form (p, q, dP, dQ, qInv) and (r_i, d_i, t_i)
         of K is used, proceed as follows:

         i.    Let s_1 = m^dP mod p and s_2 = m^dQ mod q.

         ii.   If u > 2, let s_i = m^(d_i) mod r_i, i = 3, ..., u.

         iii.  Let h = (s_1 - s_2) * qInv mod p.

         iv.   Let s = s_2 + q * h.

         v.    If u > 2, let R = r_1 and for i = 3 to u do

                  1. Let R = R * r_(i-1).

                  2. Let h = (s_i - s) * t_i mod r_i.

                  3. Let s = s + R * h.

   3. Output s.

   Note.  Step 2.b can be rewritten as a single loop, provided that one
   reverses the order of p and q.  For consistency with PKCS #1 v2.0,
   however, the first two primes p and q are treated separately from the
   additional primes.

5.2.2 RSAVP1

   RSAVP1 ((n, e), s)

   Input:
   (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

   Error: "signature representative out of range"

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

   Steps:

   1. If the signature representative s is not between 0 and n - 1,
      output "signature representative out of range" and stop.

   2. Let m = s^e mod n.

   3. Output m.