## Appendix A. ASN.1 syntax

## A.1 RSA 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 [46], and PKCS #12 [47]. The object identifier rsaEncryption identifies RSA public and private keys as defined in Appendices A.1.1 and A.1.2. The parameters field associated with this OID in a value of type AlgorithmIdentifier shall have a value of type NULL. rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 } The definitions in this section have been extended to support multi- prime RSA, but are backward compatible with previous versions.## A.1.1 RSA public key syntax

An RSA public key should be represented with the ASN.1 type RSAPublicKey: RSAPublicKey ::= SEQUENCE { modulus INTEGER, -- n publicExponent INTEGER -- e } The fields of type RSAPublicKey have the following meanings: * modulus is the RSA modulus n. * publicExponent is the RSA public exponent e.

## A.1.2 RSA private key syntax

An RSA private key should be represented with the ASN.1 type RSAPrivateKey: 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 otherPrimeInfos OtherPrimeInfos OPTIONAL } 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, unless multi-prime is used, in which case it shall be 1. Version ::= INTEGER { two-prime(0), multi(1) } (CONSTRAINED BY {-- version must be multi if otherPrimeInfos present --}) * modulus is the RSA modulus n. * publicExponent is the RSA public exponent e. * privateExponent is the RSA 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 CRT coefficient q^(-1) mod p. * otherPrimeInfos contains the information for the additional primes r_3, ..., r_u, in order. It shall be omitted if version is 0 and shall contain at least one instance of OtherPrimeInfo if version is 1.

OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo OtherPrimeInfo ::= SEQUENCE { prime INTEGER, -- ri exponent INTEGER, -- di coefficient INTEGER -- ti } The fields of type OtherPrimeInfo have the following meanings: * prime is a prime factor r_i of n, where i >= 3. * exponent is d_i = d mod (r_i - 1). * coefficient is the CRT coefficient t_i = (r_1 * r_2 * ... * r_(i- 1))^(-1) mod r_i. Note. It is important to protect the RSA private key against both disclosure and modification. Techniques for such protection are outside the scope of this document. Methods for storing and distributing private keys and other cryptographic data are described in PKCS #12 and #15.## A.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. Here are type identifier definitions for the PKCS #1 OIDs: PKCS1Algorithms ALGORITHM-IDENTIFIER ::= { { OID rsaEncryption PARAMETERS NULL } | { OID md2WithRSAEncryption PARAMETERS NULL } | { OID md5WithRSAEncryption PARAMETERS NULL } | { OID sha1WithRSAEncryption PARAMETERS NULL } | { OID sha256WithRSAEncryption PARAMETERS NULL } | { OID sha384WithRSAEncryption PARAMETERS NULL } | { OID sha512WithRSAEncryption PARAMETERS NULL } | { OID id-RSAES-OAEP PARAMETERS RSAES-OAEP-params } | PKCS1PSourceAlgorithms | { OID id-RSASSA-PSS PARAMETERS RSASSA-PSS-params } , ... -- Allows for future expansion -- }

## A.2.1 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 a value of type AlgorithmIdentifier shall have a value of type RSAES-OAEP-params: RSAES-OAEP-params ::= SEQUENCE { hashAlgorithm [0] HashAlgorithm DEFAULT sha1, maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1, pSourceAlgorithm [2] PSourceAlgorithm DEFAULT pSpecifiedEmpty } The fields of type RSAES-OAEP-params have the following meanings: * hashAlgorithm identifies the hash function. It shall be an algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms. For a discussion of supported hash functions, see Appendix B.1. HashAlgorithm ::= AlgorithmIdentifier { {OAEP-PSSDigestAlgorithms} } OAEP-PSSDigestAlgorithms ALGORITHM-IDENTIFIER ::= { { OID id-sha1 PARAMETERS NULL }| { OID id-sha256 PARAMETERS NULL }| { OID id-sha384 PARAMETERS NULL }| { OID id-sha512 PARAMETERS NULL }, ... -- Allows for future expansion -- } The default hash function is SHA-1: sha1 HashAlgorithm ::= { algorithm id-sha1, parameters SHA1Parameters : NULL } SHA1Parameters ::= NULL * maskGenAlgorithm 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 Appendix B.2.1). The parameters field associated with id-mgf1

shall be an algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms, identifying the hash function on which MGF1 is based. MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} } PKCS1MGFAlgorithms ALGORITHM-IDENTIFIER ::= { { OID id-mgf1 PARAMETERS HashAlgorithm }, ... -- Allows for future expansion -- } The default mask generation function is MGF1 with SHA-1: mgf1SHA1 MaskGenAlgorithm ::= { algorithm id-mgf1, parameters HashAlgorithm : sha1 } * pSourceAlgorithm identifies the source (and possibly the value) of the label L. 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 label is specified explicitly. The parameters field associated with id-pSpecified shall have a value of type OCTET STRING, containing the label. In previous versions of this specification, the term "encoding parameters" was used rather than "label", hence the name of the type below. PSourceAlgorithm ::= AlgorithmIdentifier { {PKCS1PSourceAlgorithms} } PKCS1PSourceAlgorithms ALGORITHM-IDENTIFIER ::= { { OID id-pSpecified PARAMETERS EncodingParameters }, ... -- Allows for future expansion -- } id-pSpecified OBJECT IDENTIFIER ::= { pkcs-1 9 } EncodingParameters ::= OCTET STRING(SIZE(0..MAX))

The default label is an empty string (so that lHash will contain the hash of the empty string): pSpecifiedEmpty PSourceAlgorithm ::= { algorithm id-pSpecified, parameters EncodingParameters : emptyString } emptyString EncodingParameters ::= ''H 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 RSAES-AlgorithmIdentifier ::= { algorithm id-RSAES-OAEP, parameters RSAES-OAEP-params : { hashAlgorithm sha1, maskGenAlgorithm mgf1SHA1, pSourceAlgorithm pSpecifiedEmpty } } RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier { {PKCS1Algorithms} }## A.2.2 RSAES-PKCS1-v1_5

The object identifier rsaEncryption (see Appendix A.1) identifies the RSAES-PKCS1-v1_5 encryption scheme. The parameters field associated with this OID in a value of type AlgorithmIdentifier shall have a value of type NULL. This is the same as in PKCS #1 v1.5. rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 }## A.2.3 RSASSA-PSS

The object identifier id-RSASSA-PSS identifies the RSASSA-PSS encryption scheme. id-RSASSA-PSS OBJECT IDENTIFIER ::= { pkcs-1 10 }

The parameters field associated with this OID in a value of type AlgorithmIdentifier shall have a value of type RSASSA-PSS-params: RSASSA-PSS-params ::= SEQUENCE { hashAlgorithm [0] HashAlgorithm DEFAULT sha1, maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1, saltLength [2] INTEGER DEFAULT 20, trailerField [3] TrailerField DEFAULT trailerFieldBC } The fields of type RSASSA-PSS-params have the following meanings: * hashAlgorithm identifies the hash function. It shall be an algorithm ID with an OID in the set OAEP-PSSDigestAlgorithms (see Appendix A.2.1). The default hash function is SHA-1. * maskGenAlgorithm identifies the mask generation function. It shall be an algorithm ID with an OID in the set PKCS1MGFAlgorithms (see Appendix A.2.1). The default mask generation function is MGF1 with SHA-1. For MGF1 (and more generally, for other mask generation functions based on a hash function), it is recommended that the underlying hash function be the same as the one identified by hashAlgorithm; see Note 2 in Section 9.1 for further comments. * saltLength is the octet length of the salt. It shall be an integer. For a given hashAlgorithm, the default value of saltLength is the octet length of the hash value. Unlike the other fields of type RSASSA-PSS-params, saltLength does not need to be fixed for a given RSA key pair. * trailerField is the trailer field number, for compatibility with the draft IEEE P1363a [27]. It shall be 1 for this version of the document, which represents the trailer field with hexadecimal value 0xbc. Other trailer fields (including the trailer field HashID || 0xcc in IEEE P1363a) are not supported in this document. TrailerField ::= INTEGER { trailerFieldBC(1) } If the default values of the hashAlgorithm, maskGenAlgorithm, and trailerField fields of RSASSA-PSS-params are used, then the algorithm identifier will have the following value:

rSASSA-PSS-Default-Identifier RSASSA-AlgorithmIdentifier ::= { algorithm id-RSASSA-PSS, parameters RSASSA-PSS-params : { hashAlgorithm sha1, maskGenAlgorithm mgf1SHA1, saltLength 20, trailerField trailerFieldBC } } RSASSA-AlgorithmIdentifier ::= AlgorithmIdentifier { {PKCS1Algorithms} } Note. In some applications, the hash function underlying a signature scheme is identified separately from the rest of the operations in the signature scheme. For instance, in PKCS #7 [45], a hash function identifier is placed before the message and a "digest encryption" algorithm identifier (indicating the rest of the operations) is carried with the signature. In order for PKCS #7 to support the RSASSA-PSS signature scheme, an object identifier would need to be defined for the operations in RSASSA-PSS after the hash function (analogous to the RSAEncryption OID for the RSASSA-PKCS1-v1_5 scheme). S/MIME CMS [25] takes a different approach. Although a hash function identifier is placed before the message, an algorithm identifier for the full signature scheme may be carried with a CMS signature (this is done for DSA signatures). Following this convention, the id-RSASSA-PSS OID can be used to identify RSASSA-PSS signatures in CMS. Since CMS is considered the successor to PKCS #7 and new developments such as the addition of support for RSASSA-PSS will be pursued with respect to CMS rather than PKCS #7, an OID for the "rest of" RSASSA-PSS is not defined in this version of PKCS #1.## A.2.4 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, SHA-1, SHA-256, SHA-384, or SHA-512. 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 a value of type AlgorithmIdentifier shall have a value of type NULL. The OID should be chosen in accordance with the following table: Hash algorithm OID -------------------------------------------------------- MD2 md2WithRSAEncryption ::= {pkcs-1 2} MD5 md5WithRSAEncryption ::= {pkcs-1 4} SHA-1 sha1WithRSAEncryption ::= {pkcs-1 5} SHA-256 sha256WithRSAEncryption ::= {pkcs-1 11}

SHA-384 sha384WithRSAEncryption ::= {pkcs-1 12} SHA-512 sha512WithRSAEncryption ::= {pkcs-1 13} The EMSA-PKCS1-v1_5 encoding method includes an ASN.1 value of type DigestInfo, where the type DigestInfo has the syntax DigestInfo ::= SEQUENCE { digestAlgorithm DigestAlgorithm, digest OCTET STRING } digestAlgorithm identifies the hash function and shall be an algorithm ID with an OID in the set PKCS1-v1-5DigestAlgorithms. For a discussion of supported hash functions, see Appendix B.1. DigestAlgorithm ::= AlgorithmIdentifier { {PKCS1-v1-5DigestAlgorithms} } PKCS1-v1-5DigestAlgorithms ALGORITHM-IDENTIFIER ::= { { OID id-md2 PARAMETERS NULL }| { OID id-md5 PARAMETERS NULL }| { OID id-sha1 PARAMETERS NULL }| { OID id-sha256 PARAMETERS NULL }| { OID id-sha384 PARAMETERS NULL }| { OID id-sha512 PARAMETERS NULL } }## Appendix B. Supporting techniques

This section gives several examples of underlying functions supporting the encryption schemes in Section 7 and the encoding methods in Section 9. A range of techniques is given here to allow compatibility with existing applications as well as migration to new techniques. While these supporting techniques are appropriate for applications to implement, none of them is required to be implemented. It is expected that profiles for PKCS #1 v2.1 will be developed that specify particular supporting techniques. This section also gives object identifiers for the supporting techniques.## B.1 Hash functions

Hash functions are used in the operations contained in Sections 7 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 and 9 should generally 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. In addition to the requirements, the hash function should yield a mask generation function (Appendix B.2) with pseudorandom output. Six hash functions are given as examples for the encoding methods in this document: MD2 [33], MD5 [41], SHA-1 [38], and the proposed algorithms SHA-256, SHA-384, and SHA-512 [39]. For the RSAES-OAEP encryption scheme and EMSA-PSS encoding method, only SHA-1 and SHA- 256/384/512 are recommended. For the EMSA-PKCS1-v1_5 encoding method, SHA-1 or SHA-256/384/512 are recommended for new applications. MD2 and MD5 are recommended only for compatibility with existing applications based on PKCS #1 v1.5. The object identifiers id-md2, id-md5, id-sha1, id-sha256, id-sha384, and id-sha512, identify the respective hash functions: id-md2 OBJECT IDENTIFIER ::= { iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 2 } id-md5 OBJECT IDENTIFIER ::= { iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 5 } id-sha1 OBJECT IDENTIFIER ::= { iso(1) identified-organization(3) oiw(14) secsig(3) algorithms(2) 26 } id-sha256 OBJECT IDENTIFIER ::= { joint-iso-itu-t(2) country(16) us(840) organization(1) gov(101) csor(3) nistalgorithm(4) hashalgs(2) 1 } id-sha384 OBJECT IDENTIFIER ::= { joint-iso-itu-t(2) country(16) us(840) organization(1) gov(101) csor(3) nistalgorithm(4) hashalgs(2) 2 }

id-sha512 OBJECT IDENTIFIER ::= { joint-iso-itu-t(2) country(16) us(840) organization(1) gov(101) csor(3) nistalgorithm(4) hashalgs(2) 3 } The parameters field associated with id-md2 and id-md5 in a value of type AlgorithmIdentifier shall have a value of type NULL. The parameters field associated with id-sha1, id-sha256, id-sha384, and id-sha512 should be omitted, but if present, shall have a value of type NULL. Note. Version 1.5 of PKCS #1 also allowed for the use of MD4 in signature schemes. The cryptanalysis of MD4 has progressed significantly in the intervening years. For example, Dobbertin [18] demonstrated how to find collisions for MD4 and that the first two rounds of MD4 are not one-way [20]. Because of these results and others (e.g., [8]), 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 [43] 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., [9][19]). To address these concerns, SHA-1, SHA-256, SHA-384, or SHA-512 are recommended for new applications. As of today, the best (known) collision attacks against these hash functions are generic attacks with complexity 2^(L/2), where L is the bit length of the hash output. For the signature schemes in this document, a collision attack is easily translated into a signature forgery. Therefore, the value L / 2 should be at least equal to the desired security level in bits of the signature scheme (a security level of B bits means that the best attack has complexity 2^B). The same rule of thumb can be applied to RSAES-OAEP; it is recommended that the bit length of the seed (which is equal to the bit length of the hash output) be twice the desired security level in bits.## B.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: Given one part of the output but not the input, it should be infeasible to

predict another part of the output. The provable security of RSAES- OAEP and RSASSA-PSS 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 given here: MGF1, which is based on a hash function. MGF1 coincides with the mask generation functions defined in IEEE Std 1363-2000 [26] and the draft ANSI X9.44 [1]. Future versions of this document may define other mask generation functions.## B.2.1 MGF1

MGF1 is a Mask Generation Function based on a hash function. MGF1 (mgfSeed, maskLen) Options: Hash hash function (hLen denotes the length in octets of the hash function output) Input: mgfSeed seed from which mask is generated, an octet string maskLen intended length in octets of the mask, at most 2^32 hLen Output: mask mask, an octet string of length maskLen Error: "mask too long" Steps: 1. If maskLen > 2^32 hLen, output "mask too long" and stop. 2. Let T be the empty octet string. 3. For counter from 0 to \ceil (maskLen / hLen) - 1, do the following: a. Convert counter to an octet string C of length 4 octets (see Section 4.1): C = I2OSP (counter, 4) . b. Concatenate the hash of the seed mgfSeed and C to the octet string T: T = T || Hash(mgfSeed || C) .

4. Output the leading maskLen octets of T as the octet string mask. The object identifier id-mgf1 identifies the MGF1 mask generation function: id-mgf1 OBJECT IDENTIFIER ::= { pkcs-1 8 } The parameters field associated with this OID in a value of type AlgorithmIdentifier shall have a value of type hashAlgorithm, identifying the hash function on which MGF1 is based.## Appendix C. ASN.1 module

PKCS-1 { iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs-1(1) modules(0) pkcs-1(1) } -- $ Revision: 2.1r1 $ -- This module has been checked for conformance with the ASN.1 -- standard by the OSS ASN.1 Tools DEFINITIONS EXPLICIT TAGS ::= BEGIN -- EXPORTS ALL -- All types and values defined in this module are exported for use -- in other ASN.1 modules. IMPORTS id-sha256, id-sha384, id-sha512 FROM NIST-SHA2 { joint-iso-itu-t(2) country(16) us(840) organization(1) gov(101) csor(3) nistalgorithm(4) modules(0) sha2(1) }; -- ============================ -- Basic object identifiers -- ============================ -- The DER encoding of this in hexadecimal is: -- (0x)06 08 -- 2A 86 48 86 F7 0D 01 01 -- pkcs-1 OBJECT IDENTIFIER ::= {

iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) 1 } -- -- When rsaEncryption is used in an AlgorithmIdentifier the -- parameters MUST be present and MUST be NULL. -- rsaEncryption OBJECT IDENTIFIER ::= { pkcs-1 1 } -- -- When id-RSAES-OAEP is used in an AlgorithmIdentifier the -- parameters MUST be present and MUST be RSAES-OAEP-params. -- id-RSAES-OAEP OBJECT IDENTIFIER ::= { pkcs-1 7 } -- -- When id-pSpecified is used in an AlgorithmIdentifier the -- parameters MUST be an OCTET STRING. -- id-pSpecified OBJECT IDENTIFIER ::= { pkcs-1 9 } -- When id-RSASSA-PSS is used in an AlgorithmIdentifier the -- parameters MUST be present and MUST be RSASSA-PSS-params. -- id-RSASSA-PSS OBJECT IDENTIFIER ::= { pkcs-1 10 } -- -- When the following OIDs are used in an AlgorithmIdentifier the -- parameters MUST be present and MUST be NULL. -- md2WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 2 } md5WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 4 } sha1WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 5 } sha256WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 11 } sha384WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 12 } sha512WithRSAEncryption OBJECT IDENTIFIER ::= { pkcs-1 13 } -- -- This OID really belongs in a module with the secsig OIDs. -- id-sha1 OBJECT IDENTIFIER ::= { iso(1) identified-organization(3) oiw(14) secsig(3) algorithms(2) 26 } -- -- OIDs for MD2 and MD5, allowed only in EMSA-PKCS1-v1_5. --

id-md2 OBJECT IDENTIFIER ::= { iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 2 } id-md5 OBJECT IDENTIFIER ::= { iso(1) member-body(2) us(840) rsadsi(113549) digestAlgorithm(2) 5 } -- -- When id-mgf1 is used in an AlgorithmIdentifier the parameters MUST -- be present and MUST be a HashAlgorithm, for example sha1. -- id-mgf1 OBJECT IDENTIFIER ::= { pkcs-1 8 } -- ================ -- Useful types -- ================ ALGORITHM-IDENTIFIER ::= CLASS { &id OBJECT IDENTIFIER UNIQUE, &Type OPTIONAL } WITH SYNTAX { OID &id [PARAMETERS &Type] } -- -- Note: the parameter InfoObjectSet in the following definitions -- allows a distinct information object set to be specified for sets -- of algorithms such as: -- DigestAlgorithms ALGORITHM-IDENTIFIER ::= { -- { OID id-md2 PARAMETERS NULL }| -- { OID id-md5 PARAMETERS NULL }| -- { OID id-sha1 PARAMETERS NULL } -- } -- AlgorithmIdentifier { ALGORITHM-IDENTIFIER:InfoObjectSet } ::= SEQUENCE { algorithm ALGORITHM-IDENTIFIER.&id({InfoObjectSet}), parameters ALGORITHM-IDENTIFIER.&Type({InfoObjectSet}{@.algorithm}) OPTIONAL } -- ============== -- Algorithms -- ============== --

-- Allowed EME-OAEP and EMSA-PSS digest algorithms. -- OAEP-PSSDigestAlgorithms ALGORITHM-IDENTIFIER ::= { { OID id-sha1 PARAMETERS NULL }| { OID id-sha256 PARAMETERS NULL }| { OID id-sha384 PARAMETERS NULL }| { OID id-sha512 PARAMETERS NULL }, ... -- Allows for future expansion -- } -- -- Allowed EMSA-PKCS1-v1_5 digest algorithms. -- PKCS1-v1-5DigestAlgorithms ALGORITHM-IDENTIFIER ::= { { OID id-md2 PARAMETERS NULL }| { OID id-md5 PARAMETERS NULL }| { OID id-sha1 PARAMETERS NULL }| { OID id-sha256 PARAMETERS NULL }| { OID id-sha384 PARAMETERS NULL }| { OID id-sha512 PARAMETERS NULL } } -- When id-md2 and id-md5 are used in an AlgorithmIdentifier the -- parameters MUST be present and MUST be NULL. -- When id-sha1, id-sha256, id-sha384 and id-sha512 are used in an -- AlgorithmIdentifier the parameters (which are optional) SHOULD -- be omitted. However, an implementation MUST also accept -- AlgorithmIdentifier values where the parameters are NULL. sha1 HashAlgorithm ::= { algorithm id-sha1, parameters SHA1Parameters : NULL -- included for compatibility -- with existing implementations } HashAlgorithm ::= AlgorithmIdentifier { {OAEP-PSSDigestAlgorithms} } SHA1Parameters ::= NULL -- -- Allowed mask generation function algorithms. -- If the identifier is id-mgf1, the parameters are a HashAlgorithm. -- PKCS1MGFAlgorithms ALGORITHM-IDENTIFIER ::= { { OID id-mgf1 PARAMETERS HashAlgorithm }, ... -- Allows for future expansion -- }

-- -- Default AlgorithmIdentifier for id-RSAES-OAEP.maskGenAlgorithm and -- id-RSASSA-PSS.maskGenAlgorithm. -- mgf1SHA1 MaskGenAlgorithm ::= { algorithm id-mgf1, parameters HashAlgorithm : sha1 } MaskGenAlgorithm ::= AlgorithmIdentifier { {PKCS1MGFAlgorithms} } -- -- Allowed algorithms for pSourceAlgorithm. -- PKCS1PSourceAlgorithms ALGORITHM-IDENTIFIER ::= { { OID id-pSpecified PARAMETERS EncodingParameters }, ... -- Allows for future expansion -- } EncodingParameters ::= OCTET STRING(SIZE(0..MAX)) -- -- This identifier means that the label L is an empty string, so the -- digest of the empty string appears in the RSA block before -- masking. -- pSpecifiedEmpty PSourceAlgorithm ::= { algorithm id-pSpecified, parameters EncodingParameters : emptyString } PSourceAlgorithm ::= AlgorithmIdentifier { {PKCS1PSourceAlgorithms} } emptyString EncodingParameters ::= ''H -- -- Type identifier definitions for the PKCS #1 OIDs. -- PKCS1Algorithms ALGORITHM-IDENTIFIER ::= { { OID rsaEncryption PARAMETERS NULL } | { OID md2WithRSAEncryption PARAMETERS NULL } | { OID md5WithRSAEncryption PARAMETERS NULL } | { OID sha1WithRSAEncryption PARAMETERS NULL } | { OID sha256WithRSAEncryption PARAMETERS NULL } | { OID sha384WithRSAEncryption PARAMETERS NULL } | { OID sha512WithRSAEncryption PARAMETERS NULL } | { OID id-RSAES-OAEP PARAMETERS RSAES-OAEP-params } | PKCS1PSourceAlgorithms |

{ OID id-RSASSA-PSS PARAMETERS RSASSA-PSS-params } , ... -- Allows for future expansion -- } -- =================== -- Main structures -- =================== RSAPublicKey ::= SEQUENCE { modulus INTEGER, -- n publicExponent INTEGER -- e } -- -- Representation of RSA private key with information for the CRT -- algorithm. -- 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 otherPrimeInfos OtherPrimeInfos OPTIONAL } Version ::= INTEGER { two-prime(0), multi(1) } (CONSTRAINED BY { -- version must be multi if otherPrimeInfos present -- }) OtherPrimeInfos ::= SEQUENCE SIZE(1..MAX) OF OtherPrimeInfo OtherPrimeInfo ::= SEQUENCE { prime INTEGER, -- ri exponent INTEGER, -- di coefficient INTEGER -- ti } -- -- AlgorithmIdentifier.parameters for id-RSAES-OAEP. -- Note that the tags in this Sequence are explicit. -- RSAES-OAEP-params ::= SEQUENCE {

hashAlgorithm [0] HashAlgorithm DEFAULT sha1, maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1, pSourceAlgorithm [2] PSourceAlgorithm DEFAULT pSpecifiedEmpty } -- -- Identifier for default RSAES-OAEP algorithm identifier. -- The DER Encoding of this is in hexadecimal: -- (0x)30 0D -- 06 09 -- 2A 86 48 86 F7 0D 01 01 07 -- 30 00 -- Notice that the DER encoding of default values is "empty". -- rSAES-OAEP-Default-Identifier RSAES-AlgorithmIdentifier ::= { algorithm id-RSAES-OAEP, parameters RSAES-OAEP-params : { hashAlgorithm sha1, maskGenAlgorithm mgf1SHA1, pSourceAlgorithm pSpecifiedEmpty } } RSAES-AlgorithmIdentifier ::= AlgorithmIdentifier { {PKCS1Algorithms} } -- -- AlgorithmIdentifier.parameters for id-RSASSA-PSS. -- Note that the tags in this Sequence are explicit. -- RSASSA-PSS-params ::= SEQUENCE { hashAlgorithm [0] HashAlgorithm DEFAULT sha1, maskGenAlgorithm [1] MaskGenAlgorithm DEFAULT mgf1SHA1, saltLength [2] INTEGER DEFAULT 20, trailerField [3] TrailerField DEFAULT trailerFieldBC } TrailerField ::= INTEGER { trailerFieldBC(1) } -- -- Identifier for default RSASSA-PSS algorithm identifier -- The DER Encoding of this is in hexadecimal: -- (0x)30 0D -- 06 09 -- 2A 86 48 86 F7 0D 01 01 0A -- 30 00 -- Notice that the DER encoding of default values is "empty".

-- rSASSA-PSS-Default-Identifier RSASSA-AlgorithmIdentifier ::= { algorithm id-RSASSA-PSS, parameters RSASSA-PSS-params : { hashAlgorithm sha1, maskGenAlgorithm mgf1SHA1, saltLength 20, trailerField trailerFieldBC } } RSASSA-AlgorithmIdentifier ::= AlgorithmIdentifier { {PKCS1Algorithms} } -- -- Syntax for the EMSA-PKCS1-v1_5 hash identifier. -- DigestInfo ::= SEQUENCE { digestAlgorithm DigestAlgorithm, digest OCTET STRING } DigestAlgorithm ::= AlgorithmIdentifier { {PKCS1-v1-5DigestAlgorithms} } END -- PKCS1Definitions## Appendix D. Intellectual Property Considerations

The RSA public-key cryptosystem is described in U.S. Patent 4,405,829, which expired on September 20, 2000. RSA Security Inc. makes no other patent claims on the constructions described in this document, although specific underlying techniques may be covered. Multi-prime RSA is described in U.S. Patent 5,848,159. The University of California has indicated that it has a patent pending on the PSS signature scheme [5]. It has also provided a letter to the IEEE P1363 working group stating that if the PSS signature scheme is included in an IEEE standard, "the University of California will, when that standard is adopted, FREELY license any conforming implementation of PSS as a technique for achieving a digital signature with appendix" [23]. The PSS signature scheme is specified in the IEEE P1363a draft [27], which was in ballot resolution when this document was published.

License to copy this document is granted provided that it is identified as "RSA Security Inc. Public-Key Cryptography Standards (PKCS)" in all material mentioning or referencing this document. RSA Security Inc. makes no other representations regarding intellectual property claims by other parties. Such determination is the responsibility of the user.## Appendix E. 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 incorporated 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 1.5 was republished as IETF RFC 2313. Version 2.0 Version 2.0 incorporated major editorial changes in terms of the document structure and introduced the RSAES-OAEP encryption scheme. This version continued to support the encryption and signature processes in version 1.5, although the hash algorithm MD4 was no longer allowed due to cryptanalytic advances in the intervening years. Version 2.0 was republished as IETF RFC 2437 [35].

Version 2.1 Version 2.1 introduces multi-prime RSA and the RSASSA-PSS signature scheme with appendix along with several editorial improvements. This version continues to support the schemes in version 2.0.## Appendix F. References

[1] ANSI X9F1 Working Group. ANSI X9.44 Draft D2: Key Establishment Using Integer Factorization Cryptography. Working Draft, March 2002. [2] M. Bellare, A. Desai, D. Pointcheval and P. Rogaway. Relations Among Notions of Security for Public-Key Encryption Schemes. In H. Krawczyk, editor, Advances in Cryptology - Crypto '98, volume 1462 of Lecture Notes in Computer Science, pp. 26 - 45. Springer Verlag, 1998. [3] M. Bellare and P. Rogaway. Optimal Asymmetric Encryption - How to Encrypt with RSA. In A. De Santis, editor, Advances in Cryptology - Eurocrypt '94, volume 950 of Lecture Notes in Computer Science, pp. 92 - 111. Springer Verlag, 1995. [4] M. Bellare and P. Rogaway. The Exact Security of Digital Signatures - How to Sign with RSA and Rabin. In U. Maurer, editor, Advances in Cryptology - Eurocrypt '96, volume 1070 of Lecture Notes in Computer Science, pp. 399 - 416. Springer Verlag, 1996. [5] M. Bellare and P. Rogaway. PSS: Provably Secure Encoding Method for Digital Signatures. Submission to IEEE P1363 working group, August 1998. Available from http://grouper.ieee.org/groups/1363/. [6] D. Bleichenbacher. Chosen Ciphertext Attacks Against Protocols Based on the RSA Encryption Standard PKCS #1. In H. Krawczyk, editor, Advances in Cryptology - Crypto '98, volume 1462 of Lecture Notes in Computer Science, pp. 1 - 12. Springer Verlag, 1998. [7] D. Bleichenbacher, B. Kaliski and J. Staddon. Recent Results on PKCS #1: RSA Encryption Standard. RSA Laboratories' Bulletin No. 7, June 1998.

[8] B. den Boer and A. Bosselaers. An Attack on the Last Two Rounds of MD4. In J. Feigenbaum, editor, Advances in Cryptology - Crypto '91, volume 576 of Lecture Notes in Computer Science, pp. 194 - 203. Springer Verlag, 1992. [9] B. den Boer and A. Bosselaers. Collisions for the Compression Function of MD5. In T. Helleseth, editor, Advances in Cryptology - Eurocrypt '93, volume 765 of Lecture Notes in Computer Science, pp. 293 - 304. Springer Verlag, 1994. [10] D. Coppersmith, M. Franklin, J. Patarin and M. Reiter. Low- Exponent RSA with Related Messages. In U. Maurer, editor, Advances in Cryptology - Eurocrypt '96, volume 1070 of Lecture Notes in Computer Science, pp. 1 - 9. Springer Verlag, 1996. [11] D. Coppersmith, S. Halevi and C. Jutla. ISO 9796-1 and the New Forgery Strategy. Presented at the rump session of Crypto '99, August 1999. [12] J.-S. Coron. On the Exact Security of Full Domain Hashing. In M. Bellare, editor, Advances in Cryptology - Crypto 2000, volume 1880 of Lecture Notes in Computer Science, pp. 229 - 235. Springer Verlag, 2000. [13] J.-S. Coron. Optimal Security Proofs for PSS and Other Signature Schemes. In L. Knudsen, editor, Advances in Cryptology - Eurocrypt 2002, volume 2332 of Lecture Notes in Computer Science, pp. 272 - 287. Springer Verlag, 2002. [14] J.-S. Coron, M. Joye, D. Naccache and P. Paillier. New Attacks on PKCS #1 v1.5 Encryption. In B. Preneel, editor, Advances in Cryptology - Eurocrypt 2000, volume 1807 of Lecture Notes in Computer Science, pp. 369 - 379. Springer Verlag, 2000. [15] J.-S. Coron, D. Naccache and J. P. Stern. On the Security of RSA Padding. In M. Wiener, editor, Advances in Cryptology - Crypto '99, volume 1666 of Lecture Notes in Computer Science, pp. 1 - 18. Springer Verlag, 1999. [16] Y. Desmedt and A.M. Odlyzko. A Chosen Text Attack on the RSA Cryptosystem and Some Discrete Logarithm Schemes. In H.C. Williams, editor, Advances in Cryptology - Crypto '85, volume 218 of Lecture Notes in Computer Science, pp. 516 - 522. Springer Verlag, 1986. [17] Dierks, T. and C. Allen, "The TLS Protocol, Version 1.0", RFC 2246, January 1999.

[18] H. Dobbertin. Cryptanalysis of MD4. In D. Gollmann, editor, Fast Software Encryption '96, volume 1039 of Lecture Notes in Computer Science, pp. 55 - 72. Springer Verlag, 1996. [19] H. Dobbertin. Cryptanalysis of MD5 Compress. Presented at the rump session of Eurocrypt '96, May 1996. [20] H. Dobbertin. The First Two Rounds of MD4 are Not One-Way. In S. Vaudenay, editor, Fast Software Encryption '98, volume 1372 in Lecture Notes in Computer Science, pp. 284 - 292. Springer Verlag, 1998. [21] E. Fujisaki, T. Okamoto, D. Pointcheval and J. Stern. RSA-OAEP is Secure under the RSA Assumption. In J. Kilian, editor, Advances in Cryptology - Crypto 2001, volume 2139 of Lecture Notes in Computer Science, pp. 260 - 274. Springer Verlag, 2001. [22] H. Garner. The Residue Number System. IRE Transactions on Electronic Computers, EC-8 (6), pp. 140 - 147, June 1959. [23] M.L. Grell. Re: Encoding Methods PSS/PSS-R. Letter to IEEE P1363 working group, University of California, June 15, 1999. Available from http://grouper.ieee.org/groups/1363/P1363/patents.html. [24] J. Haastad. Solving Simultaneous Modular Equations of Low Degree. SIAM Journal of Computing, volume 17, pp. 336 - 341, 1988. [25] Housley, R., "Cryptographic Message Syntax (CMS)", RFC 3369, August 2002. Housley, R., "Cryptographic Message Syntax (CMS) Algorithms", RFC 3370, August 2002. [26] IEEE Std 1363-2000: Standard Specifications for Public Key Cryptography. IEEE, August 2000. [27] IEEE P1363 working group. IEEE P1363a D11: Draft Standard Specifications for Public Key Cryptography -- Amendment 1: Additional Techniques. December 16, 2002. Available from http://grouper.ieee.org/groups/1363/. [28] ISO/IEC 9594-8:1997: Information technology - Open Systems Interconnection - The Directory: Authentication Framework. 1997.

[29] ISO/IEC FDIS 9796-2: Information Technology - Security Techniques - Digital Signature Schemes Giving Message Recovery - Part 2: Integer Factorization Based Mechanisms. Final Draft International Standard, December 2001. [30] ISO/IEC 18033-2: Information Technology - Security Techniques - Encryption Algorithms - Part 2: Asymmetric Ciphers. V. Shoup, editor, Text for 2nd Working Draft, January 2002. [31] J. Jonsson. Security Proof for the RSA-PSS Signature Scheme (extended abstract). Second Open NESSIE Workshop. September 2001. Full version available from http://eprint.iacr.org/2001/053/. [32] J. Jonsson and B. Kaliski. On the Security of RSA Encryption in TLS. In M. Yung, editor, Advances in Cryptology - CRYPTO 2002, vol. 2442 of Lecture Notes in Computer Science, pp. 127 - 142. Springer Verlag, 2002. [33] Kaliski, B., "The MD2 Message-Digest Algorithm", RFC 1319, April 1992. [34] B. Kaliski. On Hash Function Identification in Signature Schemes. In B. Preneel, editor, RSA Conference 2002, Cryptographers' Track, volume 2271 of Lecture Notes in Computer Science, pp. 1 - 16. Springer Verlag, 2002. [35] Kaliski, B. and J. Staddon, "PKCS #1: RSA Cryptography Specifications Version 2.0", RFC 2437, October 1998. [36] J. Manger. A Chosen Ciphertext Attack on RSA Optimal Asymmetric Encryption Padding (OAEP) as Standardized in PKCS #1 v2.0. In J. Kilian, editor, Advances in Cryptology - Crypto 2001, volume 2139 of Lecture Notes in Computer Science, pp. 260 - 274. Springer Verlag, 2001. [37] A. Menezes, P. van Oorschot and S. Vanstone. Handbook of Applied Cryptography. CRC Press, 1996. [38] National Institute of Standards and Technology (NIST). FIPS Publication 180-1: Secure Hash Standard. April 1994. [39] National Institute of Standards and Technology (NIST). Draft FIPS 180-2: Secure Hash Standard. Draft, May 2001. Available from http://www.nist.gov/sha/.

[40] J.-J. Quisquater and C. Couvreur. Fast Decipherment Algorithm for RSA Public-Key Cryptosystem. Electronics Letters, 18 (21), pp. 905 - 907, October 1982. [41] Rivest, R., "The MD5 Message-Digest Algorithm", RFC 1321, April 1992. [42] 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. [43] 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 1995. [44] RSA Laboratories. PKCS #1 v2.0: RSA Encryption Standard. October 1998. [45] RSA Laboratories. PKCS #7 v1.5: Cryptographic Message Syntax Standard. November 1993. (Republished as IETF RFC 2315.) [46] RSA Laboratories. PKCS #8 v1.2: Private-Key Information Syntax Standard. November 1993. [47] RSA Laboratories. PKCS #12 v1.0: Personal Information Exchange Syntax Standard. June 1999. [48] V. Shoup. OAEP Reconsidered. In J. Kilian, editor, Advances in Cryptology - Crypto 2001, volume 2139 of Lecture Notes in Computer Science, pp. 239 - 259. Springer Verlag, 2001. [49] R. D. Silverman. A Cost-Based Security Analysis of Symmetric and Asymmetric Key Lengths. RSA Laboratories Bulletin No. 13, April 2000. Available from http://www.rsasecurity.com.rsalabs/bulletins/. [50] G. J. Simmons. Subliminal communication is easy using the DSA. In T. Helleseth, editor, Advances in Cryptology - Eurocrypt '93, volume 765 of Lecture Notes in Computer Science, pp. 218- 232. Springer-Verlag, 1993.

## Appendix G. About PKCS

The Public-Key Cryptography Standards are specifications produced by RSA Laboratories in cooperation with secure systems developers worldwide for the purpose of accelerating the deployment of public-key cryptography. First published in 1991 as a result of meetings with a small group of early adopters of public-key technology, the PKCS documents have become widely referenced and implemented. Contributions from the PKCS series have become part of many formal and de facto standards, including ANSI X9 and IEEE P1363 documents, PKIX, SET, S/MIME, SSL/TLS, and WAP/WTLS. Further development of PKCS occurs through mailing list discussions and occasional workshops, and suggestions for improvement are welcome. For more information, contact: PKCS Editor RSA Laboratories 174 Middlesex Turnpike Bedford, MA 01730 USA pkcs-editor@rsasecurity.com http://www.rsasecurity.com/rsalabs/pkcs## Appendix H. Corrections Made During RFC Publication Process

The following corrections were made in converting the PKCS #1 v2.1 document to this RFC: * The requirement that the parameters in an AlgorithmIdentifier value for id-sha1, id-sha256, id-sha384, and id-sha512 be NULL was changed to a recommendation that the parameters be omitted (while still allowing the parameters to be NULL). This is to align with the definitions originally promulgated by NIST. Implementations MUST accept AlgorithmIdentifier values both without parameters and with NULL parameters. * The notes after RSADP and RSASP1 (Secs. 5.1.2 and 5.2.1) were corrected to refer to step 2.b rather than 2.a. * References [25], [27] and [32] were updated to reflect new publication data. These corrections will be reflected in future editions of PKCS #1 v2.1.## Security Considerations

Security issues are discussed throughout this memo.

## Acknowledgements

This document is based on a contribution of RSA Laboratories, the research center of RSA Security Inc. Any substantial use of the text from this document must acknowledge RSA Security Inc. RSA Security Inc. requests that all material mentioning or referencing this document identify this as "RSA Security Inc. PKCS #1 v2.1".## Authors' Addresses

Jakob Jonsson Philipps-Universitaet Marburg Fachbereich Mathematik und Informatik Hans Meerwein Strasse, Lahnberge DE-35032 Marburg Germany Phone: +49 6421 28 25672 EMail: jonsson@mathematik.uni-marburg.de Burt Kaliski RSA Laboratories 174 Middlesex Turnpike Bedford, MA 01730 USA Phone: +1 781 515 7073 EMail: bkaliski@rsasecurity.com

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