RFC 8017
PKCS #1: RSA Cryptography Specifications Version 2.2
Internet Engineering Task Force (IETF) K. Moriarty, Ed. Request for Comments: 8017 EMC Corporation Obsoletes: 3447 B. Kaliski Category: Informational Verisign ISSN: 2070-1721 J. Jonsson Subset AB A. Rusch RSA November 2016 PKCS #1: RSA Cryptography Specifications Version 2.2Abstract
This document provides recommendations for the implementation of public-key cryptography based on the RSA algorithm, covering cryptographic primitives, encryption schemes, signature schemes with appendix, and ASN.1 syntax for representing keys and for identifying the schemes. This document represents a republication of PKCS #1 v2.2 from RSA Laboratories' Public-Key Cryptography Standards (PKCS) series. By publishing this RFC, change control is transferred to the IETF. This document also obsoletes RFC 3447. Status of This Memo This document is not an Internet Standards Track specification; it is published for informational purposes. This document is a product of the Internet Engineering Task Force (IETF). It represents the consensus of the IETF community. It has received public review and has been approved for publication by the Internet Engineering Steering Group (IESG). Not all documents approved by the IESG are a candidate for any level of Internet Standard; see Section 2 of RFC 7841. Information about the current status of this document, any errata, and how to provide feedback on it may be obtained at http://www.rfc-editor.org/info/rfc8017.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1. Requirements Language . . . . . . . . . . . . . . . . . . 5 2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Key Types . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1. RSA Public Key . . . . . . . . . . . . . . . . . . . . . 8 3.2. RSA Private Key . . . . . . . . . . . . . . . . . . . . . 9 4. Data Conversion Primitives . . . . . . . . . . . . . . . . . 11 4.1. I2OSP . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2. OS2IP . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5. Cryptographic Primitives . . . . . . . . . . . . . . . . . . 12 5.1. Encryption and Decryption Primitives . . . . . . . . . . 12 5.1.1. RSAEP . . . . . . . . . . . . . . . . . . . . . . . . 13 5.1.2. RSADP . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2. Signature and Verification Primitives . . . . . . . . . . 15 5.2.1. RSASP1 . . . . . . . . . . . . . . . . . . . . . . . 15 5.2.2. RSAVP1 . . . . . . . . . . . . . . . . . . . . . . . 16 6. Overview of Schemes . . . . . . . . . . . . . . . . . . . . . 17 7. Encryption Schemes . . . . . . . . . . . . . . . . . . . . . 18 7.1. RSAES-OAEP . . . . . . . . . . . . . . . . . . . . . . . 19 7.1.1. Encryption Operation . . . . . . . . . . . . . . . . 22 7.1.2. Decryption Operation . . . . . . . . . . . . . . . . 25 7.2. RSAES-PKCS1-v1_5 . . . . . . . . . . . . . . . . . . . . 27 7.2.1. Encryption Operation . . . . . . . . . . . . . . . . 28 7.2.2. Decryption Operation . . . . . . . . . . . . . . . . 29 8. Signature Scheme with Appendix . . . . . . . . . . . . . . . 31 8.1. RSASSA-PSS . . . . . . . . . . . . . . . . . . . . . . . 32 8.1.1. Signature Generation Operation . . . . . . . . . . . 33 8.1.2. Signature Verification Operation . . . . . . . . . . 34 8.2. RSASSA-PKCS1-v1_5 . . . . . . . . . . . . . . . . . . . . 35 8.2.1. Signature Generation Operation . . . . . . . . . . . 36 8.2.2. Signature Verification Operation . . . . . . . . . . 37 9. Encoding Methods for Signatures with Appendix . . . . . . . . 39 9.1. EMSA-PSS . . . . . . . . . . . . . . . . . . . . . . . . 40 9.1.1. Encoding Operation . . . . . . . . . . . . . . . . . 42 9.1.2. Verification Operation . . . . . . . . . . . . . . . 44 9.2. EMSA-PKCS1-v1_5 . . . . . . . . . . . . . . . . . . . . . 45 10. Security Considerations . . . . . . . . . . . . . . . . . . . 47 11. References . . . . . . . . . . . . . . . . . . . . . . . . . 48 11.1. Normative References . . . . . . . . . . . . . . . . . . 48 11.2. Informative References . . . . . . . . . . . . . . . . . 48
Appendix A. ASN.1 Syntax . . . . . . . . . . . . . . . . . . . . 54 A.1. RSA Key Representation . . . . . . . . . . . . . . . . . 54 A.1.1. RSA Public Key Syntax . . . . . . . . . . . . . . . . 54 A.1.2. RSA Private Key Syntax . . . . . . . . . . . . . . . 55 A.2. Scheme Identification . . . . . . . . . . . . . . . . . . 57 A.2.1. RSAES-OAEP . . . . . . . . . . . . . . . . . . . . . 57 A.2.2. RSAES-PKCS-v1_5 . . . . . . . . . . . . . . . . . . . 60 A.2.3. RSASSA-PSS . . . . . . . . . . . . . . . . . . . . . 60 A.2.4. RSASSA-PKCS-v1_5 . . . . . . . . . . . . . . . . . . 62 Appendix B. Supporting Techniques . . . . . . . . . . . . . . . 63 B.1. Hash Functions . . . . . . . . . . . . . . . . . . . . . 63 B.2. Mask Generation Functions . . . . . . . . . . . . . . . . 66 B.2.1. MGF1 . . . . . . . . . . . . . . . . . . . . . . . . 67 Appendix C. ASN.1 Module . . . . . . . . . . . . . . . . . . . . 68 Appendix D. Revision History of PKCS #1 . . . . . . . . . . . . 76 Appendix E. About PKCS . . . . . . . . . . . . . . . . . . . . . 77 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 78 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 781. Introduction
This document provides recommendations for the implementation of public-key cryptography based on the RSA algorithm [RSA], covering the following aspects: o Cryptographic primitives o Encryption schemes o Signature schemes with appendix o 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 standards IEEE 1363 [IEEE1363], IEEE 1363a [IEEE1363A], and ANSI X9.44 [ANSIX944]. This document supersedes PKCS #1 version 2.1 [RFC3447] but includes compatible techniques. The organization of this document is as follows: o Section 1 is an introduction. o Section 2 defines some notation used in this document.
o Section 3 defines the RSA public and private key types. o Sections 4 and 5 define several primitives, or basic mathematical operations. Data conversion primitives are in Section 4, and cryptographic primitives (encryption-decryption and signature- verification) are in Section 5. o 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 encryption scheme based on Optimal Asymmetric Encryption Padding (OAEP) [OAEP], and Section 8 defines a signature scheme with appendix based on the Probabilistic Signature Scheme (PSS) [RSARABIN] [PSS]. o Section 9 defines the encoding methods for the signature schemes in Section 8. o Appendix A defines the ASN.1 syntax for the keys defined in Section 3 and the schemes in Sections 7 and 8. o Appendix B defines the hash functions and the mask generation function (MGF) used in this document, including ASN.1 syntax for the techniques. o Appendix C gives an ASN.1 module. o Appendices D and E outline the revision history of PKCS #1 and provide general information about the Public-Key Cryptography Standards. This document represents a republication of PKCS #1 v2.2 [PKCS1_22] from RSA Laboratories' Public-Key Cryptography Standards (PKCS) series.1.1. Requirements Language
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119].
2. Notation
The notation in this document includes: 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
values 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 Chinese Remainder Theorem (CRT) can be applied in a non-
recursive as well as a recursive way. In this document, a recursive
approach following Garner's algorithm [GARNER] 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 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 [SILVERMAN].
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 [GARNER] (see also Algorithm 14.71 in [HANDBOOK]).
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 [FASTDEC] observed the benefit of
applying the CRT to RSA operations.
4. Data Conversion Primitives
Two data conversion primitives are employed in the schemes defined in this document: o I2OSP - Integer-to-Octet-String primitive o 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: RSA Encryption Primitive (RSAEP) / RSA Decryption Primitive (RSADP). RSAEP and RSADP involve the same mathematical operation, with different keys as input. The primitives defined here are the same as Integer Factorization Encryption Primitive using RSA (IFEP-RSA) / Integer Factorization Decryption Primitive using RSA (IFDP-RSA) in IEEE 1363 [IEEE1363] (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: RSA Signature Primitive, version 1 (RSASP1) / RSA Verification Primitive, version 1 (RSAVP1). The primitives defined here are the same as Integer Factorization Signature Primitive using RSA, version 1 (IFSP-RSA1) / Integer Factorization Verification Primitive using RSA, version 1 (IFVP-RSA1) in IEEE 1363 [IEEE1363] (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:
1. Let s_1 = m^dP mod p and s_2 = m^dQ mod q.
2. If u > 2, let s_i = m^(d_i) mod r_i, i = 3, ..., u.
3. Let h = (s_1 - s_2) * qInv mod p.
4. Let s = s_2 + q * h.
5. If u > 2, let R = r_1 and for i = 3 to u do
a. Let R = R * r_(i-1).
b. Let h = (s_i - s) * t_i mod r_i.
c. 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.
6. Overview of Schemes
A scheme combines cryptographic primitives and other techniques to
achieve a particular security goal. Two types of scheme are
specified in this document: encryption schemes and signature schemes
with appendix.
The schemes specified in this document are limited in scope in that
their operations consist only of steps to process data with an RSA
public or private key, and they do not include steps for obtaining or
validating the key. Thus, in addition to the scheme operations, an
application will typically include key management operations by which
parties may select RSA public and private keys for a scheme
operation. The specific additional operations and other details are
outside the scope of this document.
As was the case for the cryptographic primitives (Section 5), the
specifications of scheme operations assume that certain conditions
are met by the inputs, in particular that RSA public and private keys
are valid. The behavior of an implementation is thus unspecified
when a key is invalid. The impact of such unspecified behavior
depends on the application. Possible means of addressing key
validation include explicit key validation by the application; key
validation within the public-key infrastructure; and assignment of
liability for operations performed with an invalid key to the party
who generated the key.
A generally good cryptographic practice is to employ a given RSA key
pair in only one scheme. This avoids the risk that vulnerability in
one scheme may compromise the security of the other and may be
essential to maintain provable security. While RSAES-PKCS1-v1_5
(Section 7.2) and RSASSA-PKCS1-v1_5 (Section 8.2) have traditionally been employed together without any known bad interactions (indeed, this is the model introduced by PKCS #1 v1.5), such a combined use of an RSA key pair is NOT RECOMMENDED for new applications. To illustrate the risks related to the employment of an RSA key pair in more than one scheme, suppose an RSA key pair is employed in both RSAES-OAEP (Section 7.1) and RSAES-PKCS1-v1_5. Although RSAES-OAEP by itself would resist attack, an opponent might be able to exploit a weakness in the implementation of RSAES-PKCS1-v1_5 to recover messages encrypted with either scheme. As another example, suppose an RSA key pair is employed in both RSASSA-PSS (Section 8.1) and RSASSA-PKCS1-v1_5. Then the security proof for RSASSA-PSS would no longer be sufficient since the proof does not account for the possibility that signatures might be generated with a second scheme. Similar considerations may apply if an RSA key pair is employed in one of the schemes defined here and in a variant defined elsewhere.
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