Algorithm Identifiers for HSS and XMSS for Use in the Internet X.509 Public Key Infrastructure
draft-vangeest-x509-hash-sigs-00
This document is an Internet-Draft (I-D).
Anyone may submit an I-D to the IETF.
This I-D is not endorsed by the IETF and has no formal standing in the
IETF standards process.
The information below is for an old version of the document.
Document | Type |
This is an older version of an Internet-Draft whose latest revision state is "Replaced".
|
|
---|---|---|---|
Authors | Daniel Van Geest , Scott Fluhrer | ||
Last updated | 2018-10-10 | ||
Replaced by | draft-gazdag-x509-hash-sigs | ||
RFC stream | (None) | ||
Formats | |||
Stream | Stream state | (No stream defined) | |
Consensus boilerplate | Unknown | ||
RFC Editor Note | (None) | ||
IESG | IESG state | I-D Exists | |
Telechat date | (None) | ||
Responsible AD | (None) | ||
Send notices to | (None) |
draft-vangeest-x509-hash-sigs-00
Network Working Group D. Van Geest Internet-Draft ISARA Corporation Intended status: Standards Track S. Fluhrer Expires: April 13, 2019 Cisco Systems October 10, 2018 Algorithm Identifiers for HSS and XMSS for Use in the Internet X.509 Public Key Infrastructure draft-vangeest-x509-hash-sigs-00 Abstract This document specifies algorithm identifiers and ASN.1 encoding formats for the Hierarchical Signature System (HSS), eXtended Merkle Signature Scheme (XMSS), and XMSS^MT, a multi-tree variant of XMSS. This specification applies to the Internet X.509 Public Key infrastructure (PKI) when digital signatures are used to sign certificates and certificate revocation lists (CRLs). Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at https://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on April 13, 2019. Copyright Notice Copyright (c) 2018 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must Van Geest & Fluhrer Expires April 13, 2019 [Page 1] Internet-Draft Hash-Based Signatures for X.509 October 2018 include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Subject Public Key Algorithms . . . . . . . . . . . . . . . . 3 2.1. HSS Public Keys . . . . . . . . . . . . . . . . . . . . . 3 2.2. XMSS Public Keys . . . . . . . . . . . . . . . . . . . . 4 2.3. XMSS^MT Public Keys . . . . . . . . . . . . . . . . . . . 4 3. Key Usage Bits . . . . . . . . . . . . . . . . . . . . . . . 5 4. Signature Algorithms . . . . . . . . . . . . . . . . . . . . 5 4.1. HSS Signature Algorithm . . . . . . . . . . . . . . . . . 6 4.2. XMSS Signature Algorithm . . . . . . . . . . . . . . . . 6 4.3. XMSS^MT Signature Algorithm . . . . . . . . . . . . . . . 7 5. ASN.1 Module . . . . . . . . . . . . . . . . . . . . . . . . 7 6. Security Considerations . . . . . . . . . . . . . . . . . . . 10 6.1. Algorithm Security Considerations . . . . . . . . . . . . 10 6.2. Implementation Security Considerations . . . . . . . . . 10 7. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 11 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 11 9. References . . . . . . . . . . . . . . . . . . . . . . . . . 11 9.1. Normative References . . . . . . . . . . . . . . . . . . 11 9.2. Informative References . . . . . . . . . . . . . . . . . 12 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 13 1. Introduction The Hierarchical Signature System (HSS) is described in [I-D.mcgrew-hash-sigs]. The eXtended Merkle Signature Scheme (XMSS), and its multi-tree variant XMSS^MT, are described in [RFC8391]. These signature algorithms are based on well-studied Hash Based Signature (HBS) schemes, which can withstand known attacks using quantum computers. They combine Merkle Trees with One Time Signature (OTS) schemes in order to create signature systems which can sign a large but limited number of messages per private key. The private keys are stateful; a key's state must be updated and persisted after signing to prevent reuse of OTS keys. If an OTS key is reused, cryptographic security is not guaranteed for that key. Due to the statefulness of the private key and the limited number of signatures that can be created, these signature algorithms might not be appropriate for use in interactive protocols. While the right selection of algorithm parameters would allow a private key to sign a Van Geest & Fluhrer Expires April 13, 2019 [Page 2] Internet-Draft Hash-Based Signatures for X.509 October 2018 virtually unbounded number of messages (e.g. 2^60), this is at the cost of a larger signature size and longer signing time. Since these algorithms are already known to be secure against quantum attacks, and because roots of trust are generally long-lived and can take longer to be deployed than end-entity certificates, these signature algorithms are more appropriate to be used in root and subordinate CA certificates. They are also appropriate in non-interactive contexts such as code signing. 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. Subject Public Key Algorithms Certificates conforming to [RFC5280] can convey a public key for any public key algorithm. The certificate indicates the algorithm through an algorithm identifier. An algorithm identifier consists of an OID and optional parameters. In this document, we define two new OIDs for identifying the different hash-based signature algorithms. A third OID is defined in [I-D.ietf-lamps-cms-hash-sig] and repeated here for convenience. For all of the OIDs, the parameters MUST be absent. 2.1. HSS Public Keys The object identifier and public key algorithm identifier for HSS is defined in [I-D.ietf-lamps-cms-hash-sig]. The definitions are repeated here for reference. The object identifier for an HSS public key is id-alg-hss-lms- hashsig: id-alg-hss-lms-hashsig OBJECT IDENTIFIER ::= { iso(1) member-body(2) us(840) rsadsi(113549) pkcs(1) pkcs9(9) smime(16) alg(3) 17 } Note that the id-alg-hss-lms-hashsig algorithm identifier is also referred to as id-alg-mts-hashsig. This synonym is based on the terminology used in an early draft of the document that became [I-D.mcgrew-hash-sigs]. The HSS public key's properties are defined as follows: Van Geest & Fluhrer Expires April 13, 2019 [Page 3] Internet-Draft Hash-Based Signatures for X.509 October 2018 pk-HSS-LMS-HashSig PUBLIC-KEY ::= { IDENTIFIER id-alg-hss-lms-hashsig KEY HSS-LMS-HashSig-PublicKey PARAMS ARE absent CERT-KEY-USAGE { digitalSignature, nonRepudiation, keyCertSign, cRLSign } } HSS-LMS-HashSig-PublicKey ::= OCTET STRING [I-D.ietf-lamps-cms-hash-sig] contains more information on the contents and format of an HSS public key. 2.2. XMSS Public Keys The object identifier for an XMSS public key is id-xmss: id-xmss OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmss(13) 0 } The XMSS public key's properties are defined as follows: pk-xmss PUBLIC-KEY ::= { IDENTIFIER id-xmss KEY XMSS-PublicKey PARAMS ARE absent CERT-KEY-USAGE { digitalSignature, nonRepudiation, keyCertSign, cRLSign } } XMSS-PublicKey ::= OCTET STRING The format of an XMSS public key is is formally defined using XDR [RFC4506] and is defined in Appendix B.3 of [RFC8391]. In particular, the first 4 bytes represents the big-ending encoding of the XMSS algorithm type. 2.3. XMSS^MT Public Keys The object identifier for an XMSS^MT public key is id-xmssmt: id-xmssmt OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmssmt(14) 0 } The XMSS^MT public key's properties are defined as follows: Van Geest & Fluhrer Expires April 13, 2019 [Page 4] Internet-Draft Hash-Based Signatures for X.509 October 2018 pk-xmssmt PUBLIC-KEY ::= { IDENTIFIER id-xmssmt KEY XMSSMT-PublicKey PARAMS ARE absent CERT-KEY-USAGE { digitalSignature, nonRepudiation, keyCertSign, cRLSign } } XMSSMT-PublicKey ::= OCTET STRING The format of an XMSS^MT public key is is formally defined using XDR [RFC4506] and is defined in Appendix C.3 of [RFC8391]. In particular, the first 4 bytes represents the big-ending encoding of the XMSS^MT algorithm type. 3. Key Usage Bits The intended application for the key is indicated in the keyUsage certificate extension. If the keyUsage extension is present in an end-entity certificate that indicates id-xmss or id-xmssmt in SubjectPublicKeyInfo, then the keyUsage extension MUST contain one or both of the following values: nonRepudiation; and digitalSignature. If the keyUsage extension is present in a certification authority certificate that indicates id-xmss or id-xmssmt, then the keyUsage extension MUST contain one or more of the following values: nonRepudiation; digitalSignature; keyCertSign; and cRLSign. [I-D.ietf-lamps-cms-hash-sig] defines the key usage for id-alg-hss- lms-hashsig, which is the same as for the keys above. 4. Signature Algorithms Certificates and CRLs conforming to [RFC5280] may be signed with any public key signature algorithm. The certificate or CRL indicates the algorithm through an algorithm identifier which appears in the signatureAlgorithm field within the Certificate or CertificateList. This algorithm identifier is an OID and has optionally associated parameters. This section identifies algorithm identifiers that MUST be used in the signatureAlgorithm field in a Certificate or CertificateList. Van Geest & Fluhrer Expires April 13, 2019 [Page 5] Internet-Draft Hash-Based Signatures for X.509 October 2018 Signature algorithms are always used in conjunction with a one-way hash function. This section identifies OIDs for HSS, XMSS, and XMSS^MT. When these algorithm identifiers appear in the algorithm field as an AlgorithmIdentifier, the encoding MUST omit the parameters field. That is, the AlgorithmIdentifier SHALL be a SEQUENCE of one component, one of the OIDs defined below. The data to be signed (e.g., the one-way hash function output value) is directly signed by the hash-based signature algorithms without any additional formatting necessary. The signature values is a large OCTET STRING. This signature value is then ASN.1 encoded as a BIT STRING and included in the Certificate or CertificateList in the signature field. 4.1. HSS Signature Algorithm The ASN.1 OIDs used to specify that an HSS signature was generated on a SHA-256 or SHA-512 hash of an object are, respectively: hss-with-SHA256 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) hss(12) 2 } hss-with-SHA512 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) hss(12) 1 } [I-D.ietf-lamps-cms-hash-sig] contains more information on the contents and format of an HSS signature. 4.2. XMSS Signature Algorithm The ASN.1 OIDs used to specify that an XMSS signature was generated on a SHA-256 or SHA-512 hash of an object are, respectively: xmss-with-SHA256 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmss(13) 2 } xmss-with-SHA512 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmss(13) 1 } Van Geest & Fluhrer Expires April 13, 2019 [Page 6] Internet-Draft Hash-Based Signatures for X.509 October 2018 The format of an XMSS signature is is formally defined using XDR [RFC4506] and is defined in Appendix B.2 of [RFC8391]. 4.3. XMSS^MT Signature Algorithm The ASN.1 OIDs used to specify that an XMSS^MT signature was generated on a SHA-256 or SHA-512 hash of an object are, respectively: xmssmt-with-SHA256 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmssmt(14) 2 } xmssmt-with-SHA512 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmssmt(14) 1 } The format of an XMSS^MT signature is is formally defined using XDR [RFC4506] and is defined in Appendix C.2 of [RFC8391]. 5. ASN.1 Module For reference purposes, the ASN.1 syntax is presented as an ASN.1 module here. -- ASN.1 Module Hashsigs-pkix-0 -- TBD - IANA assigned module OID DEFINITIONS EXPLICIT TAGS ::= BEGIN IMPORTS PUBLIC-KEY FROM AlgorithmInformation-2009 {iso(1) identified-organization(3) dod(6) internet(1) security(5) mechanisms(5) pkix(7) id-mod(0) id-mod-algorithmInformation-02(58)} ; Van Geest & Fluhrer Expires April 13, 2019 [Page 7] Internet-Draft Hash-Based Signatures for X.509 October 2018 -- -- HSS Signatures -- -- OID for HSS signature generated with SHA-256 hash hss-with-SHA256 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) hss(12) 2 } -- OID for HSS signature generated with SHA-512 hash hss-with-SHA512 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) hss(12) 1 } -- -- XMSS Keys and Signatures -- -- OID for XMSS public keys id-xmss OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmss(13) 0 } -- XMSS Public Keys pk-xmss PUBLIC-KEY ::= { IDENTIFIER id-xmss KEY XMSS-PublicKey PARAMS ARE absent CERT-KEY-USAGE { digitalSignature, nonRepudiation, keyCertSign, cRLSign } } XMSS-PublicKey ::= OCTET STRING -- OID for XMSS signature generated with SHA-256 hash xmss-with-SHA256 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmss(13) 2 } Van Geest & Fluhrer Expires April 13, 2019 [Page 8] Internet-Draft Hash-Based Signatures for X.509 October 2018 -- OID for XMSS signature generated with SHA-512 hash xmss-with-SHA512 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmss(13) 1 } -- -- XMSS^MT Keys and Signatures -- -- OID for XMSS^MT public keys id-xmssmt OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmssmt(14) 0 } -- XMSS^MT Public Keys pk-xmssmt PUBLIC-KEY ::= { IDENTIFIER id-xmssmt KEY XMSSMT-PublicKey PARAMS ARE absent CERT-KEY-USAGE { digitalSignature, nonRepudiation, keyCertSign, cRLSign } } XMSSMT-PublicKey ::= OCTET STRING -- OID for XMSS^MT signature generated with SHA-256 hash xmssmt-with-SHA256 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmssmt(14) 2 } -- OID for XMSS^MT signature generated with SHA-512 hash xmssmt-with-SHA512 OBJECT IDENTIFIER ::= { itu-t(0) identified-organization(4) etsi(0) reserved(127) etsi-identified-organization(0) isara(15) algorithms(1) asymmetric(1) xmssmt(14) 1 } END Van Geest & Fluhrer Expires April 13, 2019 [Page 9] Internet-Draft Hash-Based Signatures for X.509 October 2018 6. Security Considerations 6.1. Algorithm Security Considerations The cryptographic security of the signatures generated by the algorithms mentioned in this document depends only on the hash algorithms used within the signature algorithms and the pre-hash algorithm used to create an X.509 certificate's message digest. Grover's algorithm [Grover96] is a quantum search algorithm which gives a quadratic improvement in search time to brute-force pre-image attacks. The results of [BBBV97] show that this improvement is optimal, however [Fluhrer17] notes that Grover's algorithm doesn't parallelize well. Thus, given a bounded amount of time to perform the attack and using a conservative estimate of the performance of a real quantum computer, the pre-image quantum security of SHA-256 is closer to 190 bits. All parameter sets for the signature algorithms in this document currently use SHA-256 internally and thus have at least 128 bits of quantum pre-image resistance, or 190 bits using the security assumptions in [Fluhrer17]. [Zhandry15] shows that hash collisions can be found using an algorithm with a lower bound on the number of oracle queries on the order of 2^(n/3) on the number of bits, however [DJB09] demonstrates that the quantum memory requirements would be much greater. Therefore a pre-hash using SHA-256 would have at least 128 bits of quantum collision-resistance as well as the pre-image resistance mentioned in the previous paragraph. Given the quantum collision and pre-image resistance of SHA-256 estimated above, the algorithm identifiers hss-with-SHA256, xmss- with-SHA256 and xmssmt-with-SHA256 defined in this document provide 128 bits or more of quantum security. This is believed to be secure enough to protect X.509 certificates for well beyond any reasonable certificate lifetime, although the SHA-512 variants could be used if there are any doubts. The algorithm identifiers hss-with-SHA512, xmss-with-SHA512 and xmssmt-with-SHA512 are defined in order to provide 256 bits of classical security (256 bits of brute-force pre-image resistance with the signature algorithms' SHA-256 and 256 bits of birthday attack collision resistance with the SHA-512 pre-hash). 6.2. Implementation Security Considerations Implementations must protect the private keys. Compromise of the private keys may result in the ability to forge signatures. Along with the private key, the implementation must keep track of which leaf nodes in the tree have been used. Loss of integrity of this Van Geest & Fluhrer Expires April 13, 2019 [Page 10] Internet-Draft Hash-Based Signatures for X.509 October 2018 tracking data can cause an one-time key to be used more than once. As a result, when a private key and the tracking data are stored on non-volatile media or stored in a virtual machine environment, care must be taken to preserve confidentiality and integrity. The generation of private keys relies on random numbers. The use of inadequate pseudo-random number generators (PRNGs) to generate these values can result in little or no security. An attacker may find it much easier to reproduce the PRNG environment that produced the keys, searching the resulting small set of possibilities, rather than brute force searching the whole key space. The generation of quality random numbers is difficult. [RFC4086] offers important guidance in this area. The generation of hash-based signatures also depends on random numbers. While the consequences of an inadequate pseudo-random number generator (PRNGs) to generate these values is much less severe than the generation of private keys, the guidance in [RFC4086] remains important. 7. Acknowledgements This document uses a lot of text from similar documents ([RFC3279] and [RFC8410]) as well as [I-D.ietf-lamps-cms-hash-sig]. Thanks go to the authors of those documents. "Copying always makes things easier and less error prone" - [RFC8411]. 8. IANA Considerations IANA is requested to assign a module OID from the "SMI for PKIX Module Identifier" registry for the ASN.1 module in Section 5. 9. References 9.1. Normative References [I-D.ietf-lamps-cms-hash-sig] Housley, R., "Use of the HSS/LMS Hash-based Signature Algorithm in the Cryptographic Message Syntax (CMS)", draft-ietf-lamps-cms-hash-sig-01 (work in progress), September 2018. [I-D.mcgrew-hash-sigs] McGrew, D., Curcio, M., and S. Fluhrer, "Hash-Based Signatures", draft-mcgrew-hash-sigs-13 (work in progress), September 2018. Van Geest & Fluhrer Expires April 13, 2019 [Page 11] Internet-Draft Hash-Based Signatures for X.509 October 2018 [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, <https://www.rfc-editor.org/info/rfc2119>. [RFC4506] Eisler, M., Ed., "XDR: External Data Representation Standard", STD 67, RFC 4506, DOI 10.17487/RFC4506, May 2006, <https://www.rfc-editor.org/info/rfc4506>. [RFC5280] Cooper, D., Santesson, S., Farrell, S., Boeyen, S., Housley, R., and W. Polk, "Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile", RFC 5280, DOI 10.17487/RFC5280, May 2008, <https://www.rfc-editor.org/info/rfc5280>. [RFC8391] Huelsing, A., Butin, D., Gazdag, S., Rijneveld, J., and A. Mohaisen, "XMSS: eXtended Merkle Signature Scheme", RFC 8391, DOI 10.17487/RFC8391, May 2018, <https://www.rfc-editor.org/info/rfc8391>. 9.2. Informative References [BBBV97] Bennett, C., Bernstein, E., Brassard, G., and U. Vazirani, "Strengths and weaknesses of quantum computing", SIAM J. Comput. 26(5), 1510-1523, 1997. [DJB09] Bernstein, D., "Cost analysis of hash collisions: Will quantum computers make SHARCS obsolete?", SHARCS 9, p. 105, 2009. [Fluhrer17] Fluhrer, S., "Reassessing Grover's Algorithm", Cryptology ePrint Archive Report 2017/811, August 2017, <https://eprint.iacr.org/2017/811.pdf>. [Grover96] Grover, L., "A fast quantum mechanical algorithm for database search", 28th ACM Symposium on the Theory of Computing p. 212, 1996. [RFC3279] Bassham, L., Polk, W., and R. Housley, "Algorithms and Identifiers for the Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile", RFC 3279, DOI 10.17487/RFC3279, April 2002, <https://www.rfc-editor.org/info/rfc3279>. Van Geest & Fluhrer Expires April 13, 2019 [Page 12] Internet-Draft Hash-Based Signatures for X.509 October 2018 [RFC4086] Eastlake 3rd, D., Schiller, J., and S. Crocker, "Randomness Requirements for Security", BCP 106, RFC 4086, DOI 10.17487/RFC4086, June 2005, <https://www.rfc-editor.org/info/rfc4086>. [RFC8410] Josefsson, S. and J. Schaad, "Algorithm Identifiers for Ed25519, Ed448, X25519, and X448 for Use in the Internet X.509 Public Key Infrastructure", RFC 8410, DOI 10.17487/RFC8410, August 2018, <https://www.rfc-editor.org/info/rfc8410>. [RFC8411] Schaad, J. and R. Andrews, "IANA Registration for the Cryptographic Algorithm Object Identifier Range", RFC 8411, DOI 10.17487/RFC8411, August 2018, <https://www.rfc-editor.org/info/rfc8411>. [Zhandry15] Zhandry, M., "A note on the quantum collision and set equality problems", Quantum Information & Computation 15, 7-8, 557-567, May 2015. Authors' Addresses Daniel Van Geest ISARA Corporation 560 Westmount Rd N Waterloo, Ontario N2L 0A9 Canada Email: daniel.vangeest@isara.com Scott Fluhrer Cisco Systems 170 West Tasman Drive San Jose, CA 95134 USA Email: sfluhrer@cisco.com Van Geest & Fluhrer Expires April 13, 2019 [Page 13]