Network Working Group P. Hoffman Internet-Draft ICANN Intended status: Informational July 2, 2017 Expires: January 3, 2018 The Transition from Classical to Post-Quantum Cryptography draft-hoffman-c2pq-01 Abstract Quantum computing is the study of computers that use quantum features in calculations. For over 20 years, it has been known that if large- scale quantum computers could be built, they could have a devastating effect on classical cryptographic algorithms such as RSA and elliptic curve signatures and key exchange, as well as on encryption algorithms. There has already been a great deal of study on how to create algorithms that will resist large-scale quantum computers, but so far, the properties of those algorithms make them onerous to adopt before they are needed. Small-scale quantum computers are being built today, but it is still far from clear when large-scale quantum computers that can be used to break classical algorithms with key sizes commonly used today will be available. It is important to be able to predict when large-scale quantum computers usable for cryptanalysis will be possible so that organization can change to post-quantum cryptographic algorithms well before they are needed. This document describes quantum computing, how it can be used to attack classical cryptographic algorithms, and possibly how to predict when large-scale quantum computers will become feasible. 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 http://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." Hoffman Expires January 3, 2018 [Page 1]

Internet-Draft Classical to Post-Quantum Crypto July 2017 This Internet-Draft will expire on January 3, 2018. Copyright Notice Copyright (c) 2017 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 (http://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 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 . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. Disclaimer . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. Executive Summary . . . . . . . . . . . . . . . . . . . . 3 1.3. Terminology . . . . . . . . . . . . . . . . . . . . . . . 3 1.4. Not Covered: Post-Quantum Cryptographic Algorithms . . . 4 1.5. Not Covered: Quantum Cryptography . . . . . . . . . . . . 5 1.6. Where to Read More . . . . . . . . . . . . . . . . . . . 5 2. Brief Introduction to Quantum Computers . . . . . . . . . . . 5 2.1. Quantum Computers that Discover Cryptographic Keys . . . 6 2.2. Physical Designs for Quantum Computers . . . . . . . . . 6 2.3. Challenges for Physical Designs . . . . . . . . . . . . . 7 2.4. Qubits, Error Detection, and Error Correction . . . . . . 7 3. Quantum Computers and Public Key Cryptography . . . . . . . . 8 3.1. Explanation of Shor's Algorithm . . . . . . . . . . . . . 8 3.2. Properties of Large-Scale Quantum Computers Needed for Discovering Public Keys . . . . . . . . . . . . . . . . . 8 4. Quantum Computers and Symmetric Key Cryptography . . . . . . 9 4.1. Explanation of Grover's Algorithm . . . . . . . . . . . . 10 4.2. Properties of Large-Scale Quantum Computers Needed for Discovering Symmetric Keys . . . . . . . . . . . . . . . 10 5. Predicting When Useful Cryptographic Attacks Will Be Feasible 10 5.1. Proposal: Public Measurements of Various Quantum Technologies . . . . . . . . . . . . . . . . . . . . . . 11 6. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 12 7. Security Considerations . . . . . . . . . . . . . . . . . . . 12 8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 12 9. References . . . . . . . . . . . . . . . . . . . . . . . . . 12 9.1. Normative References . . . . . . . . . . . . . . . . . . 12 9.2. Informative References . . . . . . . . . . . . . . . . . 13 Hoffman Expires January 3, 2018 [Page 2]

Internet-Draft Classical to Post-Quantum Crypto July 2017 Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 13 1. Introduction Early drafts of this document use "@@@@@" to indicate where the editor particularly want input from reviewers. The editor welcomes all types of review, but the areas marked with "@@@@@" are in the most noticeable need of new material. (The editor particularly appreciates new material that comes with references that can be included in this document as well.) 1.1. Disclaimer **** This is an early version of this draft. **** As such, it has had little in-depth review in the cryptography community. Statements in this document might be wrong; given that the entire document is about cryptography, those wrong statements might have significant security problems associated with them. Readers of this document should not rely on any statements in this version of this draft. As the draft gets more input from the cryptography community over time, this disclaimer will be softened and eventually eliminated. 1.2. Executive Summary The development of quantum computers that can break classical cryptographic keys is at a very early stage. None of the published examples of such quantum computers is useful in breaking keys that are in use today. There is a great amount of interest in this development, and researchers expect large strides in this development in the coming decade. Because the world does not know when large-scale quantum computers that can break cryptographic keys will be available, organizations should be watching this area so that they have plenty of time to either change to larger key sizes for classical cryptography or to change to post-quantum algorithms. See Section 5 for a fuller discussion of determining how to predict when large-scale quantum computers might become feasible. 1.3. Terminology The term "classical cryptography" is used to indicate the cryptographic algorithms that are in common use today. In particular, signature and key exchange algorithms that are based on the difficulty of factoring numbers into two large prime numbers, or Hoffman Expires January 3, 2018 [Page 3]

Internet-Draft Classical to Post-Quantum Crypto July 2017 are based on the difficulty of determining the discrete log of a large composite number, are considered classical cryptography. The term "post-quantum cryptography" is the invention and study of encryption, signature and key exchange algorithms that are not based on the difficulty of factoring numbers into two large prime numbers, nor on the difficulty of determining the discrete log of a large composite number. Note that these definitions apply to only one aspect of quantum computing as it relates to cryptography. It is expected that quantum computing will also be able to be used against symmetric key cryptography to make it possible to search for a secret symmetric key using far fewer operations than are needed using classical computers (see Section 4 for more detail). However, using longer keys to thwart that possibility is not normally called "post-quantum cryptography". There are many terms that are only used in the field of quantum computing, such as "qubit", "quantum algorithm", and so on. Chapter 1 of [NielsenChuang] has good definitions of such terms. The "^" symbol is used to indicate "the power of". The term "log" always means "logarithm base 2". 1.4. Not Covered: Post-Quantum Cryptographic Algorithms This document discusses when an organization would want to consider using post-quantum cryptographic algorithms, but definitely does not delve into which of those algorithms would be best to use. Post- quantum cryptography is an active field of research; in fact, it is much more active than the study of when we might want to transition from classical to post-quantum cryptography. Readers interested in post-quantum cryptographic algorithms will have no problem finding many articles proposing such algorithms, comparing the many current proposals, and so on. An excellent starting point is the web site <http://pqcrypto.org/>. Another is the article on post-quantum cryptography at Wikipedia: <https://en.wikipedia.org/wiki/Post-quantum_cryptography>. In addition, various organizations are working on standardizing the algorithms for post-quantum cryptography. For example, the US National Institute of Standards and Technology (commonly just called "NIST") is holding a competition to evaluate post-quantum cryptographic algorithms. NIST's description of that effort is currently at <http://csrc.nist.gov/groups/ST/post-quantum-crypto/>. Hoffman Expires January 3, 2018 [Page 4]

Internet-Draft Classical to Post-Quantum Crypto July 2017 1.5. Not Covered: Quantum Cryptography Outside of this section, this document does not cover "quantum cryptography". The field of quantum cryptography is related to quantum computers, but not to cryptanalysis. Quantum cryptography is used to share random values that cannot be observed by outside parties without discovery. 1.6. Where to Read More There are many reasonably accessible articles on Wikipedia, notably <https://en.wikipedia.org/wiki/Quantum_computing>. @@@@@ Note to the CFRG: please review the various pages at Wikipedia and update them if they are wrong or out of date. Doing so is incredibly helpful to the world. [NielsenChuang] is a well-regarded college textbook on quantum computers. Prerequisites for understanding the book include linear algebra and some quantum physics; however, even without those, a reader can probably get value from the introductory material in the book. @@@@@ Maybe add more references that might be useful to non-experts. 2. Brief Introduction to Quantum Computers A quantum computer is a computer that uses quantum bits (qubits) in quantum circuits to perform calculations. Quantum computers also use classical bits and regular circuits: most calculations in a quantum computer are a mix of classical and quantum bits and circuits. @@@@@ This can be expanded and made less hand-wavy. Qubits are valuable in quantum computers when they are combined in calculations. Combining qubits in a calculation requires that the qubits are correlated. Correlating qubits requires much more effort than correlating classical bits (such as in registers or volatile memory), which is one of the main reasons that developing quantum computers has proven more difficult than early development of classical computers. @@@@@ Discuss measurements and how they have to be done with correlated qubits. Hoffman Expires January 3, 2018 [Page 5]

Internet-Draft Classical to Post-Quantum Crypto July 2017 2.1. Quantum Computers that Discover Cryptographic Keys Quantum computers are expected to be useful in the future for some problems that take up too many resources on a large classical computer. However, this document only discusses how they might be used to discover cryptographic keys faster than classical computers. In order to discover cryptographic keys, a quantum computer needs to have a quantum circuit specifically designed for the type of key it is attempting to break. A quantum computer will need to have a circuit with thousands of qubits to be useful to discover the type and size keys that are in common use today. Smaller quantum computers (those with fewer qubits and simpler circuits) are not useful for using Shor's algorithm (as discussed in Section 3.1) at all. That is, no one has devised a way to combine a bunch of smaller quantum computers to perform the same attacks on cryptographic keys via Shor's algorithm as a properly- sized quantum computer. This is why this document uses the term "large-scale quantum computer" when describing ones that can be used to break keys: there will certainly be small-scale quantum computers built first, but those computers cannot be used to discover the type and size keys that are in common use today. A straight-forward application of Shor's algorithm may not be the only way for large-scale quantum computers to attack RSA keys. [LowResource] describes how to combine quantum computers with classical methods for breaking RSA keys at speeds faster than just using the classical methods. 2.2. Physical Designs for Quantum Computers Quantum computers can be built using many different physical technologies. Deciding which physical technologies are best to pursue is an extremely active research topic. A few physical technologies (particularly trapped ions, super-conduction using Josephson junctions, and nuclear magnetic resonance) are currently getting the most press, but other technologies are also showing promise. @@@@@ It would be useful to have maybe two paragraphs about each physical design that is being actively pursued. Hoffman Expires January 3, 2018 [Page 6]

Internet-Draft Classical to Post-Quantum Crypto July 2017 2.3. Challenges for Physical Designs Different designs have different challenges to overcome before the physical technology can be scaled enough to build a useful large- scale quantum computer. Some of those challenges include the following. (Note that some items on this list apply only to some of the physical technologies Temperature: Getting stable operation without extreme cooling is difficult for many of the proposed technologies. The definition of "extreme" is different for different low-temperature technologies. Stabilization: The length of time every qubit in a circuit holds is value Quantum control: Coherence and reproducibility of qubits Error detection and correction: Getting accurate results through simultaneous detection of bit-flip and phase-flip. See Section 2.4 for a longer description of this. Substrate: The material on which the qubit circuits are built. This has a large effect on the stability of the qubits. Particles: The atoms or sub-atomic particles used to make the qubits Scalability: The ability to handle the number of physical qubits needed for the desired the circuit Architecture: Ability to change quantum gates in a circuit 2.4. Qubits, Error Detection, and Error Correction Researchers building small-scale quantum computers have discovered that correlating qubits often has a large rate of error, and that error increases rapidly over time. Performing quantum calculations such as those needed to break cryptographic keys is not feasible with the current state of physical qubits. Researchers have also discovered that they do not need to rely only on the properties of physical qubits. Instead, they can build "logical qubits" from multiple physical qubits, and these logical qubits have much lower error rates over much longer lifetimes. Currently, it is estimated that it takes hundreds or thousands of physical qubits to make a logical qubit. Hoffman Expires January 3, 2018 [Page 7]

Internet-Draft Classical to Post-Quantum Crypto July 2017 @@@@@ Lots more material should goe here. We will need recent references for how many physical qubits are needed for each corrected qubit. It's OK if this section has lots of references, but hopefully they don't contradict each other. 3. Quantum Computers and Public Key Cryptography The area of quantum computing that has generated the most interest in the cryptographic community is the ability of quantum computers to find the secret keys in the RSA and Diffie-Hellman algorithms using many fewer operations than classical computers would need to use. It is widely believed that factoring large numbers and finding discrete logs using classical computers increases with the exponential size of the key. [RFC3766] describes in detail how classical computers can be used to determine keys; even though that RFC is over a decade old, no significant changes have been made to the process of classical attacks on RSA and Diffie-Hellman. @@@@@ CFRG: is that true? Does RFC 3766 need to be updated? Shor's algorithm shows that these problems can be solved on quantum computers in polynomial time, meaning that the speed of finding the keys is a polynomial function based on the size of the keys, which would require significantly fewer steps than a classical computer. The definitive paper on Shor's algorithm is [Shor97]. 3.1. Explanation of Shor's Algorithm @@@@@ Pointers to understandable articles would be good here. @@@@@ Describe period-finding and why it applies to finding prime factors and discrete logs. @@@@@ Give the steps for applying Shor's algorithm to 2048-bit RSA. Describe how many rounds of the quantum subroutine would likely be needed. Describe how many rounds of the classical loop would likely be needed. @@@@@ Give the steps for applying Shor's algorithm to 256-bit elliptic curves. Describe how many rounds of the quantum subroutine would likely be needed. Describe how many rounds of the classical loop would likely be needed. 3.2. Properties of Large-Scale Quantum Computers Needed for Discovering Public Keys Researchers have built small-scale quantum computers that implement Shor's algorithm, factoring numbers with four or five bits. These Hoffman Expires January 3, 2018 [Page 8]

Internet-Draft Classical to Post-Quantum Crypto July 2017 are used to show that Shor's algorithm is possible to realize in actual hardware. @@@@@ References are needed here. Did they implement all of Shor's algorithm, including the looping logic in the classical part and the looping logic in the quantum part? @@@@@ Numbers and explanation is needed below: A quantum computer that can determine the secret keys for 2048-bit RSA would require SOME NUMBER GOES HERE correlated qubits and SOME NUMBER GOES HERE circuit elements. A quantum computer that can determine the secret keys for 256-bt elliptic curves would require SOME NUMBER GOES HERE correlated qubits and SOME NUMBER GOES HERE circuit elements. 4. Quantum Computers and Symmetric Key Cryptography Section 3 is about Shor's algorithm and compromises to public key cryptography. There is a second quantum computing algorithm, Grover's algorithm, that is often mentioned at the same time as Shor's algorithm but, with respect to cryptanalysis, only applies to symmetric ciphers such as AES. The definitive paper on Grover's algorithm is by Grover: [Grover96]. Grover later wrote a more accessible paper about the algorithm in [QuantumSearch]. Grover's algorithm gives a way to search for keys to symmetric algorithms in the square root of the time that a normal exhaustive search would take. Thus, a large-scale quantum computer that implemented Grover's algorithm could find a secret AES-128 key in about 2^64 steps instead of the 2^128 steps that would be required for a classical computer. When it appears that it is feasible to build a large-scale quantum computer that can defeat a particular symmetric algorithm at a particular key size, the proper response would be to use keys with twice as many bits. That is, if one is using the AES-128 algorithm and there is a concern that an adversary might be able to build a large-scale quantum computer that is designed to attack AES-128 keys, move to an algorithm that has keys twice as long as AES-128, namely AES-256. It is currently expected that large-scale quantum computers that implement Grover's algorithm are expected to be built long before ones that implement Shor's algorithm are. There are two primary reasons for this: Hoffman Expires January 3, 2018 [Page 9]

Internet-Draft Classical to Post-Quantum Crypto July 2017 o Grover's algorithm is likely to be useful in areas other than cryptography. For example, a large-scale quantum computer that implements Grover's algorithm might be used to help create medicines by speeding up complex problems that involve how proteins fold. @@@@@ Add more likely examples and references here. o A large-scale quantum computer that can be used to break AES-128 will likely much smaller (and thus easier to build) than one that implements Shor's algorithm for 256-bit elliptic curves or 2048-bit RSA/DSA keys. 4.1. Explanation of Grover's Algorithm @@@@@ Give the steps for applying Grover's algorithm to AES-128. 4.2. Properties of Large-Scale Quantum Computers Needed for Discovering Symmetric Keys @@@@@ Numbers and explanation is needed below: A quantum computer that can determine the secret keys for AES-128 would require SOME NUMBER GOES HERE correlated qubits and SOME NUMBER GOES HERE circuit elements. @@@@@ <https://arxiv.org/abs/1512.04965> indicates that the quantum part of the computer would have more than 2^80 quantum gates, which might be prohibitive for physical hardware. 5. Predicting When Useful Cryptographic Attacks Will Be Feasible If quantum computers that perform useful cryptographic attacks can be built in the future, many organizations will want to start using post-quantum algorithms well before those computers can be built. However, given how few implementations of such quantum computers exist (even for tiny keys), it is impossible to predict with any accuracy when quantum computers that perform useful cryptographic attacks will be feasible. The term "useful" above is relative to the value of the material being protected by the cryptographic algorithm to the attacker. For example, if the quantum computer attacking a particular key costs US$100 billion to build, costs US$1 billion a year to run, and can extract only one key a year, it is possibly useful to some governments, but probably not useful for attacking the TLS key used to protect a small mail server. On the other hand, if later a similar computer costs US$1 billion to build, costs US$10 million a year to run, and can extract ten keys a year, many more keys become vulnerable. Hoffman Expires January 3, 2018 [Page 10]

Internet-Draft Classical to Post-Quantum Crypto July 2017 [BeReady] gives a simple way to approach the calculation of when one needs to deploy post-quantum algorithms. In short, if the sum of how long you need your keys to be secure plus how long it takes to deploy new algorithms is longer than the length of time it will take for an attacker to create a large-scale quantum computer and use it against your keys, then you waited too long. @@@@@ If the following is wrong, it would be great to have references to replace this with To date, few people have done systematic research that would give estimates for when useful quantum-based cryptographic attacks might be feasible, and at what cost. Without such research, it is easy to make wild guesses but those are not of much value to people having to decide when to start using post-quantum cryptography. For example, in [NIST8105], NIST says "researchers working on building a quantum computer have estimated that it is likely that a quantum computer capable of breaking 2000-bit RSA in a matter of hours could be built by 2030 for a budget of about a billion dollars". However, the referenced link is to a YouTube video [MariantoniYoutube] where the researcher, Matteo Mariantoni, says "maybe you should not quote me on that". [NIST8105] gives no other references for predictions on cost and availability of useful cryptographic attacks with quantum computers. 5.1. Proposal: Public Measurements of Various Quantum Technologies In order to get a rough idea of when useful cryptographic attacks with quantum computers may be feasible, researchers creating such computers can demonstrate them when they can break keys an eighth the size of those in common use. That is, given that 2048-bit RSA, 256-bit elliptic curve, and AES-128 are common today, when a research team has a computer than can break 256-bit RSA, 32-bit elliptic curve, or AES-128 where only 16 bits are unknown, they should demonstrate it. Such a demonstration could easily be made fair with trusted representatives from the cryptographic community using verifiable means to pick the keys to break and verifying the time that it takes to break each key. It might be interesting to run the same tests in classical computers at the same time to give perspective. Note that this proposal would only give an idea of how public progress is being made on quantum computers. Well-funded military agencies (and possibly even criminal enterprises) could be way ahead of the publicly-visible computers. No one should rely on just the Hoffman Expires January 3, 2018 [Page 11]

Internet-Draft Classical to Post-Quantum Crypto July 2017 public measurements when deciding how safe their keys are against quantum computers. 6. IANA Considerations None, and thus this section can be removed at final publication. 7. Security Considerations This entire document is about cryptography, and thus about security. See Section 1.1 for an important disclaimer about this document and security. This document is meant to help the reader predict when to transition from using classical cryptographic algorithms to post-quantum algorithms. That decision is ultimately up to the reader, and must be made not only based on predictions of how quantum computing is progressing but also the value of every key that the user handles. For example, a financial institution using TLS to protect its customers' transactions will probably consider its keys more valuable than a small online store, and will thus be likely to begin the transition earlier. 8. Acknowledgements The list here is meant to acknowledge input to this document. The people listed here do not necessarily agree with ideas presented. Some of the ideas in this document come from Denis Butin and Tomofumi Okubo. 9. References 9.1. Normative References [Grover96] Grover, L., "A fast quantum mechanical algorithm for database search", 1996, <https://arxiv.org/abs/quant- ph/9605043>. [Shor97] Shor, P., "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer", 1997, <http://epubs.siam.org/doi/pdf/10.1137/S0097539795293172>. Hoffman Expires January 3, 2018 [Page 12]

Internet-Draft Classical to Post-Quantum Crypto July 2017 9.2. Informative References [BeReady] Mosca, M., "Cybersecurity in an era with quantum computers: will we be ready?", 2015, <http://eprint.iacr.org/2015/1075.pdf>. [LowResource] Bernstein, D., Fiassse, J., and M. Mosca, "A low-resource quantum factoring algorithm", 2017, <https://eprint.iacr.org/2017/352.pdf>. [MariantoniYoutube] Mariantoni, M., "Building a Superconducting Quantum Computer", 2014, <https://www.youtube.com/watch?v=wWHAs-- HA1c>. [NielsenChuang] Nielsen, M. and I. Chuang, "Quantum Computation and Quantum Information, 10th Anniversary Edition", ISBN 97801-107-00217-3 , 2010. [NIST8105] Chen, L. and et. al, "Report on Post-Quantum Cryptography", 2016, <http://nvlpubs.nist.gov/nistpubs/ir/2016/ NIST.IR.8105.pdf>. [QuantumSearch] Grover, L., "From Schrodinger's Equation to the Quantum Search Algorithm", 2001, <https://arxiv.org/abs/quant- ph/0109116>. [RFC3766] Orman, H. and P. Hoffman, "Determining Strengths For Public Keys Used For Exchanging Symmetric Keys", BCP 86, RFC 3766, DOI 10.17487/RFC3766, April 2004, <http://www.rfc-editor.org/info/rfc3766>. Author's Address Paul Hoffman ICANN Email: paul.hoffman@icann.org Hoffman Expires January 3, 2018 [Page 13]