NISTIR 8105

Report on Post-Quantum Cryptography
Lily Chen
Stephen Jordan
Yi-Kai Liu
Dustin Moody
Rene Peralta
Ray Perlner
Daniel Smith-Tone

This publication is available free of charge from:http://dx.doi.org/10.6028/NIST.IR.8105

NISTIR 8105

Report on Post-Quantum Cryptography
Lily Chen
Stephen Jordan
Yi-Kai Liu
Dustin Moody
Rene Peralta
Ray Perlner
Daniel Smith-Tone
Computer Security Division
Applied and Computational Mathematics Division
Information Technology Laboratory

This publication is available free of charge from:http://dx.doi.org/10.6028/NIST.IR.8105

April 2016

U.S. Department of Commerce
Penny Pritzker, Secretary
National Institute of Standards and Technology
Willie May, Under Secretary of Commerce for Standards and Technology and Director

National Institute of Standards and Technology Internal Report 8105
15 pages (April 2016)
This publication is available free of charge from:http://dx.doi.org/10.6028/NIST.IR.8105
Certain commercial entities, equipment, or materials may be identified in this document in order to describe an
experimental procedure or concept adequately. Such identification is not intended to imply recommendation or
endorsement by NIST, nor is it intended to imply that the entities, materials, or equipment are necessarily the best
available for the purpose.
There may be references in this publication to other publications currently under development by NIST in accordancewith its assigned statutory responsibilities. The information in this publication, including concepts and methodologies,
may be used by federal agencies even before the completion of such companion publications. Thus, until eachpublication is completed, current requirements, guidelines, and procedures, where they exist, remain operative. For
planning and transition purposes, federal agencies may wish to closely follow the development of these new
publications by NIST.
Organizations are encouraged to review all draft publications during public comment periods and provide feedback to
NIST. Many NIST cybersecurity publications, other than the ones noted above, are available at
http://csrc.nist.gov/publications.

Comments on this publication may be submitted to:
National Institute of Standards and Technology
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All comments are subject to release under the Freedom of Information Act (FOIA).

Reports on Computer Systems Technology
The Information Technology Laboratory (ITL) at the National Institute of Standards andTechnology (NIST) promotes the U.S. economy and public welfare by providing technicalleadership for the Nation’s measurement and standards infrastructure. ITL develops tests, test
methods, reference data, proof of concept implementations, and technical analyses to advance thedevelopment and productive use of information technology. ITL’s responsibilities include thedevelopment of management, administrative, technical, and physical standards and guidelines forthe cost-effective security and privacy of other than national security-related information in federal
information systems.

Abstract
In recent years, there has been a substantial amount of research on quantum computers –machines that exploit quantum mechanical phenomena to solve mathematical problems that are
difficult or intractable for conventional computers. If large-scale quantum computers are everbuilt, they will be able to break many of the public-key cryptosystems currently in use. This
would seriously compromise the confidentiality and integrity of digital communications on theInternet and elsewhere. The goal of post-quantum cryptography (also called quantum-resistantcryptography) is to develop cryptographic systems that are secure against both quantum andclassical computers, and can interoperate with existing communications protocols and networks.
This Internal Report shares the National Institute of Standards and Technology (NIST)’s current
understanding about the status of quantum computing and post-quantum cryptography, andoutlines NIST’s initial plan to move forward in this space. The report also recognizes thechallenge of moving to new cryptographic infrastructures and therefore emphasizes the need foragencies to focus on crypto agility.

Keywords
post-quantum cryptography; public key cryptography; quantum computing; quantum-resistant;quantum-safe.

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Table of Contents
1

Introduction ............................................................................................................ 1

2

An Overview of Quantum-Resistant Cryptography............................................. 3

3

Progress in Quantum Computing Hardware........................................................ 5

4

The Path Forward ................................................................................................... 6
List of Appendices

Appendix A— References ............................................................................................ 8iii

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Introduction

In the last three decades, public key cryptography has become an indispensable component ofour global communication digital infrastructure. These networks support a plethora ofapplications that are important to our economy, our security, and our way of life, such as mobile
phones, internet commerce, social networks, and cloud computing. In such a connected world,
the ability of individuals, businesses and governments to communicate securely is of the utmost
importance.
Many of our most crucial communication protocols rely principally on three core cryptographic
functionalities: public key encryption, digital signatures, and key exchange 1. Currently, thesefunctionalities are primarily implemented using Diffie-Hellman key exchange, the RSA (RivestShamir-Adleman) cryptosystem, and elliptic curve cryptosystems. The security of these depends
on the difficulty of certain number theoretic problems such as Integer Factorization or the
Discrete Log Problem over various groups.
In 1994, Peter Shor of Bell Laboratories showed that quantum computers, a new technology
leveraging the physical properties of matter and energy to perform calculations, can efficiently
solve each of these problems, thereby rendering all public key cryptosystems based on suchassumptions impotent [1]. Thus a sufficiently powerful quantum computer will put many formsof modern communication—from key exchange to encryption to digital authentication—in peril.
The discovery that quantum computers could be utilized to solve certain problems faster thanclassical computers has inspired great interest in quantum computing. Is quantum complexity
fundamentally different from classical complexity? When will large-scale quantum computers bebuilt? Is there a way to resist both a quantum and a classical computing adversary? Researchers
are working on these questions.
In the twenty years since Shor’s discovery, the theory of quantum algorithms has developed
significantly. Quantum algorithms achieving exponential speedup have been discovered forseveral problems relating to physics simulation, number theory, and topology. Nevertheless, thelist of problems admitting exponential speedup by quantum computation remains relatively
small. In contrast, more modest speedups have been developed for broad classes of problems
related to searching, collision finding, and evaluation of Boolean formulae. In particular,
Grover’s search algorithm proffers a quadratic speedup on unstructured search problems. While
such a speedup does not render cryptographic technologies obsolete, it can have the effect of
requiring larger key sizes, even in the symmetric key case. See Table 1 for a summary of the
impact of large-scale quantum computers on common cryptographic algorithms, such as RSA
and the Advanced Encryption Standard (AES). It is not known how far these quantum
advantages can be pushed, nor how wide is the gap between feasibility in the classical andquantum models.

1 NIST standardized digital signature schemes in [FIPS 186-4], as well as public key-based key establishment schemes in[SP800-56A] (using key exchange) and [SP800-56B] (using public key encryption).

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The question of when a large-scale quantum computer will be built is complicated andcontentious. While in the past it was less clear that large quantum computers are a physicalpossibility, many scientists now believe it to be merely a significant engineering challenge. Someexperts even predict that within the next 20 or so years, sufficiently large quantum computerswill be built to break essentially all public key schemes currently in use [2]. It has taken almost
20 years to deploy our modern public key cryptography infrastructure. It will take significant
effort to ensure a smooth and secure migration from the current widely used cryptosystems totheir quantum computing resistant counterparts. Therefore, regardless of whether we canestimate the exact time of the arrival of the quantum computing era, we must begin now to
prepare our information security systems to be able to resist quantum computing.
Table 1 - Impact of Quantum Computing on Common Cryptographic Algorithms
Cryptographic Algorithm

Type

Purpose

Impact from large-scalequantum computer

AES

Symmetric key

Encryption

Larger key sizes needed

SHA-2, SHA-3---------------Hash functions

Larger output needed

RSA

Public key

Signatures, key
establishment

No longer secure

ECDSA, ECDH
(Elliptic Curve
Cryptography)

Public key

Signatures, key
exchange

No longer secure

DSA(Finite Field Cryptography)

Public key

Signatures, key
exchange

No longer secure

A large international community has emerged to address the issue of information security in aquantum computing future, in the hope that our public key infrastructure may remain intact byutilizing new quantum-resistant primitives. In the academic world, this new science bears thename “Post-Quantum Cryptography 2.” This is an active area of research, with its own conference
series, PQCrypto, which started in 2006. It has received substantial support from national
funding agencies, most notably in Europe and Japan, through the European Union (EU) projects
PQCrypto and SAFEcrypto, and the CREST Crypto-Math project in Japan.
These efforts have led to advances in fundamental research, paving the way for the deploymentof post-quantum cryptosystems in the real world. In the past few years, industry and standards

2 Post-quantum cryptography should not be conflated with quantum cryptography (or quantum key-distribution), which uses

properties of quantum mechanics to create a secure communication channel. This report is only concerned with post-quantum
cryptography.

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NISTIR 8105

Report on Post-Quantum Cryptographyorganizations have started their own activities in this field: since 2013, the European
Telecommunications Standards Institute (ETSI) has held three “Quantum-Safe Cryptography"
workshops, and in 2015 NIST held a workshop on “Cybersecurity in a Post-Quantum World,”
which was attended by over 140 people from government, industry, and academia.
NIST has a unique role to play in standardizing post-quantum cryptography, as part of its broader
responsibility for the development of standards and guidelines for the protection of non-nationalsecurity federal information systems. Many NIST standards, such as the Advanced Encryption
Standard (AES), have been developed with broad participation from academia and industry, andhave been widely adopted because they are effective solutions, thus helping to protect U.S.
information and information systems. NIST standardization of post-quantum cryptography will
likely provide similar benefits.
Considering all of these sources, it is clear that the effort to develop quantum-resistanttechnologies is intensifying. Equally clear is the urgency, implied by these investments, of theneed for standardizing new post-quantum public key cryptography. It is critical to engage withthe community for NIST cryptographic standards to be endorsed by industry and other standards
organizations around the world. This Internal Report shares NIST’s current understanding about
the status of quantum computing and post-quantum cryptography, and outlines our initial plan tomove forward.

2

An Overview of Quantum-Resistant Cryptography

The most important uses of public key cryptography today are for digital signatures and key
establishment. As mentioned in Section 1, the construction of a large-scale quantum computer
would render many of these public key cryptosystems insecure. In particular, this includes those
based on the difficulty of integer factorization, such as RSA, as well as ones based on thehardness of the discrete log problem. In contrast, the impact on symmetric key systems will not
be as drastic (see Table 1). Grover’s algorithm provides a quadratic speed-up for quantum
search algorithms in comparison with search algorithms on classical computers. We don’t knowthat Grover’s algorithm will ever be practically relevant, but if it is, doubling the key size will be
sufficient to preserve security. Furthermore, it has been shown that an exponential speed up forsearch algorithms is impossible, suggesting that symmetric algorithms and hash functions should
be usable in a quantum era [3].
Consequently, the search for algorithms believed to be resistant to attacks from both classicaland quantum computers has focused on public key algorithms. In this section, we briefly give an
overview of the main families for which post-quantum primitives have been proposed. Thesefamilies include those based on lattices, codes, and multivariate polynomials, as well as ahandful of others. For further information, see [4, 5].
Lattice-based cryptography – Cryptosystems based on lattice problems have received renewedinterest, for a few reasons. Exciting new applications (such as fully homomorphic encryption,code obfuscation, and attribute-based encryption) have been made possible using lattice-based
cryptography. Most lattice-based key establishment algorithms are relatively simple, efficient,3

NISTIR 8105

Report on Post-Quantum Cryptography

and highly parallelizable. Also, the security of some lattice-based systems are provably secure
under a worst-case hardness assumption, rather than on the average case. On the other hand, it
has proven difficult to give precise estimates of the security of lattice schemes against even
known cryptanalysis techniques.
Code-based cryptography – In 1978, the McEliece cryptosystem was first proposed, and has
not been broken since. Since that time, other systems based on error-correcting codes have been
proposed. While quite fast, most code-based primitives suffer from having very large key sizes.
Newer variants have introduced more structure into the codes in an attempt to reduce the key
sizes, however the added structure has also led to successful attacks on some proposals. While
there have been some proposals for code-based signatures, code-based cryptography has seen
more success with encryption schemes.
Multivariate polynomial cryptography – These schemes are based on the difficulty of solving
systems of multivariate polynomials over finite fields. Several multivariate cryptosystems havebeen proposed over the past few decades, with many having been broken [6]. While there havebeen some proposals for multivariate encryption schemes, multivariate cryptography hashistorically been more successful as an approach to signatures.
Hash-based signatures – Hash-based signatures are digital signatures constructed using hashfunctions. Their security, even against quantum attacks, is well understood. Many of the more
efficient hash-based signature schemes have the drawback that the signer must keep a record of
the exact number of previously signed messages, and any error in this record will result in
insecurity. Another drawback is that they can produce only a limited number of signatures. The
number of signatures can be increased, even to the point of being effectively unlimited, but thisalso increases the signature size.
Other - A variety of systems have been proposed which do not fall into the above families. One
such proposal is based on evaluating isogenies on supersingular elliptic curves. While thediscrete log problem on elliptic curves can be efficiently solved by Shor’s algorithm on aquantum computer, the isogeny problem on supersingular curves has no similar quantum attack
known. Like some other proposals, for example those based on the conjugacy search problem
and related problems in braid groups, there has not been enough analysis to have muchconfidence in their security.

It seems improbable that any of the currently known algorithms can serve as a drop-in
replacement for what is in use today. One challenge that will likely need to be overcome is thatmost of the quantum-resistant algorithms have larger key sizes than the algorithms they will
replace. This may result in needing to change various Internet protocols, such as the Transport
Layer Security (TLS) protocol, or the Internet Key Exchange (IKE). The ways in which thisshould be done must be carefully considered.
We note that none of the above proposals have been shown to guarantee security against all
quantum attacks. A new quantum algorithm may be discovered which breaks some of theseschemes. However, this is similar to the state today. Although most public key cryptosystems
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Report on Post-Quantum Cryptographycome with a security proof, these proofs are based on unproven assumptions. Thus the lack of
known attacks is used to justify the security of public key cryptography currently in use.
Nonetheless, NIST believes that more research and analysis are needed before any of the above
proposed post-quantum algorithms could be recommended for use today. They have not received
nearly as much scrutiny from the cryptographic community as the currently deployed algorithms.
One exception is hash-based signatures, whose security is well-understood. For certain specificapplications, such as digital code signing, hash-based signatures could potentially be
standardized in the next few years.

3

Progress in Quantum Computing Hardware

Research into the feasibility of building large-scale quantum computers began in earnest after
Peter Shor’s 1994 discovery of a polynomial-time quantum algorithm for integer factorization[1]. At the time, it was unclear whether quantum computing would ever be a fundamentally
scalable technology. Many leading experts suggested that quantum states were too fragile andsubject to the accumulation of error for large-scale quantum computation ever to be realized.
This situation changed in the late 1990s with the development of quantum error correcting codes
and threshold theorems [7]. These threshold theorems show that if the error rate per logicaloperation (“quantum gate”) in a quantum computer can be brought below a fixed threshold thenarbitrarily long quantum computations can be carried out in a reliable and fault-tolerant mannerby incorporating error-correction steps throughout the execution of the quantum computation [8].
Over the years, experimentalists have gradually developed improved hardware with ever lower
error rates per quantum gate. Simultaneously, theorists have developed new quantum error
correction procedures yielding higher fault-tolerance thresholds. Recently, some experiments
using ion traps and superconducting circuits have demonstrated universal sets of quantum gatesthat are nominally below the highest theoretical fault-tolerance thresholds (around 1 %) [9, 10].
This is a significant milestone, which has spurred increased investment from both government
and industry. However, it is clear that substantial long-term efforts are needed to move frompresent-day laboratory demonstrations involving a few qubits up to large-scale quantum
computers involving thousands of logical qubits encoded in perhaps hundreds of thousands or
millions of physical qubits.
In parallel to the development of general-purpose digital quantum computers, there have been
efforts to develop special purpose analog quantum computers, such as quantum annealers (e.g.
the D-Wave machine), analog quantum simulators, and boson sampling devices. Some of thesedevices have been scaled up to far larger numbers of qubits than digital quantum computers
have. However, due to their specialized nature, these analog quantum devices are not believed to
be of relevance to cryptanalysis.

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Report on Post-Quantum Cryptography

The Path Forward

The need for stronger cryptography is driven by advances in both classical and quantum
computing technologies. To maintain security against classical attacks, NIST has alreadyrecommended transitions from key sizes and algorithms that provide 80 bits of security, to key
sizes and algorithms that provide 112 or 128 bits of security [SP 800-131A]. To provide security
against quantum attacks, NIST will have to facilitate a more difficult transition, to new postquantum cryptosystems.
It is unclear when scalable quantum computers will be available. However, in the past year or so,
researchers working on building a quantum computer have estimated that it is likely that aquantum 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 [11]. This is a serious long-term threat to thecryptosystems currently standardized by NIST.
It is useful to compare the above predictions with the cost of breaking these cryptosystems usingclassical computers. Cryptosystems offering 80 bits of security or less, which were phased out in2011-2013, are at the greatest risk: they can be broken now at a cost ranging from tens of
thousands to hundreds of millions of dollars [12, 13, 14, 15]. Cryptosystems offering 112 bits of
security are likely to remain secure for some time: they may be breakable for a budget of abillion dollars in 30 to 40 years 3 (using classical computers).
Thus, transitioning from 112 to 128 (or higher) bits of security is perhaps less urgent thantransitioning from existing cryptosystems to post-quantum cryptosystems. This post-quantum
transition raises many fundamental challenges.
Previous transitions from weaker to stronger cryptography have been based on the bits-ofsecurity paradigm, which measures the security of an algorithm based on the time-complexity of
attacking it with a classical computer (e.g. an algorithm is said to have 128 bits of security if the
difficulty of attacking it with a classical computer is comparable to the time and resources
required to brute-force search for a 128-bit cryptographic key.) NIST Special Publication (SP)
800-57 Part 1 [SP800-57] classifies the algorithms standardized by NIST as of January 2016 into
80, 112, 128, 192 and 256 bits of security. It further recommended that the 80-bit security levelis no longer considered sufficiently secure, and the 112-bit security level be phased out by 2031.
Unfortunately, the bits-of-security paradigm does not take into account the security of algorithms
against quantum cryptanalysis, so it is inadequate to guide our transition to quantum-resistantcryptography. There is not yet a consensus view on what key lengths will provide acceptablelevels of security against quantum attacks. For symmetric key systems, one simple heuristic is to
double the key lengths to compensate for the quadratic speedup achieved by Grover’s algorithm.
But this recommendation may be overly conservative, as quantum computing hardware will
likely be more expensive to build than classical hardware. At the same time, thisrecommendation does not take into account the possibility of more sophisticated quantum attacks

3 This is based on an extrapolation assuming Moore’s law, assuming roughly 90 bits of security is breakable now for one billion

dollars and 18 months per bit of security at a given cost.

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NISTIR 8105

Report on Post-Quantum Cryptography[16, 17, 18]. Our understanding of quantum cryptanalysis remains rather limited, and more
research in this area is urgently needed.
The development of standards for post-quantum cryptography will require significant resources
to analyze candidate quantum-resistant schemes, and will require significant public engagement
to assure trust in the algorithms NIST chooses to standardize. Interest in the areas of quantum
computing and quantum-resistant cryptography has recently increased, due to milestones in thedevelopment of quantum computing hardware and the National Security Agency’s (NSA) recentchanges to its Suite B guidance [19]. This provides an opportunity for engagement with theresearch community that may not come again before practical quantum computing is truly
imminent. Consequently, NIST is beginning to prepare for the transition to quantum-resistantcryptography now.
NIST is taking the following steps to initiate a standardization effort in post-quantum
cryptography. NIST plans to specify preliminary evaluation criteria for quantum-resistant publickey cryptography standards. The criteria will include security and performance requirements.
The draft criteria will be released for public comments in 2016 and hopefully finalized by theend of the year. At that time NIST will begin accepting proposals for quantum-resistant public
key encryption, digital signature, and key exchange algorithms. NIST intends to select at least
one algorithm providing each of these functionalities for standardization. NIST will establish asubmission deadline late in 2017 for algorithms to be considered, allowing the proposals to be
subject to 3 to 5 years of public scrutiny before they are standardized.
While this process will have many commonalities with the processes that led to thestandardization of AES [20] and SHA3 [21], this is not a competition. NIST sees its role as
managing a process of achieving community consensus in a transparent and timely manner.
Ideally, several algorithms will emerge as “good choices.” NIST may pick one or more of thesefor standardization in each category. In this respect, NIST’s process for standardizing quantumresistant public key cryptography will be similar to the ongoing block cipher modes development
process [22].
When standards for quantum-resistant public key cryptography become available, NIST willreassess the imminence of the threat of quantum computers to existing standards, and may decideto deprecate or withdraw the affected standards thereafter as a result. Agencies should therefore
be prepared to transition away from these algorithms as early as 10 years from now. As thereplacements for currently standardized public key algorithms are not yet ready, a focus on
maintaining crypto agility is imperative. Until new quantum-resistant algorithms are
standardized, agencies should continue to use the recommended algorithms currently specified in
NIST standards.

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Appendix A—References[FIPS 186-4]

Federal Information Processing Standards (FIPS) 186-4, Digital SignatureStandard (DSS), National Institute of Standards and Technology,
Gaithersburg, Maryland, July 2013, 130pp.
http://dx.doi.org/10.6028/nist.fips.186-4.

[SP800-56A]

NIST Special Publication (SP) 800-56A Revision 2, Recommendations for
Pair-Wise Key-Establishment Schemes Using Discrete Logarithm
Cryptography, National Institute of Standards and Technology,
Gaithersburg, Maryland, May 2013, 138pp.
http://dx.doi.org/10.6028/nist.sp.800-56ar2.

[SP800-56B]

NIST Special Publication (SP) 800-56B Revision 1, Recommendations for
Pair-Wise Key-Establishment Schemes Using Integer Factorization
Cryptography, National Institute of Standards and Technology,
Gaithersburg, Maryland, September 2014, 131p.
http://dx.doi.org/10.6028/nist.sp.800-56br1.

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Report on Post-Quantum Cryptography[7]

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[SP 800131A]NIST Special Publication (SP) 800-131A Revision 1, Transitions:
Recommendation for Transitioning the Use of Cryptographic Algorithms and
Key Lengths, National Institute of Standards and Technology, Gaithersburg,
Maryland, November 2015, 23pp.
http://dx.doi.org/10.6028/nist.sp.800-131ar1.

[11]

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Talk PQCrypto 2014, October 2014 Waterloo, Canada.
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Leyland, Factoring Estimates for a 1024-bit RSA Modulus, Advances in
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security of 1024-bit RSA and 160-bit Elliptic Curve Cryptography, IACR
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Report on Post-Quantum Cryptography[SP 800-57]

NIST Special Publication (SP) 800-57 Part 1 Revision 4, Recommendation
for Key Management – Part 1: General, National Institute of Standards andTechnology, Gaithersburg, Maryland, January 2016, 160pp.
http://dx.doi.org/10.6028/NIST.SP.800-57pt1r4.

[16]

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[20]

NIST, AES Competition [Web page], http://csrc.nist.gov/archive/aes/.

[21]

NIST, SHA-3 Competition [Web page],
http://csrc.nist.gov/groups/ST/hash/sha-3/.

[22]

NIST, Modes Development [Web page],
http://csrc.nist.gov/groups/ST/toolkit/BCM/modes_development.html.

10