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 accordance 

with 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 each 

publication 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

Attn: Computer Security Division, Information Technology Laboratory

100 Bureau Drive (Mail Stop 8930) Gaithersburg, MD 20899-8930

Email: NISTIR8105-comments@nist.gov 

All comments are subject to release under the Freedom of Information Act (FOIA).
ii

Reports on Computer Systems Technology

The Information Technology Laboratory (ITL) at the National Institute of Standards and 

Technology (NIST) promotes the U.S. economy and public welfare by providing technical 

leadership for the Nation’s measurement and standards infrastructure. ITL develops tests, test 

methods, reference data, proof of concept implementations, and technical analyses to advance the 

development and productive use of information technology. ITL’s responsibilities include the 

development of management, administrative, technical, and physical standards and guidelines for 

the 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 ever 

built, 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 the 

Internet and elsewhere. The goal of post-quantum cryptography (also called quantum-resistant 

cryptography) is to develop cryptographic systems that are secure against both quantum and 

classical 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, and

outlines NIST’s initial plan to move forward in this space. The report also recognizes the 

challenge of moving to new cryptographic infrastructures and therefore emphasizes the need for 

agencies to focus on crypto agility.

Keywords

post-quantum cryptography; public key cryptography; quantum computing; quantum-resistant; 

quantum-safe.
NISTIR 8105 Report on Post-Quantum Cryptography

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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 ............................................................................................ 8
NISTIR 8105 Report on Post-Quantum Cryptography

1

1 Introduction

In the last three decades, public key cryptography has become an indispensable component of 

our global communication digital infrastructure. These networks support a plethora of 

applications 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 exchange1. Currently, these 

functionalities 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 such 

assumptions impotent [1]. Thus a sufficiently powerful quantum computer will put many forms 

of 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 than 

classical computers has inspired great interest in quantum computing. Is quantum complexity 

fundamentally different from classical complexity? When will large-scale quantum computers be 

built? 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 for 

several problems relating to physics simulation, number theory, and topology. Nevertheless, the 

list 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 and 

quantum 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).
NISTIR 8105 Report on Post-Quantum Cryptography

2

The question of when a large-scale quantum computer will be built is complicated and 

contentious. While in the past it was less clear that large quantum computers are a physical 

possibility, many scientists now believe it to be merely a significant engineering challenge. Some 

experts even predict that within the next 20 or so years, sufficiently large quantum computers 

will 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 to 

their quantum computing resistant counterparts. Therefore, regardless of whether we can 

estimate 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-scale 

quantum 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 a 

quantum computing future, in the hope that our public key infrastructure may remain intact by 

utilizing new quantum-resistant primitives. In the academic world, this new science bears the 

name “Post-Quantum Cryptography2

.” 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 deployment 

of 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.
NISTIR 8105 Report on Post-Quantum Cryptography

3

organizations 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, and 

have 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-resistant 

technologies is intensifying. Equally clear is the urgency, implied by these investments, of the 

need for standardizing new post-quantum public key cryptography. It is critical to engage with 

the 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 to 

move 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 the 

hardness 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 know 

that 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 for 

search 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 classical 

and 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. These 

families include those based on lattices, codes, and multivariate polynomials, as well as a 

handful of others. For further information, see [4, 5].

Lattice-based cryptography – Cryptosystems based on lattice problems have received renewed 

interest, 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, 
NISTIR 8105 Report on Post-Quantum Cryptography

4

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 have 

been proposed over the past few decades, with many having been broken [6]. While there have 

been some proposals for multivariate encryption schemes, multivariate cryptography has 

historically been more successful as an approach to signatures. 

Hash-based signatures – Hash-based signatures are digital signatures constructed using hash 

functions. 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 this 

also 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 the 

discrete log problem on elliptic curves can be efficiently solved by Shor’s algorithm on a 

quantum 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 much 

confidence 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 that 

most 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 this 

should 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 these 

schemes. However, this is similar to the state today. Although most public key cryptosystems 
NISTIR 8105 Report on Post-Quantum Cryptography

5

come 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 specific 

applications, 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 and 

subject 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 logical 

operation (“quantum gate”) in a quantum computer can be brought below a fixed threshold then 

arbitrarily long quantum computations can be carried out in a reliable and fault-tolerant manner 

by 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 gates 

that 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 from 

present-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 these 

devices 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.
NISTIR 8105 Report on Post-Quantum Cryptography

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4 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 already 

recommended 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 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 [11]. This is a serious long-term threat to the 

cryptosystems currently standardized by NIST.

It is useful to compare the above predictions with the cost of breaking these cryptosystems using 

classical computers. Cryptosystems offering 80 bits of security or less, which were phased out in 

2011-2013, 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 a 

billion dollars in 30 to 40 years3 (using classical computers).

Thus, transitioning from 112 to 128 (or higher) bits of security is perhaps less urgent than 

transitioning 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 level 

is 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-resistant 

cryptography. There is not yet a consensus view on what key lengths will provide acceptable 

levels 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, this 

recommendation 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.
NISTIR 8105 Report on Post-Quantum Cryptography

7

[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 the 

development of quantum computing hardware and the National Security Agency’s (NSA) recent 

changes to its Suite B guidance [19]. This provides an opportunity for engagement with the 

research community that may not come again before practical quantum computing is truly 

imminent. Consequently, NIST is beginning to prepare for the transition to quantum-resistant 

cryptography 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 public 

key 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 the 

end 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 a 

submission 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 the 

standardization 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 these 

for 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 will 

reassess the imminence of the threat of quantum computers to existing standards, and may decide 

to 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 the 

replacements 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.
NISTIR 8105 Report on Post-Quantum Cryptography

8

Appendix A—References

[FIPS 186-4] Federal Information Processing Standards (FIPS) 186-4, Digital Signature 

Standard (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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Logarithms on a Quantum Computer, SIAM J. Comput., 26 (5), 1997, pp. 

1484–1509. http://dx.doi.org/10.1137/s0036144598347011.

[2] M. Mosca, Cybersecurity in an era with quantum computers: will we be 

ready? IACR Cryptology ePrint Archive Report 2015/1075, 2015. 

http://eprint.iacr.org/2015/1075.

[3] C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani, Strengths and 

weaknesses of quantum computing, SIAM J. Comput., 26 (5), 1997, pp. 

1510–1523. http://dx.doi.org/10.1137/s0097539796300933

[4] European Telecommunications Standards Institute White Paper No. 8, 

Quantum Safe Cryptography and Security: An Introduction, Benefits, 

Enablers and Challenges, June 2015. 

https://portal.etsi.org/Portals/0/TBpages/QSC/Docs/Quantum_Safe_Whitepa

per_1_0_0.pdf [accessed 4/15/2016].

[5] R. Perlner and D. Cooper, Quantum resistant public key cryptography: a 

survey, In Proc. of IDtrust, ACM, 2009, pp. 85-93. 

http://dx.doi.org/10.1145/1527017.1527028.

[6] V. Dubois, P. Fouque, A. Shamir and J. Stern, Practical cryptanalysis of 

SFLASH, Advances in Cryptology — CRYPTO 2007, Lecture Notes in 

Comput. Sci. 4622, Springer-Verlag, 2007, pp. 1–12. 

http://dx.doi.org/10.1007/978-3-540-74143-5_1.
NISTIR 8105 Report on Post-Quantum Cryptography

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[7] J. Preskill, Reliable Quantum Computers, Proc. Roy. Soc. London A, 454, 

1998, pp. 385–410. http://dx.doi.org/10.1098/rspa.1998.0167.

[8] D. Lidar, T. Brun, eds., Quantum Error Correction, Cambridge University 

Press, 2013. http://dx.doi.org/10.1017/cbo9781139034807.

[9] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, Y. Chen, B. 

Chiaro, J. Mutus, C. Neil, Superconducting quantum circuits at the surface 

code threshold for fault tolerance, Nature 508 (7497), 2014, pp. 500–503. 

http://dx.doi.org/10.1038/nature13171.

[10] T.P. Harty, D.T.C. Allcock, C.J. Balance, L. Guidoni, H.A. Janacek, N.M. 

Linke, D.N. Stacey, D.M. Lucas, High-Fidelity Preparation, Gates, Memory, 

and Readout of a Trapped-Ion Quantum Bit, Phys. Rev. Lett. 113 (22), 2014. 

http://dx.doi.org/10.1103/PhysRevLett.113.220501.

[SP 800-

131A]

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] M. Mariantoni, Building a Superconducting Quantum Computer, Invited 

Talk PQCrypto 2014, October 2014 Waterloo, Canada. 

https://www.youtube.com/watch?v=wWHAs--HA1c [accessed 4/20/2016].

[12] A. Lenstra, E. Tromer, A. Shamir, W. Kortsmit, B. Dodson, J. Hughes, P. 

Leyland, Factoring Estimates for a 1024-bit RSA Modulus, Advances in 

Cryptology - ASIACRYPT 2003, Lecture Notes in Comput. Sci. 2894, 

Springer-Verlag, 2003, pp. 55–74. http://dx.doi.org/10.1007/978-3-540-

40061-5_4.

[13] M. Stevens, P. Karpman, T Peyrin, Freestart Collision on Full SHA-1, IACR 

Cryptology ePrint Archive 2015/967, 2015. http://eprint.iacr.org/2015/967.

[14] J. Bos, M. Kaihara, T. Kleinjung, A. Lenstra, P. Montgomery, On the 

security of 1024-bit RSA and 160-bit Elliptic Curve Cryptography, IACR 

Cryptology ePrint Archive 2009/389, 2009. http://eprint.iacr.org/2009/389.

[15] D. Adrian, K. Bhargavan, Z. Durumeric, P. Gaudry, M. Green, J. A. 

Halderman, N. Heninger, D. Springall, E. Thomé, L. Valenta, B. 

VanderSloot, E. Wustrow, S. Zanella-Béguelin, P. Zimmermann, Imperfect 

Forward Secrecy: How Diffie-Hellman Fails in Practice, in: Proc. of the 

22nd ACM Conference on Computer and Communications Security, Oct. 

2015. http://dx.doi.org/10.1145/2810103.2813707.
NISTIR 8105 Report on Post-Quantum Cryptography

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[SP 800-57] NIST Special Publication (SP) 800-57 Part 1 Revision 4, Recommendation 

for Key Management – Part 1: General, National Institute of Standards and 

Technology, Gaithersburg, Maryland, January 2016, 160pp. 

http://dx.doi.org/10.6028/NIST.SP.800-57pt1r4. 

[16] P. Campbell, M. Groves, D. Shepherd, Soliloquy: A Cautionary Tale, ETSI 

Workshop on Quantum-Safe Cryptography, 2014. 

https://docbox.etsi.org/workshop/2014/201410_CRYPTO/S07_Systems_and

_Attacks/S07_Groves_Annex.pdf.

[17] M. Kaplan, G. Leurent, A. Leverrier, M. Naya-Plasencia, Quantum 

Differential and Linear Cryptanalysis, arXiv preprint ArXiv: 1510.05836, 

2015. http://arxiv.org/abs/1510.05836. 

[18] H. Kuwakado, M. Morii, Quantum distinguisher between the 3-round Feistel 

cipher and the random permutation, In Proc. of 2010 IEEE International 

Symposium on Information Theory (ISIT), IEEE, 2010, pp. 2682-2685. 

http://dx.doi.org/10.1109/isit.2010.5513654.

[19] National Security Agency, Cryptography Today, report, August 2015. 

https://www.nsa.gov/ia/programs/suiteb_cryptography/ [accessed 

4/20/2016]. Also at: https://www.iad.gov/iad/programs/iad-initiatives/cnsasuite.cfm.

[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.