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Schneier on Security: Blog Entries Tagged quantum computing

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Quantum Computing and Cryptography

Quantum computing is a new way of computing -- one that could allow humankind to perform computations that are simply impossible using today's computing technologies. It allows for very fast searching, something that would break some of the encryption algorithms we use today. And it allows us to easily factor large numbers, something that would break the RSA cryptosystem for any key length.

This is why cryptographers are hard at work designing and analyzing "quantum-resistant" public-key algorithms. Currently, quantum computing is too nascent for cryptographers to be sure of what is secure and what isn't. But even assuming aliens have developed the technology to its full potential, quantum computing doesn't spell the end of the world for cryptography. Symmetric cryptography is easy to make quantum-resistant, and we're working on quantum-resistant public-key algorithms. If public-key cryptography ends up being a temporary anomaly based on our mathematical knowledge and computational ability, we'll still survive. And if some inconceivable alien technology can break all of cryptography, we still can have secrecy based on information theory -- albeit with significant loss of capability.

At its core, cryptography relies on the mathematical quirk that some things are easier to do than to undo. Just as it's easier to smash a plate than to glue all the pieces back together, it's much easier to multiply two prime numbers together to obtain one large number than it is to factor that large number back into two prime numbers. Asymmetries of this kind -- one-way functions and trap-door one-way functions -- underlie all of cryptography.

To encrypt a message, we combine it with a key to form ciphertext. Without the key, reversing the process is more difficult. Not just a little more difficult, but astronomically more difficult. Modern encryption algorithms are so fast that they can secure your entire hard drive without any noticeable slowdown, but that encryption can't be broken before the heat death of the universe.

With symmetric cryptography -- the kind used to encrypt messages, files, and drives -- that imbalance is exponential, and is amplified as the keys get larger. Adding one bit of key increases the complexity of encryption by less than a percent (I'm hand-waving here) but doubles the cost to break. So a 256-bit key might seem only twice as complex as a 128-bit key, but (with our current knowledge of mathematics) it's 340,282,366,920,938,463,463,374,607,431,768,211,456 times harder to break.

Public-key encryption (used primarily for key exchange) and digital signatures are more complicated. Because they rely on hard mathematical problems like factoring, there are more potential tricks to reverse them. So you'll see key lengths of 2,048 bits for RSA, and 384 bits for algorithms based on elliptic curves. Here again, though, the costs to reverse the algorithms with these key lengths are beyond the current reach of humankind.

This one-wayness is based on our mathematical knowledge. When you hear about a cryptographer "breaking" an algorithm, what happened is that they've found a new trick that makes reversing easier. Cryptographers discover new tricks all the time, which is why we tend to use key lengths that are longer than strictly necessary. This is true for both symmetric and public-key algorithms; we're trying to future-proof them.

Quantum computers promise to upend a lot of this. Because of the way they work, they excel at the sorts of computations necessary to reverse these one-way functions. For symmetric cryptography, this isn't too bad. Grover's algorithm shows that a quantum computer speeds up these attacks to effectively halve the key length. This would mean that a 256-bit key is as strong against a quantum computer as a 128-bit key is against a conventional computer; both are secure for the foreseeable future.

For public-key cryptography, the results are more dire. Shor's algorithm can easily break all of the commonly used public-key algorithms based on both factoring and the discrete logarithm problem. Doubling the key length increases the difficulty to break by a factor of eight. That's not enough of a sustainable edge.

There are a lot of caveats to those two paragraphs, the biggest of which is that quantum computers capable of doing anything like this don't currently exist, and no one knows when -- or even if ­- we'll be able to build one. We also don't know what sorts of practical difficulties will arise when we try to implement Grover's or Shor's algorithms for anything but toy key sizes. (Error correction on a quantum computer could easily be an unsurmountable problem.) On the other hand, we don't know what other techniques will be discovered once people start working with actual quantum computers. My bet is that we will overcome the engineering challenges, and that there will be many advances and new techniques­but they're going to take time to discover and invent. Just as it took decades for us to get supercomputers in our pockets, it will take decades to work through all the engineering problems necessary to build large-enough quantum computers.

In the short term, cryptographers are putting considerable effort into designing and analyzing quantum-resistant algorithms, and those are likely to remain secure for decades. This is a necessarily slow process, as both good cryptanalysis transitioning standards take time. Luckily, we have time. Practical quantum computing seems to always remain "ten years in the future," which means no one has any idea.

After that, though, there is always the possibility that those algorithms will fall to aliens with better quantum techniques. I am less worried about symmetric cryptography, where Grover's algorithm is basically an upper limit on quantum improvements, than I am about public-key algorithms based on number theory, which feel more fragile. It's possible that quantum computers will someday break all of them, even those that today are quantum resistant.

If that happens, we will face a world without strong public-key cryptography. That would be a huge blow to security and would break a lot of stuff we currently do, but we could adapt. In the 1980s, Kerberos was an all-symmetric authentication and encryption system. More recently, the GSM cellular standard does both authentication and key distribution -- at scale -- with only symmetric cryptography. Yes, those systems have centralized points of trust and failure, but it's possible to design other systems that use both secret splitting and secret sharing to minimize that risk. (Imagine that a pair of communicants get a piece of their session key from each of five different key servers.) The ubiquity of communications also makes things easier today. We can use out-of-band protocols where, for example, your phone helps you create a key for your computer. We can use in-person registration for added security, maybe at the store where you buy your smartphone or initialize your Internet service. Advances in hardware may also help to secure keys in this world. I'm not trying to design anything here, only to point out that there are many design possibilities. We know that cryptography is all about trust, and we have a lot more techniques to manage trust than we did in the early years of the Internet. Some important properties like forward secrecy will be blunted and far more complex, but as long as symmetric cryptography still works, we'll still have security.

It's a weird future. Maybe the whole idea of number theory­-based encryption, which is what our modern public-key systems are, is a temporary detour based on our incomplete model of computing. Now that our model has expanded to include quantum computing, we might end up back to where we were in the late 1970s and early 1980s: symmetric cryptography, code-based cryptography, Merkle hash signatures. That would be both amusing and ironic.

Yes, I know that quantum key distribution is a potential replacement for public-key cryptography. But come on -- does anyone expect a system that requires specialized communications hardware and cables to be useful for anything but niche applications? The future is mobile, always-on, embedded computing devices. Any security for those will necessarily be software only.

There's one more future scenario to consider, one that doesn't require a quantum computer. While there are several mathematical theories that underpin the one-wayness we use in cryptography, proving the validity of those theories is in fact one of the great open problems in computer science. Just as it is possible for a smart cryptographer to find a new trick that makes it easier to break a particular algorithm, we might imagine aliens with sufficient mathematical theory to break all encryption algorithms. To us, today, this is ridiculous. Public- key cryptography is all number theory, and potentially vulnerable to more mathematically inclined aliens. Symmetric cryptography is so much nonlinear muddle, so easy to make more complex, and so easy to increase key length, that this future is unimaginable. Consider an AES variant with a 512-bit block and key size, and 128 rounds. Unless mathematics is fundamentally different than our current understanding, that'll be secure until computers are made of something other than matter and occupy something other than space.

But if the unimaginable happens, that would leave us with cryptography based solely on information theory: one-time pads and their variants. This would be a huge blow to security. One-time pads might be theoretically secure, but in practical terms they are unusable for anything other than specialized niche applications. Today, only crackpots try to build general-use systems based on one-time pads -- and cryptographers laugh at them, because they replace algorithm design problems (easy) with key management and physical security problems (much, much harder). In our alien-ridden science-fiction future, we might have nothing else.

Against these godlike aliens, cryptography will be the only technology we can be sure of. Our nukes might refuse to detonate and our fighter jets might fall out of the sky, but we will still be able to communicate securely using one-time pads. There's an optimism in that.

This essay originally appeared in IEEE Security and Privacy.

Posted on September 14, 2018 at 6:15 AMView Comments

GCHQ on Quantum Key Distribution

The UK's GCHQ delivers a brutally blunt assessment of quantum key distribution:

QKD protocols address only the problem of agreeing keys for encrypting data. Ubiquitous on-demand modern services (such as verifying identities and data integrity, establishing network sessions, providing access control, and automatic software updates) rely more on authentication and integrity mechanisms -- such as digital signatures -- than on encryption.

QKD technology cannot replace the flexible authentication mechanisms provided by contemporary public key signatures. QKD also seems unsuitable for some of the grand future challenges such as securing the Internet of Things (IoT), big data, social media, or cloud applications.

I agree with them. It's a clever idea, but basically useless in practice. I don't even think it's anything more than a niche solution in a world where quantum computers have broken our traditional public-key algorithms.

Read the whole thing. It's short.

Posted on August 1, 2018 at 2:07 PMView Comments

Post-Quantum RSA

Interesting research on a version of RSA that is secure against a quantum computer:

Post-quantum RSA

Daniel J. Bernstein, Nadia Heninger, Paul Lou, and Luke Valenta

Abstract: This paper proposes RSA parameters for which (1) key generation, encryption, decryption, signing, and verification are feasible on today's computers while (2) all known attacks are infeasible, even assuming highly scalable quantum computers. As part of the performance analysis, this paper introduces a new algorithm to generate a batch of primes. As part of the attack analysis, this paper introduces a new quantum factorization algorithm that is often much faster than Shor's algorithm and much faster than pre-quantum factorization algorithms. Initial pqRSA implementation results are provided.

Posted on May 31, 2017 at 6:31 AMView Comments

Google's Post-Quantum Cryptography

News has been bubbling about an announcement by Google that it's starting to experiment with public-key cryptography that's resistant to cryptanalysis by a quantum computer. Specifically, it's experimenting with the New Hope algorithm.

It's certainly interesting that Google is thinking about this, and probably okay that it's available in the Canary version of Chrome, but this algorithm is by no means ready for operational use. Secure public-key algorithms are very hard to create, and this one has not had nearly enough analysis to be trusted. Lattice-based public-key cryptosystems such as New Hope are particularly subtle -- and we cryptographers are still learning a lot about how they can be broken.

Targets are important in cryptography, and Google has turned New Hope into a good one. Consider this an opportunity to advance our cryptographic knowledge, not an offer of a more-secure encryption option. And this is the right time for this area of research, before quantum computers make discrete-logarithm and factoring algorithms obsolete.

Posted on July 12, 2016 at 12:53 PMView Comments

NIST Starts Planning for Post-Quantum Cryptography

Last year, the NSA announced its plans for transitioning to cryptography that is resistant to a quantum computer. Now, it's NIST's turn. Its just-released report talks about the importance of algorithm agility and quantum resistance. Sometime soon, it's going to have a competition for quantum-resistant public-key algorithms:

Creating those newer, safer algorithms is the longer-term goal, Moody says. A key part of this effort will be an open collaboration with the public, which will be invited to devise and vet cryptographic methods that -- to the best of experts' knowledge -- ­will be resistant to quantum attack. NIST plans to launch this collaboration formally sometime in the next few months, but in general, Moody says it will resemble past competitions such as the one for developing the SHA-3 hash algorithm, used in part for authenticating digital messages.

"It will be a long process involving public vetting of quantum-resistant algorithms," Moody said. "And we're not expecting to have just one winner. There are several systems in use that could be broken by a quantum computer­ -- public-key encryption and digital signatures, to take two examples­ -- and we will need different solutions for each of those systems."

The report rightly states that we're okay in the symmetric cryptography world; the key lengths are long enough.

This is an excellent development. NIST has done an excellent job with their previous cryptographic standards, giving us a couple of good, strong, well-reviewed, and patent-free algorithms. I have no doubt this process will be equally excellent. (If NIST is keeping a list, aside from post-quantum public-key algorithms, I would like to see competitions for a larger-block-size block cipher and a super-fast stream cipher as well.)

Two news articles.

Posted on May 9, 2016 at 6:19 AMView Comments

More Details on the NSA Switching to Quantum-Resistant Cryptography

The NSA is publicly moving away from cryptographic algorithms vulnerable to cryptanalysis using a quantum computer. It just published a FAQ about the process:

Q: Is there a quantum resistant public-key algorithm that commercial vendors should adopt?

A: While a number of interesting quantum resistant public key algorithms have been proposed external to NSA, nothing has been standardized by NIST, and NSA is not specifying any commercial quantum resistant standards at this time. NSA expects that NIST will play a leading role in the effort to develop a widely accepted, standardized set of quantum resistant algorithms. Once these algorithms have been standardized, NSA will require vendors selling to NSS operators to provide FIPS validated implementations in their products. Given the level of interest in the cryptographic community, we hope that there will be quantum resistant algorithms widely available in the next decade. NSA does not recommend implementing or using non-standard algorithms, and the field of quantum resistant cryptography is no exception.

[...]

Q: When will quantum resistant cryptography be available?

A: For systems that will use unclassified cryptographic algorithms it is vital that NSA use cryptography that is widely accepted and widely available as part of standard commercial offerings vetted through NIST's cryptographic standards development process. NSA will continue to support NIST in the standardization process and will also encourage work in the vendor and larger standards communities to help produce standards with broad support for deployment in NSS. NSA believes that NIST can lead a robust and transparent process for the standardization of publicly developed and vetted algorithms, and we encourage this process to begin soon. NSA believes that the external cryptographic community can develop quantum resistant algorithms and reach broad agreement for standardization within a few years.

Lots of other interesting stuff in the Q&A.

Posted on February 2, 2016 at 7:11 AMView Comments

Why Is the NSA Moving Away from Elliptic Curve Cryptography?

In August, I wrote about the NSA's plans to move to quantum-resistant algorithms for its own cryptographic needs.

Cryptographers Neal Koblitz and Alfred Menezes just published a long paper speculating as to the government's real motives for doing this. They range from some new cryptanalysis of ECC to a political need after the DUAL_EC_PRNG disaster -- to the stated reason of quantum computing fears.

Read the whole paper. (Feel free to skip over the math if it gets too hard, but keep going until the end.)

EDITED TO ADD (11/15): A commentary and critique of the paper by Matthew Green.

Posted on October 28, 2015 at 2:11 PMView Comments

NSA Plans for a Post-Quantum World

Quantum computing is a novel way to build computers -- one that takes advantage of the quantum properties of particles to perform operations on data in a very different way than traditional computers. In some cases, the algorithm speedups are extraordinary.

Specifically, a quantum computer using something called Shor's algorithm can efficiently factor numbers, breaking RSA. A variant can break Diffie-Hellman and other discrete log-based cryptosystems, including those that use elliptic curves. This could potentially render all modern public-key algorithms insecure. Before you panic, note that the largest number to date that has been factored by a quantum computer is 143. So while a practical quantum computer is still science fiction, it's not stupid science fiction.

(Note that this is completely different from quantum cryptography, which is a way of passing bits between two parties that relies on physical quantum properties for security. The only thing quantum computation and quantum cryptography have to do with each other is their first words. It is also completely different from the NSA's QUANTUM program, which is its code name for a packet-injection system that works directly in the Internet backbone.)

Practical quantum computation doesn't mean the end of cryptography. There are lesser-known public-key algorithms such as McEliece and lattice-based algorithms that, while less efficient than the ones we use, are currently secure against a quantum computer. And quantum computation only speeds up a brute-force keysearch by a factor of a square root, so any symmetric algorithm can be made secure against a quantum computer by doubling the key length.

We know from the Snowden documents that the NSA is conducting research on both quantum computation and quantum cryptography. It's not a lot of money, and few believe that the NSA has made any real advances in theoretical or applied physics in this area. My guess has been that we'll see a practical quantum computer within 30 to 40 years, but not much sooner than that.

This all means that now is the time to think about what living in a post-quantum world would be like. NIST is doing its part, having hosted a conference on the topic earlier this year. And the NSA announced that it is moving towards quantum-resistant algorithms.

Earlier this week, the NSA's Information Assurance Directorate updated its list of Suite B cryptographic algorithms. It explicitly talked about the threat of quantum computers:

IAD will initiate a transition to quantum resistant algorithms in the not too distant future. Based on experience in deploying Suite B, we have determined to start planning and communicating early about the upcoming transition to quantum resistant algorithms. Our ultimate goal is to provide cost effective security against a potential quantum computer. We are working with partners across the USG, vendors, and standards bodies to ensure there is a clear plan for getting a new suite of algorithms that are developed in an open and transparent manner that will form the foundation of our next Suite of cryptographic algorithms.

Until this new suite is developed and products are available implementing the quantum resistant suite, we will rely on current algorithms. For those partners and vendors that have not yet made the transition to Suite B elliptic curve algorithms, we recommend not making a significant expenditure to do so at this point but instead to prepare for the upcoming quantum resistant algorithm transition.

Suite B is a family of cryptographic algorithms approved by the NSA. It's all part of the NSA's Cryptographic Modernization Program. Traditionally, NSA algorithms were classified and could only be used in specially built hardware modules. Suite B algorithms are public, and can be used in anything. This is not to say that Suite B algorithms are second class, or breakable by the NSA. They're being used to protect US secrets: "Suite A will be used in applications where Suite B may not be appropriate. Both Suite A and Suite B can be used to protect foreign releasable information, US-Only information, and Sensitive Compartmented Information (SCI)."

The NSA is worried enough about advances in the technology to start transitioning away from algorithms that are vulnerable to a quantum computer. Does this mean that the agency is close to a working prototype in their own classified labs? Unlikely. Does this mean that they envision practical quantum computers sooner than my 30-to-40-year estimate? Certainly.

Unlike most personal and corporate applications, the NSA routinely deals with information it wants kept secret for decades. Even so, we should all follow the NSA's lead and transition our own systems to quantum-resistant algorithms over the next decade or so -- possibly even sooner.

The essay previously appeared on Lawfare.

EDITED TO ADD: The computation that factored 143 also accidentally "factored much larger numbers such as 3599, 11663, and 56153, without the awareness of the authors of that work," which shows how weird this all is.

EDITED TO ADD: Seems that I need to be clearer: I do not stand by my 30-40-year prediction. The NSA is acting like practical quantum computers will exist long before then, and I am deferring to their expertise.

Posted on August 21, 2015 at 12:36 PMView Comments

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