Entries Tagged "RSA"

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Breaking RSA with a Quantum Computer

A group of Chinese researchers have just published a paper claiming that they can—although they have not yet done so—break 2048-bit RSA. This is something to take seriously. It might not be correct, but it’s not obviously wrong.

We have long known from Shor’s algorithm that factoring with a quantum computer is easy. But it takes a big quantum computer, on the orders of millions of qbits, to factor anything resembling the key sizes we use today. What the researchers have done is combine classical lattice reduction factoring techniques with a quantum approximate optimization algorithm. This means that they only need a quantum computer with 372 qbits, which is well within what’s possible today. (The IBM Osprey is a 433-qbit quantum computer, for example. Others are on their way as well.)

The Chinese group didn’t have that large a quantum computer to work with. They were able to factor 48-bit numbers using a 10-qbit quantum computer. And while there are always potential problems when scaling something like this up by a factor of 50, there are no obvious barriers.

Honestly, most of the paper is over my head—both the lattice-reduction math and the quantum physics. And there’s the nagging question of why the Chinese government didn’t classify this research. But…wow…maybe…and yikes! Or not.

Factoring integers with sublinear resources on a superconducting quantum processor

Abstract: Shor’s algorithm has seriously challenged information security based on public key cryptosystems. However, to break the widely used RSA-2048 scheme, one needs millions of physical qubits, which is far beyond current technical capabilities. Here, we report a universal quantum algorithm for integer factorization by combining the classical lattice reduction with a quantum approximate optimization algorithm (QAOA). The number of qubits required is O(logN/loglogN ), which is sublinear in the bit length of the integer N , making it the most qubit-saving factorization algorithm to date. We demonstrate the algorithm experimentally by factoring integers up to 48 bits with 10 superconducting qubits, the largest integer factored on a quantum device. We estimate that a quantum circuit with 372 physical qubits and a depth of thousands is necessary to challenge RSA-2048 using our algorithm. Our study shows great promise in expediting the application of current noisy quantum computers, and paves the way to factor large integers of realistic cryptographic significance.

In email, Roger Grimes told me: “Apparently what happened is another guy who had previously announced he was able to break traditional asymmetric encryption using classical computers…but reviewers found a flaw in his algorithm and that guy had to retract his paper. But this Chinese team realized that the step that killed the whole thing could be solved by small quantum computers. So they tested and it worked.”

EDITED TO ADD: One of the issues with the algorithm is that it relies on a recent factoring paper by Claus Schnorr. It’s a controversial paper; and despite the “this destroys the RSA cryptosystem” claim in the abstract, it does nothing of the sort. Schnorr’s algorithm works well with smaller moduli—around the same order as ones the Chinese group has tested—but falls apart at larger sizes. At this point, nobody understands why. The Chinese paper claims that their quantum techniques get around this limitation (I think that’s what’s behind Grimes’s comment) but don’t give any details—and they haven’t tested it with larger moduli. So if it’s true that the Chinese paper depends on this Schnorr technique that doesn’t scale, the techniques in this Chinese paper won’t scale, either. (On the other hand, if it does scale then I think it also breaks a bunch of lattice-based public-key cryptosystems.)

I am much less worried that this technique will work now. But this is something the IBM quantum computing people can test right now.

EDITED TO ADD (1/4): A reporter just asked me my gut feel about this. I replied that I don’t think this will break RSA. Several times a year the cryptography community received “breakthroughs” from people outside the community. That’s why we created the RSA Factoring Challenge: to force people to provide proofs of their claims. In general, the smart bet is on the new techniques not working. But someday, that bet will be wrong. Is it today? Probably not. But it could be. We’re in the worst possible position right now: we don’t have the facts to know. Someone needs to implement the quantum algorithm and see.

EDITED TO ADD (1/5): Scott Aaronson’s take is a “no”:

In the new paper, the authors spend page after page saying-without-saying that it might soon become possible to break RSA-2048, using a NISQ (i.e., non-fault-tolerant) quantum computer. They do so via two time-tested strategems:

  1. the detailed exploration of irrelevancies (mostly, optimization of the number of qubits, while ignoring the number of gates), and
  2. complete silence about the one crucial point.

Then, finally, they come clean about the one crucial point in a single sentence of the Conclusion section:

It should be pointed out that the quantum speedup of the algorithm is unclear due to the ambiguous convergence of QAOA.

“Unclear” is an understatement here. It seems to me that a miracle would be required for the approach here to yield any benefit at all, compared to just running the classical Schnorr’s algorithm on your laptop. And if the latter were able to break RSA, it would’ve already done so.

All told, this is one of the most actively misleading quantum computing papers I’ve seen in 25 years, and I’ve seen … many.

EDITED TO ADD (1/7): More commentary. Again: no need to panic.

EDITED TO ADD (1/12): Peter Shor has suspicions.

Posted on January 3, 2023 at 12:38 PMView Comments

Breaking RSA through Insufficiently Random Primes

Basically, the SafeZone library doesn’t sufficiently randomize the two prime numbers it used to generate RSA keys. They’re too close to each other, which makes them vulnerable to recovery.

There aren’t many weak keys out there, but there are some:

So far, Böck has identified only a handful of keys in the wild that are vulnerable to the factorization attack. Some of the keys are from printers from two manufacturers, Canon and Fujifilm (originally branded as Fuji Xerox). Printer users can use the keys to generate a Certificate Signing Request. The creation date for the all the weak keys was 2020 or later. The weak Canon keys are tracked as CVE-2022-26351.

Böck also found four vulnerable PGP keys, typically used to encrypt email, on SKS PGP key servers. A user ID tied to the keys implied they were created for testing, so he doesn’t believe they’re in active use.

Posted on March 16, 2022 at 11:35 AMView Comments

No, RSA Is Not Broken

I have been seeing this paper by cryptographer Peter Schnorr making the rounds: “Fast Factoring Integers by SVP Algorithms.” It describes a new factoring method, and its abstract ends with the provocative sentence: “This destroys the RSA cryptosystem.”

It does not. At best, it’s an improvement in factoring—and I’m not sure it’s even that. The paper is a preprint: it hasn’t been peer reviewed. Be careful taking its claims at face value.

Some discussion here.

I’ll append more analysis links to this post when I find them.

EDITED TO ADD (3/12): The latest version of the paper does not have the words “This destroys the RSA cryptosystem” in the abstract. Some more discussion.

Posted on March 5, 2021 at 10:48 AMView Comments

Brexit Deal Mandates Old Insecure Crypto Algorithms

In what is surely an unthinking cut-and-paste issue, page 921 of the Brexit deal mandates the use of SHA-1 and 1024-bit RSA:

The open standard s/MIME as extension to de facto e-mail standard SMTP will be deployed to encrypt messages containing DNA profile information. The protocol s/MIME (V3) allows signed receipts, security labels, and secure mailing lists… The underlying certificate used by s/MIME mechanism has to be in compliance with X.509 standard…. The processing rules for s/MIME encryption operations… are as follows:

  1. the sequence of the operations is: first encryption and then signing,
  2. the encryption algorithm AES (Advanced Encryption Standard) with 256 bit key length and RSA with 1,024 bit key length shall be applied for symmetric and asymmetric encryption respectively,
  3. the hash algorithm SHA-1 shall be applied.
  4. s/MIME functionality is built into the vast majority of modern e-mail software packages including Outlook, Mozilla Mail as well as Netscape Communicator 4.x and inter-operates among all major e-mail software packages.

And s/MIME? Bleah.

Posted on December 31, 2020 at 6:19 AMView Comments

RSA-250 Factored

RSA-250 has been factored.

This computation was performed with the Number Field Sieve algorithm,
using the open-source CADO-NFS software.

The total computation time was roughly 2700 core-years, using Intel Xeon Gold 6130 CPUs as a reference (2.1GHz):

RSA-250 sieving: 2450 physical core-years
RSA-250 matrix: 250 physical core-years

The computation involved tens of thousands of machines worldwide, and was completed in a few months.

News article. On the factoring challenges.

Posted on April 8, 2020 at 6:37 AMView Comments

Chinese Hackers Bypassing Two-Factor Authentication

Interesting story of how a Chinese state-sponsored hacking group is bypassing the RSA SecurID two-factor authentication system.

How they did it remains unclear; although, the Fox-IT team has their theory. They said APT20 stole an RSA SecurID software token from a hacked system, which the Chinese actor then used on its computers to generate valid one-time codes and bypass 2FA at will.

Normally, this wouldn’t be possible. To use one of these software tokens, the user would need to connect a physical (hardware) device to their computer. The device and the software token would then generate a valid 2FA code. If the device was missing, the RSA SecureID software would generate an error.

The Fox-IT team explains how hackers might have gone around this issue:

The software token is generated for a specific system, but of course this system specific value could easily be retrieved by the actor when having access to the system of the victim.

As it turns out, the actor does not actually need to go through the trouble of obtaining the victim’s system specific value, because this specific value is only checked when importing the SecurID Token Seed, and has no relation to the seed used to generate actual 2-factor tokens. This means the actor can actually simply patch the check which verifies if the imported soft token was generated for this system, and does not need to bother with stealing the system specific value at all.

In short, all the actor has to do to make use of the 2 factor authentication codes is to steal an RSA SecurID Software Token and to patch 1 instruction, which results in the generation of valid tokens.

Posted on December 26, 2019 at 6:19 AMView Comments

RSA-240 Factored

This just in:

We are pleased to announce the factorization of RSA-240, from RSA’s challenge list, and the computation of a discrete logarithm of the same size (795 bits):

RSA-240 = 12462036678171878406583504460810659043482037465167880575481878888328 966680118821085503603957027250874750986476843845862105486553797025393057189121 768431828636284694840530161441643046806687569941524699318570418303051254959437 1372159029236099 = 509435952285839914555051023580843714132648382024111473186660296521821206469746 700620316443478873837606252372049619334517 * 244624208838318150567813139024002896653802092578931401452041221336558477095178 155258218897735030590669041302045908071447

[…]

The previous records were RSA-768 (768 bits) in December 2009 [2], and a 768-bit prime discrete logarithm in June 2016 [3].

It is the first time that two records for integer factorization and discrete logarithm are broken together, moreover with the same hardware and software.

Both computations were performed with the Number Field Sieve algorithm, using the open-source CADO-NFS software [4].

The sum of the computation time for both records is roughly 4000 core-years, using Intel Xeon Gold 6130 CPUs as a reference (2.1GHz). A rough breakdown of the time spent in the main computation steps is as follows.

RSA-240 sieving: 800 physical core-years
RSA-240 matrix: 100 physical core-years
DLP-240 sieving: 2400 physical core-years
DLP-240 matrix: 700 physical core-years

The computation times above are well below the time that was spent with the previous 768-bit records. To measure how much of this can be attributed to Moore’s law, we ran our software on machines that are identical to those cited in the 768-bit DLP computation [3], and reach the conclusion that sieving for our new record size on these old machines would have taken 25% less time than the reported sieving time of the 768-bit DLP computation.

EDITED TO ADD (12/4): News article. Dan Goodin points out that the speed improvements were more due to improvements in the algorithms than from Moore’s Law.

Posted on December 3, 2019 at 2:12 PMView Comments

Factoring 2048-bit Numbers Using 20 Million Qubits

This theoretical paper shows how to factor 2048-bit RSA moduli with a 20-million qubit quantum computer in eight hours. It’s interesting work, but I don’t want overstate the risk.

We know from Shor’s Algorithm that both factoring and discrete logs are easy to solve on a large, working quantum computer. Both of those are currently beyond our technological abilities. We barely have quantum computers with 50 to 100 qubits. Extending this requires advances not only in the number of qubits we can work with, but in making the system stable enough to read any answers. You’ll hear this called “error rate” or “coherence”—this paper talks about “noise.”

Advances are hard. At this point, we don’t know if they’re “send a man to the moon” hard or “faster-than-light travel” hard. If I were guessing, I would say they’re the former, but still harder than we can accomplish with our current understanding of physics and technology.

I write about all this generally, and in detail, here. (Short summary: Our work on quantum-resistant algorithms is outpacing our work on quantum computers, so we’ll be fine in the short run. But future theoretical work on quantum computing could easily change what “quantum resistant” means, so it’s possible that public-key cryptography will simply not be possible in the long run. That’s not terrible, though; we have a lot of good scalable secret-key systems that do much the same things.)

Posted on October 14, 2019 at 6:58 AMView Comments

Videos and Links from the Public-Interest Technology Track at the RSA Conference

Yesterday at the RSA Conference, I gave a keynote talk about the role of public-interest technologists in cybersecurity. (Video here).

I also hosted a one-day mini-track on the topic. We had six panels, and they were all great. If you missed it live, we have videos:

  • How Public Interest Technologists are Changing the World: Matt Mitchell, Tactical Tech; Bruce Schneier, Fellow and Lecturer, Harvard Kennedy School; and J. Bob Alotta, Astraea Foundation (Moderator). (Video here.)
  • Public Interest Tech in Silicon Valley: Mitchell Baker, Chairwoman, Mozilla Corporation; Cindy Cohn, EFF; and Lucy Vasserman, Software Engineer, Google. (Video here.)
  • Working in Civil Society: Sarah Aoun, Digital Security Technologist; Peter Eckersley, Partnership on AI; Harlo Holmes, Director of Newsroom Digital Security, Freedom of the Press Foundation; and John Scott-Railton, Senior Researcher, Citizen Lab. (Video here.)
  • Government Needs You: Travis Moore, TechCongress; Hashim Mteuzi, Senior Manager, Network Talent Initiative, Code for America; Gigi Sohn, Distinguished Fellow, Georgetown Law Institute for Technology, Law and Policy; and Ashkan Soltani, Independent Consultant. (Video here.)
  • Changing Academia: Latanya Sweeney, Harvard; Dierdre Mulligan, UC Berkeley; and Danny Weitzner, MIT CSAIL. (Video here.)
  • The Future of Public Interest Tech: Bruce Schneier, Fellow and Lecturer, Harvard Kennedy School; Ben Wizner, ACLU; and Jenny Toomey, Director, Internet Freedom, Ford Foundation (Moderator). (Video here.)

I also conducted eight short video interviews with different people involved in public-interest technology: independent security technologist Sarah Aoun, TechCongress’s Travis Moore, Ford Foundation’s Jenny Toomey, CitizenLab’s John-Scott Railton, Dierdre Mulligan from UC Berkeley, ACLU’s Jon Callas, Matt Mitchell of TacticalTech, and Kelley Misata from Sightline Security.

Here is my blog post about the event. Here’s Ford Foundation’s blog post on why they helped me organize the event.

We got some good press coverage about the event. (Hey MeriTalk: you spelled my name wrong.)

Related: Here’s my longer essay on the need for public-interest technologists in Internet security, and my public-interest technology resources page.

And just so we have all the URLs in one place, here is a page from the RSA Conference website with links to all of the videos.

If you liked this mini-track, please rate it highly on your RSA Conference evaluation form. I’d like to do it again next year.

Posted on March 8, 2019 at 2:24 PMView Comments

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Sidebar photo of Bruce Schneier by Joe MacInnis.