The Approximate GCD Problem

Steven Galbraith once told me that he expects mathematicians to teach RSA long after the world has migrated to post-quantum algorithms; because it is so easy to explain. Arguably, LWE is easier to explain than RSA but the Approximate Greatest Common Divisors problem (AGCD) is even easier than that and requires only scalars. Thus, it is a nice post-quantum alternative for an undergraduate mathematics module. Someone should perhaps write an undergraduate mathematics textbook introducing cryptography using Approximate Common Divisors.

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Lecturer/Assistant Professor in Cryptography in the ISG

The ISG is recruiting a lecturer (≡ assistant professor in the US system, ≡ Juniorprofessor in the German system, ≡ Maître de conférences in the French system; that’s all the systems I know). This is a full-time, permanent research and teaching position.

Look, I know this is England post-Brexit but let me give you a personal pitch of why you should apply:

  • It’s a big group. We got ~20 permanent members of staff working across the field of information security: cryptography, systems and social. Check out our seminar programme and our publications to get a sense of what is going on in the group.
  • It’s a group with lots of cryptography going on. As mentioned in the ad below, eight permanent members of staff, five postdocs and about 15 PhD students focus on or contribute to cryptographic research. As a corollary, we also have plenty of cryptographers coming through for visits and talks. We got a weekly cryptography reading group, our students have another one and our seminar regularly has cryptography talks.
  • It’s a group with a good mix of areas and lots of interaction. UK universities don’t work like German ones where professors have their little empires which don’t interact all too much. Rather, the hierarchies are pretty flat within a department (everybody is line managed by the Head of Department) which facilitates more interaction; at least within the ISG that’s true. For example, I’m currently working on a project with someone from the systems and software security lab and one of our social scientists. I doubt this sort of collaboration would have come about if we didn’t attend the same meetings, taught the same modules, went to lunch and the pub together etc. Interdisciplinarity from above is annoying, when it emerges spontaneously it can be great.
  • It’s a nice group. People are genuinely friendly and we help each other out. It will be easy to find someone to proof read your grant applications or share previously successfully funded ones etc. I don’t know any official numbers but the unionisation level seems to be relatively high, which I also take as an indication that people don’t adopt a “everyone for themselves” approach.
  • We got funding for our Centre for Doctoral Training for the next four years (then we have to reapply). This means 10 PhD positions per year. Also, our CDT attracts strong students. My research career really took off after getting a chance to work with our amazing students.
  • The ISG is its own department (in a school with Physics, EE, Mathematics and Computer Science). All of our teaching is on information security with a focus on our Information Security MSc (which is huge). So you’ll get to teach information security. It is unlikely, though, that you will get to teach cryptography specifically.
  • The ISG has strong industry links. Thus, if that’s your cup of tea, it will be easy to get introductions etc. A side effect of these strong links is that consulting opportunities tend to pop up. Consulting is not only permitted by the employer but encouraged (they take a cut if you do it through them).
  • The ISG is a large group but Royal Holloway is a relatively small university. That means getting things done by speaking to the person in charge is often possible, i.e. it’s not some massive bureaucracy and exceptions can be negotiated.
  • It’s within one standard deviation from London. This means UCL and Surrey, and thus the cryptographers there, aren’t too far away. London Crypto Day is a thing and so are the London-ish Lattice Coding & Crypto Meetings. Also, you get to live in London (or near Egham if that’s your thing, no judgement).

I’m happy to answer informal inquiries etc. We’d appreciate any help in spreading the word.

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10 PhD Positions at Royal Holloway’s Centre for Doctoral Training in Cyber Security for the Everyday

At Royal Holloway we are again taking applications for ten fully-funded PhD positions in Information Security. See the CDT website and the ISG website for what kind of research we do. Also, check out our past and current CDT students and our research seminar schedule to get an idea of how broad and diverse the areas of information security are in which the ISG works.

More narrowly, to give you some idea of cryptographic research (and thus supervision capacity) in the ISG/at Royal Holloway: currently, there are nine permanent members of staff working on cryptography: Simon Blackburn (Maths), Carlos Cid, Keith Martin, Sean Murphy, Siaw-Lynn Ng, Rachel Player, Liz Quaglia and me. In addition, there are five postdocs working on cryptography and roughly 15 PhD students. Focus areas of cryptographic research currently are: lattice-based cryptography and applications, post-quantum cryptography, symmetric cryptography, statistics, access control, information-theoretic security and protocols.

Note that most of these positions are reserved for UK residents, which does, however, not mean nationality (see CDT website for details) and there might also be some wiggle room for EU residents (yes, still!).

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17th IMA Conference on Cryptography and Coding

IMA-CC is a crypto and coding theory conference biennially held in the UK. It was previously held in Cirencester. So you might have heard of it as the “Cirncester” conference. However, it has been moved to Oxford, so calling it Cirencester now is a bit confusing. Anyway, it is happening again this year. IMA is a small but fine conference with the added perk of being right before Christmas. This is great because around that time of the year Oxford is a fairly Christmas-y place to be.

16 – 18 December 2019, St Anne’s College, University of Oxford

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Postdoc Position at Royal Holloway on Key Exchange

Carlos and I have a postdoc position on designing cryptographic key exchange protocols that support incorporating key material from, erm, … diverse sources. This is part of a consortium that looks at integrating some quantum cryptography with post-quantum cryptography, but there is no need to think so narrowly about the problem. That is, the project is about incorporating randomness from wherever it might come and what security goals can be achieved depending on what is compromised. More generally, if you enjoy cryptographic protocols, not limited to key exchange protocols, this might be a fitting postdoc position. Get in touch with Carlos or me, if you’re unsure on whether the position is a good fit.

Location: Egham
Salary: £39,479 to £41,743 per annum – including London Allowance
Closing Date: Tuesday 12 March 2019
Interview Date: To be confirmed
Reference: 0219-048

The Information Security Group at Royal Holloway University of London is seeking to recruit a postdoctoral research assistant (PDRA) to work in the area of cryptography. The position is available for immediate start, for up to 26 months (until 31 March 2021).

The PDRA will work alongside Prof. Carlos Cid, Dr. Martin Albrecht and other cryptographic researchers at Royal Holloway on topics connected to the design and analysis of cryptographic key exchange protocols that support incorporating key material from diverse sources. This post is part of the AQuaSec project, a Innovate UK-funded research project with 17 partners from industry and academia, aiming to develop technologies for quantum-safe communications by integrating post-quantum cryptography with techniques from quantum cryptography.

Applicants for this role should have already completed, or be close to completing, a PhD in a relevant discipline, with an outstanding research track record in cryptography. Applicants should be able to demonstrate scientific creativity, research independence, and the ability to communicate their ideas effectively in written and verbal form. Salary is £39,479 per annum, inclusive of London Allowance. This post is appointed at Grade 7, Spine point 34.

Established in 1990, the Information Security Group at Royal Holloway was one of the first dedicated academic groups in the world to conduct research and teaching in information security. The ISG is today a world-leading interdisciplinary research group with 20 full-time members of staff, several postdoctoral research assistants and over 50 PhD students working on a range of subjects in cyber security, in particular cryptography.

In return we offer a highly competitive rewards and benefits package including:

  • Generous annual leave entitlement
  • Training and Development opportunities
  • Pension Scheme with generous employer contribution
  • Various schemes including Cycle to Work, Season Ticket Loans and help with the cost of Eyesight testing.
  • Free parking
  • Competitive Maternity, Adoption and Shared Parental Leave provisions

The post is based in Egham, Surrey where the College is situated in a beautiful, leafy campus near to Windsor Great Park and within commuting distance from London.

To view further details of this post and to apply please visit For queries on the application process the Human Resources Department can be contacted by email at: Informal enquiries can be made to Prof. Carlos Cid at

Please quote the reference: 0219-048

Closing Date: Midnight, 12 March 2019

Interview Date: To be confirmed

PS: I will have two more postdoc positions, on lattice-based cryptography in the next few weeks/months.

10 PhD Positions at Royal Holloway’s Centre for Doctoral Training in Cyber Security

At Royal Holloway we are now taking applications for ten fully-funded PhD positions in Information Security. See the CDT website and the ISG website for what kind of research we do. In particular, check out our past and current CDT students to get an idea of how broad and diverse the areas of information security are in which they work.

Note that most of these positions are reserved for UK residents, which does, however, not mean nationality (see CDT website for details) and there might also be some wiggle room for EU residents.

Continue reading “10 PhD Positions at Royal Holloway’s Centre for Doctoral Training in Cyber Security”

NTT Considered Harmful?

In a typical Ring-LWE-based public-key encryption scheme, Alice publishes

(a, b) = (a, a \cdot s + e) \in \mathbb{Z}_q[x]/(x^n+1)

(with n a power of two1) as the public key, where s, e are both “small” and secret. To encrypt, Bob computes

(c_{0}, c_{1}) = (v \cdot a + e', v \cdot b + e'' + \textnormal{Encode}(m))

where v, e', e'' are small, m is the message \in \{0,1\}^n and \textnormal{Encode}(\cdot) some encoding function, e.g. \sum_{i=0}^{n-1} \lfloor \frac{q}{2} \rfloor m_i x^i . To decrypt, Alice computes

c_{0} \cdot s - c_{1} = (v \cdot a + e')\cdot s - v \cdot (a\cdot s + e) + e'' + \textnormal{Encode}(m),

which is equal to e' \cdot s - v \cdot e + e'' + \textnormal{Encode}(m). Finally, Alice recovers m from the noisy encoding of m where e' \cdot s - v \cdot e + e'' is the noise. In the Module-LWE variant the elements essentially live in \left(\mathbb{Z}_q[x]/(x^n+1)\right)^k, e.g. a is not a polynomial but a vector of polynomials.

Thus, both encryption and decryption involve polynomial multiplication modulo x^n+1. Using schoolbook multiplication this costs \mathcal{O}(n^2) operations. However, when selecting parameters for Ring-LWE, we can choose q \equiv 1 \bmod 2n which permits to use an NTT to realise this multiplication (we require \equiv \bmod 2n to use the negacyclic NTT which has modular reductions modulo x^n+1 baked in). Then, using the NTT we can implement multiplication by

  1. evaluation (perform NTT),
  2. pointwise multiplication,
  3. interpolation (perform inverse NTT).

Steps (1) and (3) take \mathcal{O}(n \log n) operations by using specially chosen evaluation points (roots of one). Step (2) costs \mathcal{O}(n) operations.

This is trick is very popular. For example, many (but not all!) Ring-LWE based schemes submitted to the NIST PQC competition process use it, namely NewHope, LIMA (go LIMA!), LAC, KCL, HILA5, R.EMBLEM, Ding Key-Exchange, CRYSTALS-KYBER, CRYSTALS-DILITHIUM (sorry, if I forgot one). Note that since steps (1) and (3) are the expensive steps, it makes sense to remain in the NTT domain (i.e. after applying the NTT) and only to convert back at the very end. For example, it is faster for Alice to store s, e in NTT domain and, since the NTT maps uniform to uniform, to sample a in NTT domain directly, i.e. to just assume that a random vector a is already the output of an NTT on some other random vector.

This post is about two recent results I was involved in suggesting that this is not necessarily always the best choice (depending on your priorities.)

Warning: This is going to be one of those clickbait-y pieces where the article doesn’t live up to the promise in the headline. The NTT is fine. Some of my best friends use the NTT. In fact I’ve implemented and used the NTT myself.

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