## 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.

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

At Royal Holloway we have ten PhD positions in Information Security. The catch is that almost all of those positions are reserved for UK residents. Note that this does not mean nationality, see funding page (there might also be some wiggle room in some cases). For more information see the CDT website and the ISG website for what kind of research we do.

Welcome to the EPSRC Centre for Doctoral Training (CDT) in Cyber Security at Royal Holloway. The Centre was established in 2013, and has as its main objective to produce cohorts of highly-trained researchers with a broad understanding of cyber security.

The CDT is hosted by the Information Security Group (ISG), and provides multidisciplinary training to annual cohorts of around ten students each. The students follow a 4-year doctoral programme: the first phase consists of a taught component comprising 25 per cent of the programme. The remaining three years follow the more traditional path of doctoral studies, with each student undertaking research in an advanced topic in the field of cyber security. See the CDT Course of Study page for more information about the programme.

CDT recruitment typically runs from November to April, to select students for the CDT cohort to start the following September. Selected applicants are awarded fully-funded PhD studentships (stipend and College fees) for four years. We consider applications from candidates with undergraduate and masters qualifications in a wide range of disciplines, including, but not limited to, mathematics, computer science, and electrical and electronic engineering.

We are now open to receive applications for students to start their PhD studies in September 2018.

## Postdoc at Royal Holloway on Post-Quantum Cryptography in Hardware

Together with Carlos Cid, we have a two-year postdoc position available. The position is focused on hardware implementations of post-quantum cryptography such as lattice-based, code-based, hash-based or mq-based schemes. If you are interested, feel free to get in touch with Carlos or me. If you know of somone who might be interested, we would appreciate if you could make them aware of this position.

## Postdoc at Royal Holloway on Quantum-Safe Cryptography in Hardware

Together with Carlos Cid, we have a two-year postdoc position available. The position is focused on hardware implementations of quantum-safe cryptography such as lattice-based, code-based, hash-based or mq-based schemes. If you are interested, feel free to get in touch with Carlos or me. If you know of somone who might be interested, we would appreciate if you could make them aware of this position.

## CCA Conversions

In Tightly Secure Ring-LWE Based Key Encapsulation with Short Ciphertexts we — together with Emmanuela Orsini, Kenny Paterson, Guy Peer and Nigel Smart — give a tight reduction of Alex Dent’s IND-CCA secure KEM conversion (from an OW-CPA schemes) when the underlying scheme is (Ring-)LWE:

Abstract: We provide a tight security proof for an IND-CCA Ring-LWE based Key Encapsulation Mechanism that is derived from a generic construction of Dent (IMA Cryptography and Coding, 2003). Such a tight reduction is not known for the generic construction. The resulting scheme has shorter ciphertexts than can be achieved with other generic constructions of Dent or by using the well-known Fujisaki-Okamoto constructions (PKC 1999, Crypto 1999). Our tight security proof is obtained by reducing to the security of the underlying Ring-LWE problem, avoiding an intermediate reduction to a CPA-secure encryption scheme. The proof technique maybe of interest for other schemes based on LWE and Ring-LWE.

## 16th IMA International Conference on Cryptography and Coding

IMA-CCC 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.

12 – 14 December 2017, St Catherine’s College, University of Oxford

## Fplll Days 3: July 6 – 14, Amsterdam

We’ll have an fplll coding sprint aka “FPLLL Days” in July. This time around, we plan a slightly modified format compared to previous instances. That is, in order to encourage new developers to get involved, we plan to have a 2 day tutorial session (shorter or longer depending on participants/interest) before the start of FPLLL Days proper.