The introduction of Edwards' curves in 2007 relaunched a

deeper study of the arithmetic of elliptic curves with a

view to cryptographic applications. In particular, this

research focused on the role of the model of the curve ---

a triple consisting of a curve, base point, and projective

(or affine) embedding. From the computational perspective,

a projective (as opposed to affine) model allows one to

avoid inversions in the base field, while from the

mathematical perspective, it permits one to reduce various

arithmetical operations to linear algebra (passing through

the language of sheaves). We describe the role of the model,

particularly its classification up to linear isomorphism

and its role in the linearization of the operations of addition,

doubling, and scalar multiplication.

# Past Cryptography Seminar

The Algebraic Eraser is a cryptosystem (more precisely, a class of key

agreement schemes) introduced by Anshel, Anshel, Goldfeld and Lemieux

about 10 years ago. There is a concrete instantiation of the Algebraic

Eraser called the Colored Burau Key Agreement Protocol (CBKAP), which

uses a blend of techniques from permutation groups, matrix groups and

braid groups. SecureRF, the company owning the trademark to the

Algebraic Eraser, is marketing this system for lightweight

environments such as RFID tags and other Internet of Things

applications; they have proposed making this scheme the basis for an

ISO RFID standard.

This talk gives an introduction to the Algebraic Eraser, a brief

history of the attacks on this scheme using ideas from group-theoretic

cryptography, and describes the countermeasures that have been

proposed. I would not recommend the scheme for the proposed

applications: the talk ends with a brief sketch of a recent convincing

cryptanalysis of this scheme due to Ben-Zvi, Blackburn and Tsaban

(which appeared at CRYPTO this summer), and significant attacks

on the protocol in the proposed ISO standard due to Blackburn and

Robshaw (which appeared at ACNS earlier this year).

Linear algebra is a widely used tool both in mathematics and computer science, and cryptography is no exception to this rule. Yet, it introduces some particularities, such as dealing with linear systems that are often sparse, or, in other words, linear systems inside which a lot of coefficients are equal to zero. We propose to enlarge this notion to nearly sparse matrices, caracterized by the concatenation of a sparse matrix and some dense columns, and to design an algorithm to solve this kind of problems. Motivated by discrete logarithms computations on medium and high caracteristic finite fields, the Nearly Sparse Linear Algebra briges the gap between classical dense linear algebra problems and sparse linear algebra ones, for which specific methods have already been established. Our algorithm particularly applies on one of the three phases of NFS (Number Field Sieve) which precisely consists in finding a non trivial element of the kernel of a nearly sparse matrix.

Additive combinatorics enable one to characterise subsets S of elements in a group such that S+S has small cardinality. In particular a theorem of Vosper says that subsets of integers modulo a prime p with minimal sumsets can only be arithmetic progressions, apart from some degenerate cases. We are interested in q-analogues of these results, namely characterising subspaces S in some algebras such that the linear span of its square S^2 has small dimension. Analogues of Vosper's theorem will imply that such spaces will have bases consisting of elements in geometric progression. We derive such analogues in extensions of finite fields, where bounds on codes in the space of quadratic forms play a crucial role. We also obtain that under appropriately formulated conditions, linear codes with small squares for the component-wise product can only be generalized Reed-Solomon codes. Based on joint works with Christine Bachoc and Oriol Serra, and with Diego Mirandola.

Efficient factorization or efficient computation of class

numbers would both suffice to break RSA. However the talk lies more in

computational number theory rather than in cryptography proper. We will

address two questions: (1) How quickly can one construct a factor table

for the numbers up to x?, and (2) How quickly can one do the same for the

class numbers (of imaginary quadratic fields)? Somewhat surprisingly, the

approach we describe for the second problem is motivated by the classical

Hardy-Littlewood method.

Due to Shor's algorithm, quantum computers are a severe threat for public key cryptography. This motivated the cryptographic community to search for quantum-safe solutions. On the other hand, the impact of quantum computing on secret key cryptography is much less understood. In this paper, we consider attacks where an adversary can query an oracle implementing a cryptographic primitive in a quantum superposition of different states. This model gives a lot of power to the adversary, but recent results show that it is nonetheless possible to build secure cryptosystems in it.

We study applications of a quantum procedure called Simon's algorithm (the simplest quantum period finding algorithm) in order to attack symmetric cryptosystems in this model. Following previous works in this direction, we show that several classical attacks based on finding collisions can be dramatically sped up using Simon's algorithm: finding a collision requires Ω(2n/2) queries in the classical setting, but when collisions happen with some hidden periodicity, they can be found with only O(n) queries in the quantum model.

We obtain attacks with very strong implications. First, we show that the most widely used modes of operation for authentication and authenticated encryption (e.g. CBC-MAC, PMAC, GMAC, GCM, and OCB) are completely broken in this security model. Our attacks are also applicable to many CAESAR candidates: CLOC, AEZ, COPA, OTR, POET, OMD, and Minalpher. This is quite surprising compared to the situation with encryption modes: Anand et al. show that standard modes are secure when using a quantum-secure PRF.

Second, we show that slide attacks can also be sped up using Simon's algorithm. This is the first exponential speed up of a classical symmetric cryptanalysis technique in the quantum model.

The cube attack of Dinur and Shamir and the AIDA attack of Vielhaber have been used successfully on reduced round versions of the Trivium stream cipher and a few other ciphers. These attacks can be viewed in the framework of higher order differentiation, as introduced by Lai in the cryptographic context. We generalise these attacks from the binary case to general finite fields, showing that we would need to differentiate several times with respect to each variable in order to have a reasonable chance of a successful attack. We also investigate the notion of “fast points” for a binary polynomial function f (i.e. vectors such that the derivative of f with respect to this vector has a lower than expected degree). These were introduced by Duan and Lai, motivated by the fact that higher order differential attacks are usually more efficient if they use such points. The number of functions which admit fast points were computed by Duan et al in a few particular cases; we give explicit formulae for all remaining cases and discuss the cryptographic significance of these results.

One of the peculiar features of quantum mechanics is

entanglement. It is known that entanglement is monogamous in the sense

that a quantum system can only be strongly entangled to one other

system. In this talk, I will show how this so-called monogamy of

entanglement can be captured and quantified by a "game". We show that,

in this particular game, the monogamy completely "cancels out" the

advantage of entanglement.

As an application of our analysis, we show that - in theory - the

standard BB84 quantum-key-distribution scheme is one-sided

device-independent, meaning that one of the parties, say Bob, does not

need to trust his quantum measurement device: security is guaranteed

even if his device is completely malicious.

The talk will be fully self-contained; no prior knowledge on quantum

mechanics/cryptography is necessary.

We hope to bring together all Oxford researchers interested in Cryptography, in Quantum Computing and in the interactions between the two.

Please register at: http://oxford-cryptography-day.eventbrite.co.uk