Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Analytic Combinatorics

The best lattice reduction algorithm known in practice for high dimension is Schnorr-Euchner's BKZ: all security estimates of lattice cryptosystems are based on NTL's old implementation of BKZ. However, recent progress on lattice enumeration suggests that BKZ and its NTL implementation are no longer optimal, but the precise impact on security estimates was unclear. We assess this impact thanks to extensive experiments with BKZ 2.0, the first state-of-the-art implementation of BKZ incorporating recent improvements, such as Gama-Nguyen-Regev pruning. We propose an efficient simulation algorithm to model the behaviour of BKZ in high dimension with high blocksize ≥50, which can predict approximately both the output quality and the running time, thereby revising lattice security estimates. For instance, our simulation suggests that the smallest NTRUSign parameter set, which was claimed to provide at least 93-bit security against key-recovery lattice attacks, actually offers at most 65-bit security.

/pdf/bkz-2-0-better-lattice-security-estimates-1i0txm8k41.pdf

BKZ 2.0: better lattice security estimates

We revisit the problem of generating a ‘hard’ random lattice together with a basis of relatively short vectors. This problem has gained in importance lately due to new cryptographic schemes that use such a procedure to generate public/secret key pairs. In these applications, a shorter basis corresponds to milder underlying complexity assumptions and smaller key sizes.

The contributions of this work are twofold. First, we simplify and modularize an approach originally due to Ajtai (ICALP 1999). Second, we improve the construction and its analysis in several ways, most notably by making the output basis asymptotically as short as possible.

/pdf/generating-shorter-bases-for-hard-random-lattices-1f94xyy7rk.pdf

Generating Shorter Bases for Hard Random Lattices

Lattice enumeration algorithms are the most basic algorithms for solving hard lattice problems such as the shortest vector problem and the closest vector problem, and are often used in public-key cryptanalysis either as standalone algorithms, or as subroutines in lattice reduction algorithms. Here we revisit these fundamental algorithms and show that surprising exponential speedups can be achieved both in theory and in practice by using a new technique, which we call extreme pruning. We also provide what is arguably the first sound analysis of pruning, which was introduced in the 1990s by Schnorr et al.

/pdf/lattice-enumeration-using-extreme-pruning-4s8szd2l1o.pdf

Lattice enumeration using extreme pruning

Lattices are regular arrangements of points in n-dimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated Lenstra-Lenstra-Lovasz lattice basis reduction algorithm twenty years ago, lattices have had surprising applications in cryptology. Until recently, the applications of lattices to cryptology were only negative, as lattices were used to break various cryptographic schemes. Paradoxically, several positive cryptographic applications of lattices have emerged in the past five years: there now exist public-key cryptosystems based on the hardness of lattice problems, and lattices play a crucial role in a few security proofs. In this talk, we will try to survey the main examples of the two faces of lattices in cryptology. The full material of this talk appeared in [2]. A preliminary version can be found in [1].

/pdf/the-two-faces-of-lattices-in-cryptology-kjl3e0lsca.pdf

The Two Faces of Lattices in Cryptology

LetN(x, n, α) denote the number of integer lattice points inside then-dimensional sphere of radius (an)1/2 with center at x. This numberN(x,n, α) is studied for α fixed,n → ∞, andx varying. The average value (asx varies) ofN(x,n, α) is just the volume of the sphere, which is roughly of the form (2 βe, α)
n/2. it is shown that the maximal and minimal values ofN (x,n, α) differ from the everage by factors exponential inn, which is in contrast to the usual lattice point problems in bounded dimensions. This lattice point problem arose separately in universal quantization and in low density subset sum problems.

/pdf/lattice-points-in-high-dimensional-spheres-30z0bq6pmp.pdf

Lattice points in high-dimensional spheres

A trellis code is a “sliding window” method of encoding a binary data stream into a sequence of real numbers that are input to a noisy transmission channel. When a trellis code is used to encode data at the rate of k bits/channel symbol, each channel input will depend not only on the most recent block of k data bits to enter the encoder but will also depend on, say, the v bits preceding this block. The v bits determine the state of the encoder and the most recent block of k bits generates the channel symbol conditional on the encoder state. The performance of trellis codes, like that of block codes, depends on a suitably defined minimum-distance property of the code. In this paper we obtain upper bounds on this minimum distance that are simple functions of k and v. These results also provide a lower bound on the number of states required to achieve a specific coding gain.

Upper bounds on the minimum distance of trellis codes

Let x i be members of a stationary sequence of zero mean Gaussian random variables having correlations Ex i x j = σ2 ρ|i-j|, 0 0. We address the behavior of the averaged product q m (ρ, σ) ≡ Ex 1 x 2 ··· x 2m−1 x 2m as m becomes large. Our principal result when σ2 = 1 is that this average approaches zero (infinity) as ρ is less (greater) than the critical value ρ c = 0.563007169…. To obtain this we introduce a linear recurrence for the ρ m ·(ρ, σ), and then continue generating an entire sequence of recurrences, where the (n + 1)-st relation is a recurrence for the coefficients that appear in the nth relation. This leads to a new, simple continued fraction representation for the generating function of the q m (ρ, σ). The related problem with q m (ρ, σ) = E| x 1 ··· x m | is studied via integral equations and is shown to possess a smaller critical correlation value.

On the average product of Gauss-Markov variables

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Lattice points in high-dimensional spheres

Upper bounds on the minimum distance of trellis codes

On the average product of Gauss-Markov variables