A new architecture called integer-forcing (IF) linear receiver has been recently proposed for multiple-input multipleoutput (MIMO) fading channels, wherein an appropriate integer linear combination of the received symbols has to be computed as a part of the decoding process. In this paper, we propose a method based on Hermite-Korkine-Zolotareff (HKZ) and Minkowski lattice basis reduction algorithms to obtain the integer coefficients for the IF receiver. We show that the proposed method provides a lower bound on the ergodic rate, and achieves the full receive diversity. Suitability of complex Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm (CLLL) to solve the problem is also investigated. Furthermore, we establish the connection between the proposed IF linear receivers and lattice reduction-aided MIMO detectors (with equivalent complexity), and point out the advantages of the former class of receivers over the latter. For the 2 × 2 and 4 × 4 MIMO channels, we compare the coded-block error rate and bit error rate of the proposed approach with that of other linear receivers. Simulation results show that the proposed approach outperforms the zero-forcing (ZF) receiver, minimum mean square error (MMSE) receiver, and the lattice reduction-aided MIMO detectors.

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The Rényi divergence is a measure of closeness of two probability distributions. We show that it can often be used as an alternative to the statistical distance in security proofs for lattice-based cryptography. Using the Rényi divergence is particularly suited for security proofs of primitives in which the attacker is required to solve a search problem (e.g., forging a signature). We show that it may also be used in the case of distinguishing problems (e.g., semantic security of encryption schemes), when they enjoy a public sampleability property. The techniques lead to security proofs for schemes with smaller parameters, and sometimes to simpler security proofs than the existing ones.If the event E occurs with significant probability under D 1 , and if the SD (resp. RD) is small, then the event E also occurs with significant probability under D 2 . These properties are particularly handy when the success of an attacker against a given scheme can be described as an event whose probability should be negligible, e.g., the attacker outputs a new valid message-signature pair for a signature scheme. If in the attacker succeeds with good probability in the real scheme based on distribution D 1 , then it also succeeds with good probability in the simulated scheme (of the security proof) based on distribution D 2 .To make the SD probability preservation property useful, it must be ensured that the SD ∆(D 1 , D 2 ) is smaller than any D 1 (E) that the security proof must handle. Typically, the quantity D 1 (E) is assumed to be greater than some success probability lower bound ε, which is of the order of 1/poly(λ) where λ refers to the security parameter, or even 2 −o(λ) if the proof handles attackers whose success probabilities can be sub-exponentially small (which we believe better reflects practical objectives). As a result, the SD ∆(D 1 , D 2 ) must be < ε for the SD probability preservation property to be relevant. Similarly, the RD probability preservation property is non-vacuous when the RD R a (D 1 D 2 ) is ≤ poly(1/ε). In many cases, the latter seems less demanding than the former: in all our applications of RD, the RD between D 1 and D 2 is small while their SD is too large for the SD probability preservation to be applicable. In fact, as we will see in Subsection 2.3, the RD becomes sufficiently small to be useful before the SD when sup x D 1 (x)/D 2 (x) tends to 1. This explains the superiority of the RD in several of our applications.Although RD seems more amenable than SD for search problems, it seems less so for distinguishing problems. A typical cryptographic example is semantic security of an encryption scheme. Semantic security requires an adversary A to distinguish between the encryption distributions of two plaintext messages of its choosing: the distinguishing advantage Adv A (D 1 , D 2 ), defined as the difference of probabilities that A outputs 1 using D 1 or D 2 , should be large. In security proofs, algorithm A is often called on distributions D 1 and D 2 that are close to D 1 and D 2 (respectivel...

Abstract-A key issue in compute-and-forward for physical layer network coding scheme is to determine a good function of the received messages to be reliably estimated at the relay nodes. We show that this optimization problem can be viewed as the problem of finding the closest point of Z [i] n to a line in the n-dimensional complex Euclidean space, within a bounded region around the origin. We then use the complex version of the LLL lattice basis reduction (CLLL) algorithm to provide a reduced complexity suboptimal solution as well as an upper bound to the minimum distance of the lattice point from the line. Using this bound we are able to find a lower bound to the ergodic rate and a union bound estimate on the error performance of a lattice constellation used for lattice network coding. We compare performance of the CLLL with a more complex iterative optimization method as well as with a simple quantized search. Simulations show how CLLL can trade some performance for a lower complexity.Index Terms-Ergodic rate, compute-and-forward, CLLL algorithm, quantized error, successive refinement.

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