Quadratically Constrained Channels with Causal Adversaries

Abstract

We consider the problem of communication over a channel with a causal jamming adversary subject to quadratic constraints. A sender Alice wishes to communicate a message to a receiver Bob by transmitting a real-valued length-n codeword x = (x 1 , . . . , x n ) through a communication channel. Alice and Bob do not share common randomness. Knowing Alice's encoding strategy, a jammer James chooses a real-valued length-n adversarial noise sequence s = (s 1 , . . . , s n ) in a causal manner: each s t (1 ≤ t ≤ n) can only depend on (x 1 , . . . , x t ).Bob receives y, the sum (over R) of Alice's transmission x and James' jamming vector s, and is required to reliably estimate Alice's message from this sum.In this work we characterize the channel capacity for such a channel as the limit superior of the optimal values C n P N of a series of optimizations. Upper and lower bounds on C n P N are provided both analytically and numerically. Interestingly, unlike many communication problems, in this causal setting Alice's optimal codebook may not have a uniform power allocation -for certain SNR a codebook with a two-level uniform power allocation results in a strictly higher rate than a codebook with a uniform power allocation would.Kabatiansky and Levenshtein in [7] derived the tightest known outer bounds 2 . In Figure 2, we plot the GV-type bound in [5] and the LP bound in [7] , together with the omniscient capacity C ob , as references for our results in this work.Causal The primary focus of this work is on causal adversaries. Channels with causal adversaries can be considered as a special case of arbitrarily varying channels (AVCs) [8,9]. The causality assumption is physically reasonable in many engineering situations. In [10] (c.f., in page 224), an arbitrarily "star" varying channel is introduced such that at each time step t, the state s t can only depend on previously transmitted symbols x 1 , . . . , x t−1 . Yet, as mentioned in [10], the general problem has not been fully tackled. It turns out that techniques from previous works on AVCs cannot be applied directly to channels with causal adversaries. The capacities of channels with causal adversaries are known in some special cases. For example, recent papers by Dey et al. [11] and Chen et al. [12] characterize the capacity region of binary bit-flip channels with causal (online) adversaries. The results in [13] extend these techniques to characterize the capacities of q-ary additive-error/erasure channels. Also, the capacities of binary erasure channels with causal adversaries are known by [12, 14]. Dey et al. [15, 16] considered a "delayed" causal adversary such that the state s t is only decided by x 1 , . . . , x t−∆ where ∆ can be an arbitrarily small (but constant) fraction of the code block-length n.At the risk of missing much of the relevant literature, we summarize below the known results, parametrized by the delay parameter ∆ going from −n to n.In general, adversaries with causality constraints may be weaker than omniscient adversaries, and stronger than obliv...

Quadratically Constrained Myopic Adversarial Channels

“…10a). The z-aware attack is novel in the context of myopic channels, but is also inspired by similar ideas in [26].…”

“…Remark X.6. The argument above subsumes one way to prove the Plotkin bound (via the Cauchy-Schwarz inequality -see, say, Li et al [26]) with the additional randomness in s z .…”

“…The symmetrization argument we use in Section X-2 is inspired by the "scaled babble-and-push" attack studied by Li et al [26]. This studies the attack where James generates an independent codeword x 1 and transmits px 1 ´zq{2, and lets us prove that the capacity is zero in the regime where σ 2 {P ă 4N {P ´2.…”

“…We could potentially extend this idea, along the lines of the scaled babble-and-push attack [26] in the following manner. In a subset of codeword indices, James uses the scale-and-babble attack.…”

“…2) An improved converse using symmetrization: When the encoder is a deterministic map from the set of messages to R n , we can give a better upper bound for certain values of the NSRs. This attack is based on the scaled babble-and-push attack designed by Li et al [26] for the quadratically constrained channel with a causal adversary. The basic idea in a symmetrization argument is to make sure that Bob is equally confused between the actual transmitted codeword and a random codeword independently chosen by James.…”