We present a framework for specifying, training, evaluating, and deploying machine learning models. Our focus is on simplifying cutting edge machine learning for practitioners in order to bring such technologies into production. Recognizing the fast evolution of the field of deep learning, we make no attempt to capture the design space of all possible model architectures in a domain-specific language (DSL) or similar configuration language. We allow users to write code to define their models, but provide abstractions that guide developers to write models in ways conducive to productionization. We also provide a unifying Estimator interface, making it possible to write downstream infrastructure (e.g. distributed training, hyperparameter tuning) independent of the model implementation.

We consider three channel models: the wiretap channel with M helpers, the K-user multiple access wiretap channel, and the K-user interference channel with an external eavesdropper, when no eavesdropper's channel state information (CSI) is available at the transmitters. In each case, we establish the optimal sum secure degrees of freedom (s.d.o.f.) by providing achievable schemes and matching converses. We show that the unavailability of the eavesdropper's channel state information at the transmitter (CSIT) does not reduce the s.d.o.f. of the wiretap channel with helpers. However, there is loss in s.d.o.f. for both the multiple access wiretap channel and the interference channel with an external eavesdropper. In particular, we show that in the absence of eavesdropper's CSIT, the K-user multiple access wiretap channel reduces to a wiretap channel with (K - 1) helpers from a sum s.d.o.f. perspective, and the optimal sum s.d.o.f. reduces from K(K-1) /K(K-1)+1 to K-1 K . For the interference channel with an external eavesdropper, the optimal sum s.d.o.f. decreases from K(K-1) /2K-1 to K-1/ 2 in the absence of the eavesdropper's CSIT. Our results show that the lack of eavesdropper's CSIT does not have a significant impact on the optimal s.d.o.f. for any of the three channel models, especially when the number of users is large. This implies that physical layer security can be made robust to the unavailability of eavesdropper CSIT at high signal-to-noise ratio regimes by the careful modification of the achievable schemes as demonstrated in this paper.

In this paper, we determine the entire secure degrees of freedom (s.d.o.f.) regions of the K-user Gaussian multiple access (MAC) wiretap channel and the K-user interference channel (IC) with secrecy constraints. For the IC, we consider three secrecy constraints: K-user IC with an external eavesdropper (ICEE), K-user IC with confidential messages (IC-CM), and their combination Kuser IC with confidential messages and external eavesdropper (IC-CM-EE). The converse for the IC includes constraints both due to secrecy as well as due to interference. For the IC, although the portion of the region close to the optimum sum s.d.o.f. point is governed by the upper bounds due to secrecy constraints, the other portions of the region are governed by the upper bounds due to interference constraints. Different from the existing literature, in order to fully understand the characterization of the s.d.o.f. region of the IC, one has to study the four-user case, i.e., the twoor three-user cases do not illustrate the full generality of the problem. In order to prove the achievability, we use the polytope structure of the converse region. In both MAC and IC cases, we develop explicit schemes that achieve the extreme points of the polytope region given by the converse. In particular, the extreme points of the MAC region are achieved by an m-user MAC wiretap channel with K - m helpers, i.e., by setting K - m users' secure rates to zero and utilizing them as pure (structured) cooperative jammers. The extreme points of the IC region are achieved by a (K - m)-user IC with confidential messages, m helpers, and N external eavesdroppers, for m ≥ 1 and a finite N. A byproduct of our results in this paper is that the sum s.d.o.f. is achieved only at one extreme point of the s.d.o.f. region, which is the symmetric-rate extreme point, for both MAC and IC channel models.

We revisit the recent secure degrees of freedom (s.d.o.f.) results for one-hop multiuser wireless networks by considering three fundamental wireless network structures: Gaussian wiretap channel with helpers, Gaussian multiple access wiretap channel, and Gaussian interference channel with secrecy constraints. We present main enabling tools and resulting communication schemes in an expository manner, along with key insights and design principles emerging from them. The main achievable schemes are based on real interference alignment, channel prefixing via cooperative jamming, and structured signalling. Real interference alignment enables aligning the cooperative jamming signals together with the message carrying signals at the eavesdroppers to protect them akin to one-time-pad protecting messages in wired systems. Real interference alignment also enables decodability at the legitimate receivers by rendering message carrying and cooperative jamming signals separable, and simultaneously aligning the cooperative jamming signals in the smallest possible subspace. The main converse techniques are based on two key lemmas which quantify the secrecy penalty by showing that the net effect of an eavesdropper on the system is that it eliminates one of the independent channel inputs; and the role of a helper by developing a direct relationship between the cooperative jamming signal of a helper and the message rate. These two lemmas when applied according to the unique structure of individual networks provide tight converses. Finally, we present a blind cooperative jamming scheme for the helper network with no eavesdropper channel state information at the transmitters that achieves the same optimal s.d.o.f. as in the case of full eavesdropper channel state information.

We determine the exact sum secure degrees of freedom (d.o.f.) of the K-user Gaussian interference channel. We consider three different secrecy constraints: 1) K-user interference channel with one external eavesdropper (IC-EE); 2) K-user interference channel with confidential messages (IC-CM); and 3) K-user interference channel with confidential messages and one external eavesdropper (IC-CM-EE). We show that for all of these three cases, the exact sum secure d.o.f. is K(K - 1)/(2K - 1). We show converses for IC-EE and IC-CM, which imply a converse for IC-CM-EE. We show achievability for IC-CM-EE, which implies achievability for IC-EE and IC-CM. Our converse is based on developing a direct relationship between the differential entropies of the channel inputs and the rates of the users, and quantifying the effect of eavesdropping on the rates in terms of the differential entropies of the eavesdroppers' observations. Our achievability is based on structured signaling, structured cooperative jamming, channel prefixing, and asymptotic real interference alignment. While the traditional interference alignment provides some amount of secrecy by mixing unintended signals in a smaller subspace at every receiver, in order to attain the optimum sum secure d.o.f., we incorporate structured cooperative jamming into the achievable scheme, and intricately design the structure of all of the transmitted signals jointly.

We examine the separability of the parallel multiple access wiretap channel. Separability, when exists, is useful as it enables us to code separately over parallel channels, and still achieve the optimum overall performance. It is well-known that the parallel single-user channel, parallel multiple access channel (MAC) and parallel broadcast channel (BC) are all separable, however, the parallel interference channel (IC) is not separable in general. In this paper, we show that, while MAC is separable MAC wiretap channel is not separable in general. We prove this via a specific linear deterministic MAC wiretap channel. We then show that even the Gaussian MAC wiretap channel is inseparable in general. Finally, we show that, when the channel gains are drawn from continuous distributions, and when the secure degrees of freedom (s.d.o.f.) region is considered, then the Gaussian MAC wiretap channel is almost surely separable.

The sum secure degrees of freedom (s.d.o.f.) of the K-user interference channel (IC) with secrecy constraints has been determined recently as equation [1], [2]. In this paper, we determine the entire s.d.o.f. region of this channel model. The converse includes constraints both due to secrecy as well as due to interference. Although the portion of the region close to the optimum sum s.d.o.f. point is governed by the upper bounds due to secrecy constraints, the other portions of the region are governed by the upper bounds due to interference constraints. Different from the existing literature, in order to fully understand the characterization of the s.d.o.f. region of the IC, one has to study the 4-user case, i.e., the 2 or 3-user cases do not illustrate the generality of the problem. In order to prove the achievability, we use the polytope structure of the converse region. The extreme points of the converse region are achieved by a (K - m)-user IC with confidential messages, m helpers, and N external eavesdroppers, for m ≥ 1 and a finite N. A byproduct of our results in this paper is that the sum s.d.o.f. is achieved only at one extreme point of the s.d.o.f. region, which is the symmetric-rate extreme point.

We study the secure degrees of freedom (d.o.f.) of one-hop wireless networks by considering four fundamental wireless network structures: 1) Gaussian wiretap channel; 2) Gaussian broadcast channel with confidential messages; 3) Gaussian interference channel with confidential messages; and 4) Gaussian multiple access wiretap channel. The secrecy capacity of the canonical Gaussian wiretap channel does not scale with the transmit power, and hence, the secure d.o.f. of the Gaussian wiretap channel with no helpers is zero. It has been known that a strictly positive secure d.o.f. can be obtained in the Gaussian wiretap channel by using a helper, which sends structured cooperative signals. We show that the exact secure d.o.f. of the Gaussian wiretap channel with a helper is 1/2. Our achievable scheme is based on real interference alignment and cooperative jamming, which renders the message signal and the cooperative jamming signal separable at the legitimate receiver, but aligns them perfectly at the eavesdropper preventing any reliable decoding of the message signal. Our converse is based on two key lemmas. The first lemma quantifies the secrecy penalty by showing that the net effect of an eavesdropper on the system is that it eliminates one of the independent channel inputs. The second lemma quantifies the role of a helper by developing a direct relationship between the cooperative jamming signal of a helper and the message rate. We extend this result to the case of M helpers, and show that the exact secure d.o.f. in this case is M/M+1.We then generalize this approach to more general network structures with multiple messages. We show that the sum secure d.o.f. of the Gaussian broadcast channel with confidential messages and M helpers is 1, the sum secure d.o.f. of the twouser interference channel with confidential messages is 2/3, the sum secure d.o.f. of the two-user interference channel with confidential messages and M helpers is 1, and the sum secure d.o.f. of the K-user multiple access wiretap channel is K(K-1)/K(K-1)+1.

Abstract—We determine the exact sum secure degrees of freedom (d.o.f.) of the K-user Gaussian interference channel. We consider three different secrecy constraints: 1) K-user interference channel with one external eavesdropper (IC-EE), 2) K-user interference channel with confidential messages (IC-CM), and 3) K-user interference channel with confidential messages and one external eavesdropper (IC-CM-EE). We show that for all of these three cases, the exact sum secure d.o.f. is K(K−1)

We show that the sum secure degrees of freedom (d.o.f.) of the K-user Gaussian multiple access (MAC) wiretap channel is K(K-1)/K(K-1)+1. Our achievability is based on real interference alignment and structured cooperative jamming. Each user divides its message into K - 1 sub-messages, and sends a linear combination of signals carrying these sub-messages together with a structured cooperative jamming signal. All cooperative jamming signals are aligned in a single dimension at the legitimate receiver allowing for reliable decoding of the message carrying signals by the legitimate receiver. Each cooperative jamming signal is aligned with K-1 message signals at the eavesdropper limiting the information leakage rate to the eavesdropper. We provide a matching converse establishing the exact sum secure d.o.f. of the Gaussian MAC wiretap channel as K(K-1)/K(K-1)+1.

In this paper, we study the sum secure degrees of freedom (d.o.f.) of two-unicast layered wireless networks. Without any secrecy constraints, the sum d.o.f. of this class of networks was studied by and shown to take only one of three possible values: 1, 3/2 and 2, for all network configurations. We consider the setting where, in addition to being reliably transmitted, each message is required to be kept information-theoretically secure from the unintended receiver. We show that the sum secure d.o.f. can only take one of five possible values: 0, 2/3, 1, 3/2, 2, for all network configurations. To determine the sum secure d.o.f., we divide the class of two-unicast layered networks into several sub-classes, and propose an achievable scheme based on the specific structure of the networks in each sub-class. Our achievable schemes are based on real interference alignment, cooperative jamming, interference neutralization and cooperative jamming neutralization techniques.

In this paper, we study the sum secure degrees of freedom (d.o.f.) of two-unicast layered wireless networks. Without a secrecy constraint, the sum d.o.f. of this class of networks was studied by [1] and shown to take only one of three possible values: 1, 3/2 and 2, for all network configurations. We consider the setting where the message of each source-destination pair must be kept information-theoretically secure from the unintended receiver. We show that the sum secure d.o.f. can take 0, 1, 3/2, 2 and at most countably many other positive values, which we enumerate.

In this paper, we study the two-user one-sided interference channel with confidential messages. In this interference channel, in addition to the usual selfishness of the users, the relationship between the two pairs of users is further adversarial in the sense of both receivers' desires to eavesdrop on the communication of the other pair. We develop a game-theoretic model to study the information-theoretic secure communications in this setting. We first start with a game-theoretic model where each pair's payoff is their own secrecy rate. The analysis of the binary deterministic interference channel with this payoff function shows that self-jamming of a transmitter, which injures the eavesdropping ability of its own receiver, is not excluded by the Nash equilibria. We propose a refinement for the payoff function by explicitly accounting for the desire of the receiver to eavesdrop on the other party's communication. This payoff function captures the adversarial relationship between the two pairs of users better. We determine the Nash equilibria for the binary deterministic channel for both payoff functions.

IP radio access networks (RAN) are expected to employ CDMA as the next generation access networks (Bu et al., 2004). In the previous work, the problem about congestion of voice in IP-based CDMA RAN was examined and several policies were brought forward and evaluated with the assistance of simulations (Kasera et al., 2005). In this paper, we first develop a mathematical model to investigate the multi-region soft-handoff of the IP-based CDMA RAN and analyze its systematic characteristics. The effectiveness of the developed mathematical model is demonstrated by simulations. We then propose a max-leg-limit control policy to improve the system performance in terms of the blocking rate and the number of users. We also extend it to a more general situation, where instant video, instant data and voice are provided within an IP-based CDMA RAN. For such a multiple priority instant service system, we propose two control mechanisms, borrow-leg and router buffer preservation, to control the quality of services. Simulation shows that the control mechanisms are effective in keeping the quality of multi-priority services.