One of the most important properties of a cryptographic system is a proof of its security. In the present work, information-theoretic meth¬ ods are used for proving the security of unconditionally secure cryptosystems. The security of such systems does not depend on unproven intractability assumptions. A survey of entropy measures and their applications in cryptography is presented. A new information measure, smooth entropy, is intro¬ duced to quantify the number of almost uniform random bits that can be extracted from a source by probabilistic algorithms. Smooth entropy unifies previous work on privacy amplification in cryptography and on entropy smoothing in theoretical computer science. It enables a system¬ atic investigation of the spoiling knowledge proof technique to obtain lower bounds on smooth entropy. The Renyi entropy of order at least 2 of a random variable is a lower bound for its smooth entropy, whereas an assumption about Renyi entropy of order 1, which is equivalent to the Shannon entropy, is too weak to guarantee any non-trivial amount of smooth entropy. The gap between Renyi entropy of order 1 and 2 is closed by proving that Renyi entropy of order a between 1 and 2 is a lower bound for smooth entropy, up to a small parameter depending on a, the alphabet size, and the failure probability. The operation of many unconditionally secure cryptosystems can be divided into the three phases advantage distillation, information recon¬ ciliation, and privacy amplification. The relation between privacy am¬ plification and information reconciliation is investigated, in particular, the effect of side information, obtained by an adversary through an ini¬ tial reconciliation step, on the size of the secret key that can be distilled safely by subsequent privacy amplification. It is shown that each bit of side information reduces the size of the key that can be generated by at

Entropy measures and unconditional security in cryptography

We analyze physical-layer security based on the premise that the coding mechanism for secrecy over noisy channels is tied to the notion of channel resolvability. Instead of considering capacity-based constructions, which associate to each message a subcode that operates just below the capacity of the eavesdropper's channel, we consider channel-resolvability-based constructions, which associate to each message a subcode that operates just above the resolvability of the eavesdropper's channel. Building upon the work of Csiszar and Hayashi, we provide further evidence that channel resolvability is a powerful and versatile coding mechanism for secrecy by developing results that hold for strong secrecy metrics and arbitrary channels. Specifically, we show that at least for symmetric wiretap channels, random capacity-based constructions fail to achieve the strong secrecy capacity, while channel-resolvability-based constructions achieve it. We then leverage channel resolvability to establish the secrecy-capacity region of arbitrary broadcast channels with confidential messages and a cost constraint for strong secrecy metrics. Finally, we specialize our results to study the secrecy capacity of wireless channels with perfect channel state information (CSI), mixed channels, and compound channels with receiver CSI, as well as the secret-key capacity of source models for secret-key agreement. By tying secrecy to channel resolvability, we obtain achievable rates for strong secrecy metrics with simple proofs.

Strong Secrecy From Channel Resolvability

There is a difference between the optimal rates of fixed-length source coding and intrinsic randomness when we care about the second-order asymptotics. We prove this difference for general information sources and then investigate independent and identically distributed (i.i.d.) random variables and Markovian variables as examples. The difference is demonstrated through an investigation of the second-order asymptotic behavior of the rates. A universal fixed-length source code attaining the second-order optimal rate is also proposed. The difference between the rates of fixed-length source coding and intrinsic randomness proves that the outputs of fixed-length source codes are not uniformly distributed.

Second-Order Asymptotics in Fixed-Length Source Coding and Intrinsic Randomness

We analyze physical-layer security based on the premise that the coding mechanism for secrecy over noisy channels is tied to the notion of channel resolvability. Instead of considering capacity-based constructions, which associate to each message a sub-code that operates just below the capacity of the eavesdropper's channel, we consider channel-resolvability-based constructions, which associate to each message a sub-code that operates just above the resolvability of the eavesdropper's channel. Building upon the work of Csiszar and Hayashi, we provide further evidence that channel resolvability is a powerful and versatile coding mechanism for secrecy by developing results that hold for strong secrecy metrics and arbitrary channels. 
Specifically, we show that at least for symmetric wiretap channels, random capacity-based constructions fail to achieve the strong secrecy capacity while channel-resolvability-based constructions achieve it. We then leverage channel resolvability to establish the secrecy-capacity region of arbitrary broadcast channels with confidential messages and a cost constraint for strong secrecy metrics. Finally, we specialize our results to study the secrecy capacity of wireless channels with perfect channel state information, mixed channels and compound channels with receiver Channel State Information (CSI), as well as the secret-key capacity of source models for secret-key agreement. By tying secrecy to channel resolvability, we obtain achievable rates for strong secrecy metrics with simple proofs.

Strong Secrecy from Channel Resolvability

The problem of generating a random number with an arbitrary probability distribution by using a general biased M-coin is studied. An efficient and very simple algorithm based on the successive refinement of partitions of the unit interval (0, 1), which we call the interval algorithm, is proposed. A fairly tight evaluation on the efficiency is given. Generalizations of the interval algorithm to the following cases are investigated: (1) output sequence is independent and identically distributed (i.i.d.); (2) output sequence is Markov; (3) input sequence is Markov; (4) input sequence and output sequence are both subject to arbitrary stochastic processes.

Interval algorithm for random number generation

Suppose we are given a random source and want to use it as a random number generator; at what rate can we generate fair bits from it? We address this question in an information-theoretic setting by allowing for some arbitrarily small but nonzero deviation from "ideal" random bits. We prove our results with three different measures of approximation between the ideal and the obtained probability distributions: the variational distance, the d-bar distance, and the normalized divergence. Two different contexts are studied: fixed-length and variable-length random number generation. The fixed-length results of this paper provide an operational characterization of the inf-entropy rate of a source, defined in Han and Verdu (see ibid., vol.39, no.3, p.752-772, 1993) and the variable-length results characterize the liminf of the entropy rate, thereby establishing a pleasing duality with the fundamental limits of source coding. A feature of our results is that we do not restrict ourselves to ergodic or to stationary sources. >

S. Vembu

Papers

Generating random bits from an arbitrary source: fundamental limits