Probabilistic Error Analysis of Limited-Precision Stochastic Rounding

We consider the problem of scattering n robots in a two dimensional continuous space. As this problem is impossible to solve in a deterministic manner, all solutions must be probabilistic. We investigate the amount of randomness (that is, the number of random bits used by the robots) that is required to achieve scattering. We first prove that n log n random bits are necessary to scatter n robots in any setting. Also, we give a sufficient condition for a scattering algorithm to be random bit optimal. As it turns out that previous solutions for scattering satisfy our condition, they are hence proved random bit optimal for the scattering problem. Then, we investigate the time complexity of scattering when strong multiplicity detection is not available. We prove that such algorithms cannot converge in constant time in the general case and in o(log log n) rounds for random bits optimal scattering algorithms. However, we present a family of scattering algorithms that converge as fast as needed without using multiplicity detection. Also, we put forward a specific protocol of this family that is random bit optimal (O(n log n) random bits are used) and time optimal (O(log log n) rounds are used). This improves the time complexity of previous results in the same setting by a log n factor. Aside from characterizing the random bit complexity of mobile robot scattering, our study also closes the time complexity gap with and without strong multiplicity detection (that is, O(1) time complexity is only achievable when strong multiplicity detection is available, and it is possible to approach a constant value as desired otherwise).

It is shown how to efficiently construct a small probability space on n binary random variables such that for every subset, its parity is either zero or one with “almost” equal probability. They are called $\epsilon $-biased random variables. The number of random bits needed to generate the random variables is $O(\log n + \log \frac{1}{\epsilon })$. Thus, if $\epsilon $ is polynomially small, then the size of the sample space is also polynomial. Random variables that are $\epsilon $-biased can be used to construct “almost” k-wise independent random variables where $\epsilon $ is a function of k.These probability spaces have various applications: l. Derandomization of algorithms: Many randomized algorithms that require only k-wise independence of their random bits (where k is bounded by $O(\log n)$), can be derandomized by using $\epsilon $-biased random variables. 2. Reducing the number of random bits required by certain randomized algorithms, e.g., verification of matrix multiplication. 3. Exhaustive tes...

The number of random bits required to approximate a target distribution in terms of un-normalized informational divergence is considered. It is shown that for a variable-to-variable length encoder, this number is lower bounded by the entropy of the target distribution. A fixed-to-variable length encoder is constructed using M-type quantization and Tunstall coding. It is shown that the encoder achieves in the limit an un-normalized informational divergence of zero with the number of random bits per generated symbol equal to the entropy of the target distribution. Numerical results show that the proposed encoder significantly outperforms the optimal block-to-block encoder in the finite length regime.

IoT devices have very strong requirements on all the resources such as memory, randomness, energy and execution time. This paper proposes a number of scalable shuffling techniques as countermeasures against side channel analysis. Some extensions of an existing technique called Random Start Index (RSI) are suggested in this paper. Moreover, two new shuffling techniques Reverse Shuffle (RS) and Sweep Swap Shuffle (SSS) are described within their possible extensions. Extensions of RSI, RS and SSS might be implemented in a constrained environment with a small data and time overhead. Each of them might be implemented using different amount of randomness and thus, might be fine-tuned according to requirements and constraints of a cryptographic system such as time, memory, available number of random bits, etc. RSI, RS, SSS and their extensions are described using SubBytes operation of AES-128 block cipher as an example, but they might be used with different operations of AES as well as with other algorithms. This paper also analyses RSI, RS and SSS by comparing their properties such as number of total permutations that might be generated using a fixed number of random bits, data complexity, time overhead and evaluates their resistance against some known side-channel attacks such as correlation power analysis and template attack. Several of proposed shuffling schemes are implemented on a 8-bit microcontroller that uses them to shuffle the first and the last rounds of AES-128.

Characterizing Linear Size Circuits in Terms of Privacy

We present a scheme to convert self-stabilizing algorithms that use randomization during and following convergence to self-stabilizing algorithms that use randomization only during convergence. We thus reduce the number of random bits from an infinite number to a bounded number. The scheme is applicable to the cases in which there exits a local predicate for each node, such that global consistency is implied by the union of the local predicates. We demonstrate our scheme over the token circulation algorithm of Herman and the recent constant time Byzantine self-stabilizing clock synchronization algorithm by Ben-Or, Dolev and Hoch. The application of our scheme results in the first constant time Byzantine self-stabilizing clock synchronization algorithm that uses a bounded number of random bits.

In the oblivious path selection problem, each packet in the network independently chooses a path, which is an important property if the routing algorithm is to be independent of the traffic distribution. The quality of the paths is determined by the congestion, C, the maximum number of paths crossing an edge, and the dilation, D, the maximum path length. So far, the oblivious algorithms studied in the literature have focused on minimizing the congestion while ignoring the dilation. An open problem is to give algorithms for networks in which C and D can be controlled simultaneously. Here, we solve this problem for the d-dimensional mesh. We present an oblivious algorithm for which C and D are both within O(d <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) of the optimal. The algorithm uses randomization and we show that the number of random bits required per packet is within O(d) of the minimum number of random bits required by any algorithm that obtains the same congestion. For a fixed d, our algorithm is asymptotically optimal.

We study gossip algorithms for the rumor spreading problem which asks one node to deliver a rumor to all nodes in an unknown network. We present the first protocol for any expander graph $G$ with $n$ nodes such that, the protocol informs every node in $O(\log n)$ rounds with high probability, and uses $\tilde{O}(\log n)$ random bits in total. The runtime of our protocol is tight, and the randomness requirement of $\tilde{O}(\log n)$ random bits almost matches the lower bound of $\Omega(\log n)$ random bits for dense graphs. We further show that, for many graph families, polylogarithmic number of random bits in total suffice to spread the rumor in $O(\mathrm{poly}\log n)$ rounds. These results together give us an almost complete understanding of the randomness requirement of this fundamental gossip process. Our analysis relies on unexpectedly tight connections among gossip processes, Markov chains, and branching programs. First, we establish a connection between rumor spreading processes and Markov chains, which is used to approximate the rumor spreading time by the mixing time of Markov chains. Second, we show a reduction from rumor spreading processes to branching programs, and this reduction provides a general framework to derandomize gossip processes. In addition to designing rumor spreading protocols, these novel techniques may have applications in studying parallel and multiple random walks, and randomness complexity of distributed algorithms.

We present a Monte Carlo algorithm for testing multivariate polynomial identities over any field using fewer random bits than other methods. To test if a polynomial P x xn is zero, our method uses Pn i dlog di e random bits , where di is the degree of xi in P , to obtain any inverse polynomial error in polynomial time. The algorithm applies to polynomials given as a black box or in some implicit representation such as a straight-line program. Our method works by evaluating P at truncated formal power series representing square roots of irreducible polynomials over the field. This approach is similar to that of Chen and Kao [CK97], but with the advantage that the techniques are purely algebraic and apply to any field. We also prove a lower bound showing that the number of random bits used by our algorithm is essentially optimal in the black-box model.

We consider the problem of universal delay-limited simulation of an unknown information source of a certain parametric family (e.g., the family of memoryless sources or Markov sources of a given order), given a training sequence from that source and a stream of purely random bits. In the delay-limited setting, the simulation algorithm generates a random sequence sequentially, by delivering one symbol for each training symbol that is made available after a given initial delay, whereas the random bits are assumed to be available on demand. The goal of universal simulation is that the probability law of the generated sequence be identical to that of the training sequence, with minimum mutual information between the random processes generating both sequences. We characterize the optimal delay-limited simulation scheme and upper-bound the expected number of random bits it consumes. As in the non-sequential case, this upper bound is related to the entropy rate of the source

In 1952, von Neumann introduced the rejection method for random variate generation. We revisit this algorithm when we have a source of perfect bits at our disposal. In this random bit model, there are universal lower bounds for generating a random variate with a given density to within an accuracy $\epsilon$ derived by Knuth and Yao, and refined by the authors. In general, von Neumann's method fails in this model. We propose a modification that insures proper behavior for all Riemann-integrable densities on compact sets, and show that the expected number of random bits needed behaves optimally with respect to universal lower bounds. In particular, we introduce the notion of an oracle that evaluates the supremum and infimum of a function on any rectangle of $\mathbb{R}^{d}$, and develop a quadtree-style extension of the classical rejection method.

In this paper, we analyze two Las Vegas algorithms for the problem of checking if a search item is present in an unordered array. This problem, henceforth called the Linear Search (LS) problem, is well-studied and has applications in a host of domains, including computer graphics and computational biology. While the problem can be solved deterministically by scanning the entire array, this becomes impractical for large datasets that cannot fit into the main memory. To address this issue, algorithms that operate on a portion of the input array are necessary. Likewise, minimizing array accesses in computational biology is crucial, due to the high cost of accessing an element. This paper presents two Las Vegas algorithms for the LS problem that are optimal in their respective models. We show that a partition-based Las Vegas algorithm for the LS problem (with an additional condition that will be discussed) with one bit must have expected comparison complexity at least [Formula: see text]. We also prove that any Las Vegas algorithm for the LS problem with an arbitrary number of random bits has expected comparison complexity at least [Formula: see text]. It is important to note the latter of our two algorithms is analyzed in an extremely restrictive computational model. Despite the limitations on the computational model, our approach is optimal over all Las Vegas algorithms, when the number of permitted random bits is unbounded. Finally, we analyze the expected comparison complexity of our approach when the partition size is an arbitrary value [Formula: see text], for [Formula: see text]. Our algorithms offer an efficient approach for the LS problem in various practical applications.

Random number generators (RNG) are essential elements in many cryptographic systems. True random number generators (TRNG) rely upon sources of randomness from natural processes such as those arising from quantum mechanics phenomena. We demonstrate that a quantum computer can serve as a high-quality, weakly random source for a generalized user-defined probability mass function (PMF). Specifically, QC measurement implements the process of variate sampling according to a user-specified PMF resulting in a word comprised of electronic bits that can then be processed by an extractor function to address inaccuracies due to non-ideal quantum gate operations and other system biases. We introduce an automated and flexible method for implementing a TRNG as a programmed quantum circuit that executes on commercially-available, gate-model quantum computers. The user specifies the desired word size as the number of qubits and a definition of the desired PMF. Based upon the user specification of the PMF, our compilation tool automatically synthesizes the desired TRNG as a structural OpenQASM file containing native gate operations that are optimized to reduce the circuit's quantum depth. The resulting TRNG provides multiple bits of randomness for each execution/measurement cycle; thus, the number of random bits produced in each execution is limited only by the size of the QC. We provide experimental results to illustrate the viability of this approach.

We consider two hashing models for storing a set S ⊂ {0, 1, 2, …, m - 1} in a table T of size n.The first model uses universal hashing for a partially loaded table. A set of hash functions is universal if, for any the input set, a randomly selected function has an efficient expected performance. Universal hash functions originate in [CW79], where they were used for open hashing using chaining. [CW79] poses as an open question whether comparable results can be achieved for any closed hashing schemes.The second model is perfect hashing for a full table. In preprocessing the input set is used to determine a hash function that achieves some desired performance criteria. This model was used among others in [ME82] and [FKS84].In both models a key problem is to construct a “small” set of functions, which will permit a short description (program) for each function in the set.We show, for the first time, that universal hashing can be successfully used for closed hashing and in particular for double hashing. Specifically, the set of congruential polynomials of O(log n) degree is universal for double hashing if the table load is below .75; the program size (or number of random bits generated by the algorithm) is O(log log m + log2n).For perfect hashing, we obtain nearly tight results on the size of oblivious O(1)-probe hash functions: Oblivious k-probe hash functions require O(n/k2e-k + log log m) bits of description.A probabilistic construction is presented, which shows that oblivious k-probe hash functions, can be specified in O(ne-k + log log m) bits, which nearly matches the above lower bound.We give a variation of an O(1) time 1-probe (perfect) hash function that can be specified in O(n + log log m) bits, which is tight to within a constant factor of the lower bound.In view of the adaptive schemes presented in [FNSS88], these bounds establish a significant gap between oblivious and non-oblivious O(1)-probe search.

Algorithms based on rapidly mixing Markov chains are discussed to produce nearly uniformly distributed random elements in abelian groups of finite order. Let A be an abelian group generated by set S. Then one can generate ϵ‐nearly uniform random elements of A using 4|S|log(|A|/ϵ) log(|A|) additions and the same number of random bits. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005