Michael O. Rabin

Efficient dispersal of information for security, load balancing, and fault tolerance

An Information Dispersal Algorithm (IDA) is developed that breaks a file <italic>F</italic> of length <italic>L</italic> = ↿ <italic>F</italic>↾ into <italic>n</italic> pieces <italic>F<subscrpt>i</subscrpt></italic>, l ≤ <italic>i</italic> ≤ <italic>n</italic>, each of length ↿<italic>F<subscrpt>i</subscrpt></italic>↾ = <italic>L</italic>/<italic>m</italic>, so that every <italic>m</italic> pieces suffice for reconstructing <italic>F</italic>. Dispersal and reconstruction are computationally efficient. The sum of the lengths ↿<italic>F</italic><subscrpt>i</subscrpt>↾ is (<italic>n</italic>/<italic>m</italic>) · <italic>L</italic>. Since <italic>n</italic>/<italic>m</italic> can be chosen to be close to l, the IDA is space efficient. IDA has numerous applications to secure and reliable storage of information in computer networks and even on single disks, to fault-tolerant and efficient transmission of information in networks, and to communications between processors in parallel computers. For the latter problem provably time-efficient and highly fault-tolerant routing on the <italic>n</italic>-cube is achieved, using just constant size buffers.

Finite automata and their decision problems

Finite automata are considered in this paper as instruments for classifying finite tapes. Each one-tape automaton defines a set of tapes, a two-tape automaton defines a set of pairs of tapes, et cetera. The structure of the defined sets is studied. Various generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined. Some decision problems concerning automata are shown to be solvable by effective algorithms; others turn out to be unsolvable by algorithms.

Efficient randomized pattern-matching algorithms

We present randomized algorithms to solve the following string-matching problem and some of its generalizations: Given a string X of length n (the pattern) and a string Y (the text), find the first occurrence of X as a consecutive block within Y. The algorithms represent strings of length n by much shorter strings called fingerprints, and achieve their efficiency by manipulating fingerprints instead of longer strings. The algorithms require a constant number of storage locations, and essentially run in real time. They are conceptually simple and easy to implement. The method readily generalizes to higher-dimensional patternmatching problems.

Probabilistic algorithm for testing primality

Abstract We present a practical probabilistic algorithm for testing large numbers of arbitrary form for primality. The algorithm has the feature that when it determines a number composite then the result is always true, but when it asserts that a number is prime there is a provably small probability of error. The algorithm was used to generate large numbers asserted to be primes of arbitrary and special forms, including very large numbers asserted to be twin primes. Theoretical foundations as well as details of implementation and experimental results are given.

Randomized byzantine generals

We present a randomized solution for the Byzantine Generals Problems. The solution works in the synchronous as well as the asynchronous case and produces Byzantine Agreement within a fixed small expected number of computational rounds, independent of the number n of processes and the bound t on the number of faulty processes. The solution uses A. Shamirs method for sharing secrets. It specializes to produce a simple solution for the Distributed Commit problem.

SUPER-EXPONENTIAL COMPLEXITY OF PRESBURGER ARITHMETIC

Lower bounds are established on the computational complexity of the decision problem and on the inherent lengths of proofs for two classical decidable theories of logic: the first-order theory of the real numbers under addition, and Presburger arithmetic — the first-order theory of addition on the natural numbers. There is a fixed constant c > 0 such that for every (nondeterministic) decision procedure for determining the truth of sentences of real addition and for all sufficiently large n, there is a sentence of length n for which the decision procedure runs for more than 2 cn steps. In the case of Presburger arithmetic, the corresponding bound is \({2^{{2^{cn}}}}\). These bounds apply also to the minimal lengths of proofs for any complete axiomatization in which the axioms are easily recognized.

Probabilistic Algorithms in Finite Fields

We present probabilistic algorithms for the problems of finding an irreducible polynomial of degree n over a finite field, finding roots of a polynomial, and factoring a polynomial into its irreducible factors over a finite field. All of these problems are of importance in algebraic coding theory, algebraic symbol manipulation, and number theory. These algorithms have a very transparent, easy to program structure. For finite fields of large characteristic p, so that exhaustive search through

On the advantages of free choice: a symmetric and fully distributed solution to the dining philosophers problem

{\text{Z}}_p

Weakly Definable Relations and Special Automata

, is not feasible, our algorithms are of lower order in the degrees of the polynomial and fields in question, than previously published algorithms.

Verifiable random functions

It is shown that distributed systems of probabilistic processors are essentially more powerful than distributed systems of deterministic processors, i.e., there are certain useful behaviors that can be realized only by the former. This is demonstrated on the dining philosophers problem. It is shown that, under certain natural hypotheses, there is no way the philosophers can be programmed (in a deterministic fashion) so as to guarantee the absence of deadlock (general starvation). On the other hand, if the philosophers are given some freedom of choice one may program them to guarantee that every hungry philosopher will eat (with probability one) under any circumstances (even an adversary scheduling). The solution proposed here is fully distributed and does not involve any central memory or any process with which every philosopher can communicate.