Yoshihide Igarashi

Broadcasting in Hypercubes with Randomly Distributed Byzantine Faults

We study all-to-all broadcasting in hypercubes with randomly distributed Byzantine faults. We construct an efficient broadcasting scheme BC1-n-cube running on the n-dimensional hypercube (n-cube for short) in 2n rounds, where for communication by each node of the n-cube, only one of its links is used in each round. The scheme BC1-n-cube can tolerate ⌊(n −1)/2⌋ Byzantine node and/or link faults in the worst case. If there are at most f Byzantine faulty nodes randomly distributed in the n-cube, BC1-n-cube succeeds with a probability higher than 1-(64nf/2 n )⌈n/2⌉. In other words, if 1/(64nk) of all the nodes (i.e., 2n/(64nk) nodes) fail in Byzantine manner randomly in the n-cube, then the scheme succeeds with a probability higher than 1 −k−⌈ n/2 ⌉. We also consider the case where all nodes are faultless but links may fail randomly in the n-cube. Scheme BC1-n-cube succeeds with a probability higher than 1 −k−⌈ n/2 ⌉ provided that not more than l/(64(n + 1)k) of all the links in the n-cube fail in Byzantine manner randomly. For the case where only links may fail, we give another broadcasting scheme BC2-n-cube which runs in 2n2 rounds. Broadcasting by BC2-n-cube is successful with a high probability if the number of Byzantine faulty links randomly distributed in the n-cube is not more than a constant fraction of the total number of links. That is, it succeeds with a probability higher than 1−n·k−⌈ n/2 ⌉ if l/(48k) of all the links in the n-cube fail in Byzantine manner randomly.

Break Finite Automata Public Key Cryptosystem

In this paper we break a 10-years standing public key cryptosystem, Finite Automata Public Key Cryptosystem(FAPKC for short). The security of FAPKC was mainly based on the difficulty of finding a special common left factor of two given matrix polynomials. We prove a simple but previously unknown property of the input-memory finite automata. By this property, we reduce the basis of the FAPKCs security to the same problem in module matrix polynomial rings. The problem turns out to be easily solved. Hence, we can break FAPKC by constructing decryption automata from the encryption automaton (public key). We describe a modification of FAPKC which can resist above attack.

Independent Spanning Trees of Product Graphs

A graph G is called an n-channel graph at vertex r if there are n independent spanning trees rooted at r. A graph G is called an n-channel graph if for every vertex u, G is an n-channel graph at u. Independent spanning trees of a graph play an important role in faulttolerant broadcasting in the graph. In this paper we show that if G1 is an n1-channel graph and G2 is an n2-channel graph, then G1×G2 is an (n1+n2)-channel graph. We prove this fact by a construction of n1+n2 independent spanning trees of G1 × G2 from n1 independent spanning trees of G1 and n2 independent spanning trees of G2.

Optimal Time Broadcasting in Faulty Star Networks

This paper investigates fault-tolerant broadcasting in star networks. We propose a non-adaptive single-port broadcasting scheme in the n-star network such that it tolerates n — 2 faults in the worst case and completes the broadcasting in O(n log n) time. The existence of such a broadcasting scheme was not known before. A new technique, called diffusing- and -disseminating, is introduced to design our broadcasting scheme. This technique is useful to improve the efficiency of broadcasting in star networks. We analyze the reliability of the broadcasting scheme in the case where faults are randomly distributed in the n-star network. The broadcasting scheme in the n-star network can tolerate (n!)* random faults with a high probability, where α is any constant less than 1.

Broadcasting in Star Graphs with Byzantine Failures

We study broadcasting in star graphs containing faulty nodes and/or edges of the Byzantine type. We propose a single-port broadcasting scheme that tolerates up to ⌊n−3d−1/2⌋ Byzantine failures in the n-star graph, where d is the smallest positive integer satisfying n≤d!. The broadcasting time of the scheme is logarithmic in the number of nodes of the n-star graph.

Hyper-ring connection machines

A graph G=(V,E) is called a hyper-ring with N nodes (N-HR for short) if V={0,...,N-1} and E={{u,v}|v-u modulo N is a power of 2}. We study constructions and spanners of HRs, and embeddings into HRs. The stretch factors of three types of spanners given in this paper are at most [log/sub 2/ N], 2k-1 for any 1/spl les/k/spl les/[log/sub 2/ N], and 2k-1 for any 0/spl les/k/spl les/[log/sub 2/ N]-1, respectively. The numbers of edges of these types of spanners are N-1, at most N[(log/sub 2/ N)/k] and at most N([log/sub 2/ N]-k)/(2k)+Nk, respectively. Some of these spanners are superior in both stretch factors and numbers of edges to corresponding spanners for synchronizer /spl gamma/ of HRs.<<ETX>>

Fault-tolerant file transmission by information dispersal algorithm in rotator graphs

A directed graph G=(V, E) is called the n-rotator graph if V={a/sub 1/a/sub 2//spl middot//spl middot//spl middot/a/sub n/|a/sub 1/a/sub 2//spl middot//spl middot//spl middot/a/sub n/ is a permutation of 1, 2, /spl middot//spl middot//spl middot/, n} and E={(a/sub 1/a/sub 2//spl middot//spl middot//spl middot/a/sub n/, b/sub 1/b/sub 2//spl middot//spl middot//spl middot/b/sub n/) I for some 2/spl les/i/spl les/n, b/sub 1/b/sub 2//spl middot//spl middot//spl middot/b/sub n/=a/sub 2//spl middot//spl middot//spl middot/a/sub i/a/sub 1/a/sub i+1//spl middot//spl middot//spl middot/a/sub n/}. We show that for any pair of distinct nodes in the n-rotator graph, we can construct n-1 disjoint paths, each with length less than 2n, connecting the two nodes. By using these disjoint paths and information dispersal algorithm (IDA) by M.O. Rabin (1989), we design a file transmission scheme and analyse its reliability.

Embeddings of Hyper-Rings in Hypercubes

A graph G=(V, E) with N nodes is called an N-hyper-ring if V={0,..., N−1} and E={(u, v) ¦ (u− v) modulo N is a power of 2}. We study embeddings of the 2n-hyper-ring in the n-dimensional hypercube. We show a greedy embedding with dilation 2 and congestion n+1 and a modified greedy embedding with dilation 4 and congestion 6.

A shortest path algorithm for banded matrices by a mesh connection without processor penalty

We give an efficient shortest path algorithm on a mesh-connected processor array for n/spl times/n banded matrices with bandwidth b. We use a [b/2]/spl times/[b/2] semisystolic processor array. The input data is supplied to the processors array from the host computer. The output from the processor array can be also supplied to itself through the host computer. This algorithm computes all pair shortest distances within the band in 7n-4[b/2]-1 steps.<<ETX>>

A probably optimal embedding of hyper-rings in hypercubes

Abstract A graph G = (V,E) with N nodes is called an N-hyper-ring if V = {0,…,N − 1} and E = {(u,v) ¦(v − u) modulo N is a power of 2} . We present an embedding of the 2n-hyper-ring in the n-dimensional hypercube with dilation 2 and congestion 4. This is probably an optimal embedding of hyper-rings in hypercubes.