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73
Polynomial time algorithms for multicast network code construction
 IEEE TRANS. ON INFO. THY
, 2005
"... The famous maxflow mincut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the mincut separating and. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediat ..."
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Cited by 316 (29 self)
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The famous maxflow mincut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the mincut separating and. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to reencode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.
Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity
, 1988
"... ..."
The Bit Extraction Problem or tResilient Functions
, 1985
"... \Gamma We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f : f0; 1g n 7! f0; 1g m be a function. An adversary, knowing the function f , sets t of the n input bits, while the rest (n \Gamma t input bits) are chosen at random (independently and with un ..."
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Cited by 171 (11 self)
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\Gamma We consider the following adversarial situation. Let n, m and t be arbitrary integers, and let f : f0; 1g n 7! f0; 1g m be a function. An adversary, knowing the function f , sets t of the n input bits, while the rest (n \Gamma t input bits) are chosen at random (independently and with uniform probability distribution). The adversary tries to prevent the outcome of f from being uniformly distributed in f0; 1g m . The question addressed is for what values of n, m and t does the adversary necessarily fail in biasing the outcome of f : f0; 1g n 7! f0; 1g m , when being restricted to set t of the input bits of f . We present various lower and upper bounds on m's allowing an affirmative answer. These bounds are relatively close for t n=3 and for t 2n=3. Our results have applications in the fields of faulttolerance and cryptography. 1. INTRODUCTION The bit extraction problem formulated above The bit extraction problem was suggested by Brassard and Robert [BRref] and by V...
Symbollevel Network Coding for Wireless Mesh Networks
"... This paper describes MIXIT, a system that improves the throughput of wireless mesh networks. MIXIT exploits a basic property of mesh networks: even when no node receives a packet correctly, any given bit is likely to be received by some node correctly. Instead of insisting on forwarding only correct ..."
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Cited by 82 (2 self)
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This paper describes MIXIT, a system that improves the throughput of wireless mesh networks. MIXIT exploits a basic property of mesh networks: even when no node receives a packet correctly, any given bit is likely to be received by some node correctly. Instead of insisting on forwarding only correct packets, MIXIT routers use physical layer hints to make their best guess about which bits in a corrupted packet are likely to be correct and forward them to the destination. Even though this approach inevitably lets erroneous bits through, we find that it can achieve high throughput without compromising endtoend reliability. The core component of MIXIT is a novel network code that operates on small groups of bits, called symbols. It allows the nodes to opportunistically route groups of bits to their destination with low overhead. MIXIT’s network code also incorporates an endtoend error correction component that the destination uses to correct any errors that might seep through. We have implemented MIXIT on a software radio platform running the Zigbee radio protocol. Our experiments on a 25node indoor testbed show that MIXIT has a throughput gain of 2.8 × over MORE, a stateoftheart opportunistic routing scheme, and about 3.9 × over traditional routing using the ETX metric.
Robust Protection Against FaultInjection Attacks of Smart Cards Implementing the Advanced Encryption Standard
 Proc. Int. Conference on Dependable Systems and Networks (DNS
, 2004
"... We present a method of protecting a hardware implementation of the Advanced Encryption Standard (AES) against a sidechannel attack known as Differential Fault Analysis attack. The method uses systematic nonlinear (cubic) robust error detecting codes. Errordetecting capabilities of these codes depen ..."
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Cited by 43 (18 self)
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We present a method of protecting a hardware implementation of the Advanced Encryption Standard (AES) against a sidechannel attack known as Differential Fault Analysis attack. The method uses systematic nonlinear (cubic) robust error detecting codes. Errordetecting capabilities of these codes depend not just on error patterns (as in the case of linear codes) but also on data at the output of the device which is protected by the code and this data is unknown to the attacker since it depends on the secret key. In addition to this, the proposed nonlinear (n,k)codes reduce the fraction of undetectable errors from 2 r − 2 to 2 r − as compared to the corresponding (n,k) linear code (where nk=r and k>=r). We also present results on a FPGA implementation of the proposed protection scheme for AES as well as simulation results on efficiency of the robust codes. 1.
An Efficient PseudoRandom Generator Provably as Secure as Syndrome Decoding
, 1996
"... . We show a simple and efficient construction of a pseudorandom generator based on the intractability of an NPcomplete problem from the area of errorcorrecting codes. The generator is proved as secure as a hard instance of the syndrome decoding problem. Each application of the scheme generates a l ..."
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Cited by 33 (1 self)
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. We show a simple and efficient construction of a pseudorandom generator based on the intractability of an NPcomplete problem from the area of errorcorrecting codes. The generator is proved as secure as a hard instance of the syndrome decoding problem. Each application of the scheme generates a linear amount of bits in only quadratic computing time. 1 Introduction A pseudorandom generator is an algorithm producing strings of bits that look random. The concept of "randomly looking" has been formalized by Blum and Micali [4] within the framework of complexity theory. Yao [22] has shown that the existence of a oneway permutation is sufficient to construct a pseudorandom generator. Subsequently, a long series of deep articles led to the conclusion that the existence of a oneway function is equivalent to the hypothesis that a pseudorandom generator exists [15, 10, 14]. However, the theoretical constructions proposed in these articles are often impractical. Several schemes have been ...
New Class of Nonlinear Systematic Error Detecting Codes
 IEEE Trans Info Theory, Vol
, 2004
"... We will say that code C detects error e with probability 1 Q(e), if Q(e) is a fraction of codewords y such that y, y + e C. We present a class of optimal nonlinear q ary systematic (n, k)codes (robust codes) minimizing over all (n, k) codes maxima of Q(e) over all e 0. We will also show ..."
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Cited by 30 (25 self)
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We will say that code C detects error e with probability 1 Q(e), if Q(e) is a fraction of codewords y such that y, y + e C. We present a class of optimal nonlinear q ary systematic (n, k)codes (robust codes) minimizing over all (n, k) codes maxima of Q(e) over all e 0. We will also show that any linear (n, k))code Vwith n 2k can be modified into a nonlinear (n, k)code CV with simple encoding and decoding procedures, such that set E = for CV is a (k r) dimensional subspace of V (E=q instead of q for V). For the remaining q 2 r+1 for q = 2.
Nonadaptive Group Testing: Explicit bounds and novel algorithms
, 2012
"... We present computationally efficient and provably correct algorithms with nearoptimal samplecomplexity for noisy nonadaptive group testing. Group testing involves grouping arbitrary subsets of items into pools. Each pool is then tested to identify the defective items, which are usually assumed to ..."
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Cited by 17 (4 self)
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We present computationally efficient and provably correct algorithms with nearoptimal samplecomplexity for noisy nonadaptive group testing. Group testing involves grouping arbitrary subsets of items into pools. Each pool is then tested to identify the defective items, which are usually assumed to be sparse. We consider random nonadaptive pooling where pools are selected randomly and independently of the test outcomes. Our noisy scenario accounts for both false negatives and false positives for the test outcomes. Inspired by compressive sensing algorithms we introduce three computationally efficient algorithms for group testing, namely, Combinatorial Orthogonal Matching Pursuit (COMP), Combinatorial Basis Pursuit (CBP), and CBP via Linear Programming (CBPLP) decoding. The first and third of these algorithms have several flavours, dealing separately with the noiseless and noisy measurement scenarios. We derive explicit samplecomplexity bounds—with all constants made explicit—for these algorithms as a function of the desired error probability; the noise parameters; the number of items; and the size of the defective set (or an upper bound on it). We also derive lower bounds for sample complexity based on Fano’s inequality and show that our upper and lower bounds are equal up to a constant factor.
Which Linear Codes Are AlgebraicGeometric?
 IEEE Trans. Inform. Theory
, 1991
"... An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If one imposes conditions on the degree of the divisor used, then we derive criteria for linear codes to be algebraicgeometric. In particular, we investiga ..."
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Cited by 16 (5 self)
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An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If one imposes conditions on the degree of the divisor used, then we derive criteria for linear codes to be algebraicgeometric. In particular, we investigate the family of qary Hamming codes, and prove that only those with redundancy one or two, and the binary [7; 4; 3] code are algebraicgeometric in this sense. For these codes we explicitly give a curve, rational points and a divisor. We prove that this triple is in a certain sense unique in the case of the [7; 4; 3] code. Key words: algebraicgeometric codes, algebraic curves, divisors, generalized Goppa codes, geometric Goppa codes. I. Introduction Since the early papers by Goppa [5],[6],[7], [8], algebraicgeometric codes have been in the spotlight of coding theoretic research for about a decade. As is wellknown, numerous exciting results have been achieved using Goppa's construction of li...
New Designs for Signal Sets with Low Crosscorrelation, Balance Property and Large Linear Span: GF(p) Case
 IEEE Trans. Inform. Theory
, 2000
"... New designs for families of sequences over GF (p) with low cross correlation, balance property and large linear span are presented. The key idea of the new designs is to use short pary sequences of period v with the 2level auto correlation function to construct a set of long sequences with the des ..."
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Cited by 14 (6 self)
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New designs for families of sequences over GF (p) with low cross correlation, balance property and large linear span are presented. The key idea of the new designs is to use short pary sequences of period v with the 2level auto correlation function to construct a set of long sequences with the designed properties. The resulting sequences are of interleaved sequences of period v 2 . There are v cyclically shift distinct sequences in each family. The maximal magnitude of cross/outofphase auto correlation of sequences in the family is 2v + 3 which is optimal with respect to the Welch bound. In particular, for binary case, cross/outofphase auto correlation values belong to the set f1; v; v + 2; 2v + 3; 2v 1g. Each sequence in the family is balanced and has large linear span. For binary case, any sequence in such a family where the short sequences are quadratic residue sequences achieves the maximal linear span. For nonbinary case, the new design gives the first type of signal set...