Reliable Linear, Sesquilinear and Bijective Operations On Integer Data Streams Via Numerical Entanglement

A new technique is proposed for fault-tolerant linear, sesquilinear and bijective (LSB) operations on $M$ integer data streams ($M\geq3$), such as: scaling, additions/subtractions, inner or outer vector products, permutations and convolutions. In the proposed method, the $M$ input integer data streams are linearly superimposed to form $M$ numerically-entangled integer data streams that are stored in-place of the original inputs. A series of LSB operations can then be performed directly using these entangled data streams. The results are extracted from the $M$ entangled output streams by additions and arithmetic shifts. Any soft errors affecting any single disentangled output stream are guaranteed to be detectable via a specific post-computation reliability check. In addition, when utilizing a separate processor core for each of the $M$ streams, the proposed approach can recover all outputs after any single fail-stop failure. Importantly, unlike algorithm-based fault tolerance (ABFT) methods, the number of operations required for the entanglement, extraction and validation of the results is linearly related to the number of the inputs and does not depend on the complexity of the performed LSB operations. We have validated our proposal in an Intel processor (Haswell architecture with AVX2 support) via fast Fourier transforms, circular convolutions, and matrix multiplication operations. Our analysis and experiments reveal that the proposed approach incurs between $0.03\%$ to $7\%$ reduction in processing throughput for a wide variety of LSB operations. This overhead is 5 to 1000 times smaller than that of the equivalent ABFT method that uses a checksum stream. Thus, our proposal can be used in fault-generating processor hardware or safety-critical applications, where high reliability is required without the cost of ABFT or modular redundancy.Comment: to appear in IEEE Trans. on Signal Processing, 201

Failure Mitigation in Linear, Sesquilinear and Bijective Operations On Integer Data Streams Via Numerical Entanglement

A new roll-forward technique is proposed that recovers from any single fail-stop failure in $M$ integer data streams ($M\geq3$) when undergoing linear, sesquilinear or bijective (LSB) operations, such as: scaling, additions/subtractions, inner or outer vector products and permutations. In the proposed approach, the $M$ input integer data streams are linearly superimposed to form $M$ numerically entangled integer data streams that are stored in-place of the original inputs. A series of LSB operations can then be performed directly using these entangled data streams. The output results can be extracted from any $M-1$ entangled output streams by additions and arithmetic shifts, thereby guaranteeing robustness to a fail-stop failure in any single stream computation. Importantly, unlike other methods, the number of operations required for the entanglement, extraction and recovery of the results is linearly related to the number of the inputs and does not depend on the complexity of the performed LSB operations. We have validated our proposal in an Intel processor (Haswell architecture with AVX2 support) via convolution operations. Our analysis and experiments reveal that the proposed approach incurs only $1.8\%$ to $2.8\%$ reduction in processing throughput in comparison to the failure-intolerant approach. This overhead is 9 to 14 times smaller than that of the equivalent checksum-based method. Thus, our proposal can be used in distributed systems and unreliable processor hardware, or safety-critical applications, where robustness against fail-stop failures becomes a necessity.Comment: Proc. 21st IEEE International On-Line Testing Symposium (IOLTS 2015), July 2015, Halkidiki, Greec

Generalized Numerical Entanglement For Reliable Linear, Sesquilinear And Bijective Operations On Integer Data Streams

We propose a new technique for the mitigation of fail-stop failures and/or silent data corruptions (SDCs) within linear, sesquilinear or bijective (LSB) operations on M integer data streams (M ⩾ 3). In the proposed approach, the M input streams are linearly superimposed to form M numerically entangled integer data streams that are stored in-place of the original inputs, i.e., no additional (aka. “checksum”) streams are used. An arbitrary number of LSB operations can then be performed in M processing cores using these entangled data streams. The output results can be extracted from any (M-K) entangled output streams by additions and arithmetic shifts, thereby mitigating K fail-stop failures (K ≤ ⌊(M-1)/2 ⌋ ), or detecting up to K SDCs per M-tuple of outputs at corresponding in-stream locations. Therefore, unlike other methods, the number of operations required for the entanglement, extraction and recovery of the results is linearly related to the number of the inputs and does not depend on the complexity of the performed LSB operations. Our proposal is validated within an Amazon EC2 instance (Haswell architecture with AVX2 support) via integer matrix product operations. Our analysis and experiments for failstop failure mitigation and SDC detection reveal that the proposed approach incurs 0.75% to 37.23% reduction in processing throughput in comparison to the equivalent errorintolerant processing. This overhead is found to be up to two orders of magnitude smaller than that of the equivalent checksum-based method, with increased gains offered as the complexity of the performed LSB operations is increasing. Therefore, our proposal can be used in distributed systems, unreliable multicore clusters and safety-critical applications, where robustness against failures and SDCs is a necessity

Reliable Linear, Sesquilinear, and Bijective Operations on Integer Data Streams Via Numerical Entanglement

A new technique is proposed for fault-tolerant linear, sesquilinear and bijective (LSB) operations on $M$ integer data streams ( $M \geq 3$), such as: scaling, additions/subtractions, inner or outer vector products, permutations and convolutions. In the proposed method, $M$ input integer data streams are linearly superimposed to form $M$ numerically-entangled integer data streams that are stored in-place of the original inputs. LSB operations can then be performed directly using these entangled data streams. The results are extracted from the $M$ entangled output streams by additions and arithmetic shifts. Any soft errors affecting one disentangled output stream are guaranteed to be detectable via a post-computation reliability check. Additionally, when utilizing a separate processor core for each stream, our approach can recover all outputs after any single fail-stop failure. Importantly, unlike algorithm-based fault tolerance (ABFT) methods, the number of operations required for the entire process is linearly related to the number of inputs and does not depend on the complexity of the performed LSB operations. We have validated our proposal in an Intel processor via several types of operations: fast Fourier transforms, convolutions, and matrix multiplication operations. Our analysis and experiments reveal that the proposed approach incurs between 0.03% to 7% reduction in processing throughput for numerous LSB operations. This overhead is 5 to 1000 times smaller than that of the equivalent ABFT method that uses a checksum stream. Thus, our proposal can be used in fault-generating processor hardware or safety-critical applications, where high reliability is required without the cost of ABFT or modular redundancy

Wildcard dimensions, coding theory and fault-tolerant meshes and hypercubes

Hypercubes, meshes and tori are well known interconnection networks for parallel computers. The sets of edges in those graphs can be partitioned to dimensions. It is well known that the hypercube can be extended by adding a wildcard dimension resulting in a folded hypercube that has better fault-tolerant and communication capabilities. First we prove that the folded hypercube is optimal in the sense that only a single wildcard dimension can be added to the hypercube. We then investigate the idea of adding wildcard dimensions to d-dimensional meshes and tori. Using techniques from error correcting codes we construct d-dimensional meshes and tori with wildcard dimensions. Finally, we show how these constructions can be used to tolerate edge and node faults in mesh and torus networks

Tamper-Resistant Arithmetic for Public-Key Cryptography

Cryptographic hardware has found many uses in many ubiquitous and pervasive security devices with a small form factor, e.g. SIM cards, smart cards, electronic security tokens, and soon even RFIDs. With applications in banking, telecommunication, healthcare, e-commerce and entertainment, these devices use cryptography to provide security services like authentication, identification and confidentiality to the user. However, the widespread adoption of these devices into the mass market, and the lack of a physical security perimeter have increased the risk of theft, reverse engineering, and cloning. Despite the use of strong cryptographic algorithms, these devices often succumb to powerful side-channel attacks. These attacks provide a motivated third party with access to the inner workings of the device and therefore the opportunity to circumvent the protection of the cryptographic envelope. Apart from passive side-channel analysis, which has been the subject of intense research for over a decade, active tampering attacks like fault analysis have recently gained increased attention from the academic and industrial research community. In this dissertation we address the question of how to protect cryptographic devices against this kind of attacks. More specifically, we focus our attention on public key algorithms like elliptic curve cryptography and their underlying arithmetic structure. In our research we address challenges such as the cost of implementation, the level of protection, and the error model in an adversarial situation. The approaches that we investigated all apply concepts from coding theory, in particular the theory of cyclic codes. This seems intuitive, since both public key cryptography and cyclic codes share finite field arithmetic as a common foundation. The major contributions of our research are (a) a generalization of cyclic codes that allow embedding of finite fields into redundant rings under a ring homomorphism, (b) a new family of non-linear arithmetic residue codes with very high error detection probability, (c) a set of new low-cost arithmetic primitives for optimal extension field arithmetic based on robust codes, and (d) design techniques for tamper resilient finite state machines

Mitigating Silent Data Corruptions In Integer Matrix Products: Toward Reliable Multimedia Computing On Unreliable Hardware

The generic matrix multiply (GEMM) routine comprises the compute and memory-intensive part of many information retrieval, machine learning and object recognition systems that process integer inputs. Therefore, it is of paramount importance to ensure that integer GEMM computations remain robust to silent data corruptions (SDCs), which stem from accidental voltage or frequency overscaling, or other hardware non-idealities. In this paper, we introduce a new method for SDC mitigation based on the concept of numerical packing. The key difference between our approach and all existing methods is the production of redundant results within the numerical representation of the outputs, rather than as a separate set of checksums. Importantly, unlike well-known algorithm-based fault tolerance (ABFT) approaches for GEMM, the proposed approach can reliably detect the locations of the vast majority of all possible SDCs in the results of GEMM computations. An experimental investigation of voltage-scaled integer GEMM computations for visual descriptor matching within state-of-the art image and video retrieval algorithms running on an Intel i7- 4578U 3GHz processor shows that SDC mitigation based on numerical packing leads to comparable or lower execution and energy-consumption overhead in comparison to all other alternatives