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Instituto de Informática, Programa de Pós-Graduação em Computação Universidade Federal do Rio Grande do Sul Porto Alegre, Brazil Porto Alegre, May 16, 2007 Low Overhead System Level Approaches for Mitigation of Long Duration and Multiple Transient Faults Thesis Proposal by Carlos Arthur Lang Lisbôa Advisor: Luigi Carro

Thesis Proposal - inf.ufrgs.brcalisboa/THEPROPOSAL/TheProposalSlides.pdfCarlos A. L. Lisbôa Thesis Proposal - May 16, 2007 3 Single event, multiple effects

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Instituto de Informática, Programa de Pós-Graduação em ComputaçãoUniversidade Federal do Rio Grande do Sul

Porto Alegre, Brazil

Porto Alegre, May 16, 2007

Low Overhead System Level Approaches

for Mitigation of Long Duration

and Multiple Transient Faults

Thesis Proposalby Carlos Arthur Lang Lisbôa

Advisor: Luigi Carro

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 2

Outline

• Motivation• Proposal

• Related Work

• Previous Investigations

• Proposed Approach

• Future Tasks and Schedule

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 3

Single event, multiple effects[Rossi 2005 *]

[*] Multiple Transient Faults in Logic: An Issue for Next Generation ICs ?, Daniele Rossi et al, DFT 2005

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 4

Transient pulse width scalingacross technologies [Dodd 2004 *]

[*] Production and propagation of Single-Event Transients in High-Speed Digital Logic ICs, Paul Dodd et al, IEEE Transactions on Nuclear Science, Vol. 51 No. 6, Part2, December 2004

max.1.37 x

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 5

Propagation delay vs. technologies(*)

Technology (nm)* 180 130 90 32 180/32

10-inverter chain 508.4 157.8 120.2 79.6 6.39

(*) simulated using parameters from PTM web site and HSPICE tool

in out

clk clk

32 nm

90 nm

130 nm

180 nm

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 6

Cycle time x transient pulse scalingacross technologies

6.39 x

Transientwidth scaling:max. 1.37 x

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 7

Cycle time x transient pulse scalingacross technologies

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 8

Transient pulse width scalingacross technologies [Ferlet-Cavrois 2006 *]

[*] Statistical Analysis of the Charge Collected in SOI and Bulk Devices Under Heavy Ion and Proton Irradiation, V. Ferlet-Cavoirs et al, IEEE Transactions on Nuclear Science, Vol. 53 No. 6, Part 1, Nov 2006.

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 9

Cycle time x transient pulse scalingacross technologies

Data for 70 nm technology

[Ferlet-Cavrois 2006 *]

[*] Statistical Analysis of the Charge Collected in SOI and Bulk Devices Under Heavy Ion and Proton Irradiation -Implications for Digital SETs, Ferlet-Cavrois et al, IEEE Tr. on Nuc. Sci., Vol. 53 No. 6, November 2006

Long duration transients !

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 10

4-bit ripple carry adder

a3 a2 a1 a0 b0b1b2b3

s3 s2 s1 s0s4

1 0 0 1 1 0 0 1

0 1 1 1 1

0 0 00

0 0 0

Fault model: long duration transients

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 11

a3 a2 a1 a0 b0b1b2b3

s3 s2 s1 s0s4

1 0 0 1 1 0 0 1

0 1 0 1 1

0 1 00

0 1 0

4-bit ripple carry adder

Fault model: long duration transients

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 12

a3 a2 a1 a0 b0b1b2b3

s3 s2 s1 s0s4

1 0 0 1 1 0 0 1

1 0 0 1 1

1 1 00

1 1 0

4-bit ripple carry adder

Fault model: long duration transients

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 13

Outline

• Motivation

• Proposal• Related Work

• Previous Investigations

• Proposed Approach

• Future Tasks and Schedule

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 14

Problem definition

♦ For future technologies, radiation inducedtransients pulses will last longer than thecycle time of circuits.

♦ Many current mitigation techniques, basedon time and/or space redundancy at lowlevel, will no longer be effective to cope witherrors due to those transients.

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 15

The goal of this proposal

Development of system level

mitigation techniques to cope

with long duration transient

pulses and their effects

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 16

Outline

• Motivation

• Proposal

• Related Work• Previous Investigations

• Proposed Approach

• Future Tasks and Schedule

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 17

Time redundancy [Anghel 2000 *]

[*] Cost Reduction and Evaluation of a Temporary Faults Detecting Technique, L. Anghel and M. Nicolaidis, in Proceedings of DATE 2000, March 2000

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 18

Time redundancy [Anghel 2000 *]

[*] Cost Reduction and Evaluation of a Temporary Faults Detecting Technique, L. Anghel and M. Nicolaidis, in Proceedings of DATE 2000, March 2000

Increase delay ?⇒ Higher performance

penalty !!!

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 19

Time redundancy [Mitra 2005 *]

[*] Robust System Design with Built-in Soft Error Resilience, S. Mitra, IEEE Computer, V38, N2, February 2005

Reuses scan chain,but still relies on duration

of transient pulses.

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 20

Space redundancy [Nieuwland 2006 *]

[*] Combinational Logic Soft Error Analysis and Protection, André Nieuwland et al, in Proceedings of IOLTS 2006, July 2006

Can not copewith long duration

transients !!!

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 21

Other proposed approaches

• Hardware based:• duplication and comparison

• checkers and I-IPs

• Software based:• duplication and comparison

• self-checking block signatures

• Hybrid: best features of SW and HW

High area and/or performanceoverheads !!!

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 22

Outline

• Motivation

• Proposal

• Related Work

• Previous Investigations• Proposed Approach

• Future Tasks and Schedule

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 23

MajorityLogic

Low costredundancy

Research Evolution - Overview

StochasticOperators

TMR andAnalogVoter

Bit StreamOperators

Use of Memory to Cope with Soft Errors

LongDuration

Transients

2004 2005 2006 2007

IOLTS 04

DFT 04WDES 04

LATW 06ETS 06

DFT 06

ETS 05SBCCI 05

ResearchReport

SRC 2005TechCon

ResearchReport

DATE 06PhD Forum

MemProc

DATE 07 ETS 07

ControlFlow ErrorDetection

LATW 07

OnlineHardening

Low CostChecker forMatrix Mult.

System levelapproaches

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 24

Research Approaches - 2004 / 2005

• Use of stochastic operators

• Use of bit stream operators

• Ensuring voter reliability to use n-MR

to cope with simultaneous faults -

cooperation

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 25

Research Evolution - 2004 / 2005

StochasticOperators

IOLTS2004

Stochastic multiplier circuit

1000100110011010

10010001000010111000000100001010

Stochastic Adder Circuit

01100010101

010111011001S1

S3

Sum

01010101101

0010100110101

S2

OK for s

ome

DSP

Applica

tions

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 26

DFT 2004WDES 2004

Look

ing fo

r

more sp

eed

StochasticOperators

Bit StreamOperators

Research Evolution - 2004 / 2005

A1 A0x B1 B0

B0.A1 B0.A0

B1.A1 B1.A0

p8 p7 p6 p5 p4 p3 p2 p1 p0

Conve

rsion

to

binar

y valu

e ?

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 27

Research Evolution - 2004 / 2005

Look

ing fo

r

more sp

eed

StochasticOperators

AnalogVoter

Bit StreamOperators

Looking for

tolerant converter

ETS 2005SBCCI 2005

Toler

ant to

multipl

e fau

lts in

n-MR so

lution

s

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 28

Research approaches - 2006 / 2007

• cooperation with peers• use of memory for computation

• analog voter + majority logic

• use of an I-IP to harden instructions

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 29

Research approaches - 2006 / 2007

• cooperation with peers• use of memory for computation

• analog voter + majority logic

• use of an I-IP to harden instructions

• low cost redundancy

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 30

Research approaches - 2006 / 2007

• cooperation with peers• use of memory for computation

• analog voter + majority logic

• use of an I-IP to harden instructions

• low cost redundancy

• system level approaches

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 31

Research Evolution - 2006

Use ofMemory

to Cope with Soft Errors

LATW 06ETS 06

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 32

MajorityLogic

Research Evolution - 2006

LATW 06ETS 06 DFT 06

Use ofMemory

to Cope with Soft Errors

AND gate OR gate inverter

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 33

Low costredundancy

MajorityLogic

Research Evolution - 2006

LATW 06ETS 06 DFT 06

Use ofMemory

to Cope with Soft Errors

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 34

Low costredundancy

OnlineHardening

MajorityLogic

Research Evolution - 2006

LATW 06ETS 06

DFT 06

DFT 06

Use ofMemory

to Cope with Soft Errors

µP

abus

cbus

dbus I - IP

error

IRQ

IRQ

abus

cbus

dbus

Code

Memory

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 35

Research Evolution - 2006 / 2007

Memory µPabus

cbus

dbus

errorI-IP

ControlFlow ErrorDetection

Low CostChecker forMatrix Mult.

Low costredundancy

LATW 07

LATW 07

Smaller I-IP,lower performanceoverhead, higher

efficacy

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 36

Research Evolution - 2006 / 2007

ControlFlow ErrorDetection

Low CostChecker forMatrix Mult.

Low costredundancy

LATW 07

LATW 07

C ← A * B

Cr ← C * rCrc ← C * rc

ABr ← A*(B*r)ABrc ← A*(B*rc)

output (C)

error

inputs (A, B)

Two timesFreivalds’ (25%), with 100%

detection probability

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 37

Research Evolution - 2006 / 2007

DATE 07

MemProc

ROM

RAM

MicrocodeMemory

PC

ALU

OperationMasks

Memory

ControlFlow ErrorDetection

Low CostChecker forMatrix Mult.

Low costredundancy

LATW 07

LATW 07

Smaller combinationallogic, with innovative add

and comparison ops

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 38

Low CostChecker forMatrix Mult.

Low costredundancy

Research Evolution - 2006 / 2007

System LevelApproaches

ETS 07(accepted)

LATW 07 DATE 07

MemProc

ControlFlow ErrorDetection

LATW 07

System levelapproaches

C ← A * B

Cr ← Σ CiABr ← A*(Σ Bi)

output (C)

error

inputs (A, B)

Much lower performanceand/or area overheads,

still with 100%detection probability

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 39

Outline

• Motivation

• Proposal

• Related Work

• Previous Investigations

• Proposed Approach• Future Tasks and Schedule

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 40

The goal of this proposal

Development of system level

mitigation techniques to cope

with long duration transient

pulses and their effects

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 41

Verification & Correction requirements

• Verification - very frequent

• much faster than computation

• low power consumption

• low area overhead (goal: ≤ 50%)

• Correction - very seldom (hours, weeks)

• performance is not the main issue

• static power consumption must be low or null

• low area overhead: desirable

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 42

Algorithmic level approachesVerification

Case study 1: sorting algorithm

keyi ≤ keyi+1, 1 ≤ i ≤ n-1

• verification requires only O(n) comparisons !

• low overhead verification mechanism

• best performance algorithms: O(n log n) operations

• intuitive verification condition at completion:

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 43

Algorithmic level approachesVerification

Case study 2: n × n matrix multiplication

• verification cost: O(n2) operations

• probability of error in verification: 50%

• multiplication cost: O(n3) operations

• Freivalds’ verification technique:

Cr ← C * rABr ← A*(B*r) Cr = ABr ?

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 44

Algorithmic level approachesVerification

Case study 2: matrix multiplication (cont.)

• verification cost: less than O(n2) operations

• probability of error in verification: 0%

• multiplication cost: O(n3) operations

• optimized Freivalds’ verification technique:

Cr ← Σ CiABr ← A*(Σ Bi)

Cr = ABr ?

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 45

Algorithmic level approachesVerification

Generalization of the approach• “keyi ≤ keyi+1, 1 ≤ i ≤ n-1” and “Cr = ABr”

are invariants at completion of the algorithms

• ideal solution: automated invariant determination (?)

• drawback: not every invariant is suitable:y=1;z=0;while z ~= x, z = z + 1; y = y * z;end

invariant: y = z! ⇒ high verification cost

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 46

Algorithmic level approachesError correction - main issues

• Recomputation

• Small control overhead: reuse of resources

• Must save initial state when start computation

• Power-off control circuit when not in use ?

• Circular buffer to cope with hard deadlines in

real-time systems ?

• Other (application specific) alternatives ?

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 47

Algorithmic level approachesHardware vs. software implementation

• Hardware implementation• requires access to HW description and modification

• not feasible when using COTS processors

• faster than software

• Software implementation• requires access to source code and modification

• implies memory overhead: instructions + variables

• slower than hardware

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 48

Algorithmic level approachesFuture research topics

• Invariants determination method⇒ generalization for other algorithms⇒ criteria for choice of best suited invariant

• Hardware vs. software implementation⇒ parameters for designers’ decision

• When to apply the verification mechanism ⇒ partial execution or at completion ?

• Efficient recomputation techniques

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 49

Outline

• Motivation

• Proposal

• Related Work

• Previous Investigations

• Proposed Approach

• Future Tasks and Schedule

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 50

1. Study of invariants2. Study of MP3 application3. Development of application specific techniques for MP34. Application of proposed techniques to MP3

and fault tolerance analysis5. Critical review of results, look for weak spots

and methods to cope with them

6. Comparison with arithmetic coding for mitigation of double errorsin arithmetic operators

7. Comparison with distance 6 Hamming codes for mitigation ofdouble errors in FSMs

8. Cooperation internship (Europe)9. Thesis writing10. Thesis presentation

2007 2008Task # May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

12345678910

Future Tasks and Schedule

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 51

Publications - already published

• Lisbôa, C. and Carro, L., “An Intrinsically Robust Technique for FaultTolerance Under Multiple Upsets”, 10th IEEE International Online TestSymposium - IOLTS 2004, IEEE Computer Society, Funchal, Madeira Island,Portugal, July 2004.

• Lisbôa, C. and Carro, L., “Highly Reliable Arithmetic Multipliers for FutureTechnologies”, in Proceedings of the International Workshop on DependableEmbedded Systems - WDES 2004 - in conjunction with the 23rd InternationalSymposium on Reliable Distributed Systems - SRDS 2004, pp. 13-18. Editedby Becker, L. B. and Kaiser, J., Florianópolis, October 17, 2004.

• Lisbôa, C. and Carro, L., “Arithmetic Operators Robust to MultipleSimultaneous Upsets”, in Proceedings of the 19th IEEE InternationalSymposium on Defect and Fault Tolerance in VLSI Systems - DFT 2004, pp.289-297, ISBN0-7695-2241-6. IEEE Computer Society, New York, October2004.

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 52

Publications - already published

• Lisbôa, C. A. L., Carro, L. and Cota, E., “RobOps - Arithmetic Operators forFuture Technologies”, 10th European Test Symposium - ETS 2005, Tallin,Estonia, May 2005.

• Lisbôa, C. A. L., Schüler, E. and Carro, L., “Going Beyond TMR for ProtectionAgainst Multiple Faults”, in Proceedings of the 18th Symposium on IntegratedCircuits and Systems Design - SBCCI 2005, September 2005.

• Rhod, E.; Lisbôa, C. A. L. and Carro, L., “Using Memory to Cope withSimultaneous Transient Faults”, in Proceedings of the 7th Latin-AmericanTest Workshop - LATW 2006, pp. 151-156, IEEE Computer Society, NewYork, March 2006.

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 53

Publications - already published

• Rhod, E.; Lisbôa, C. A. L.; Michels, Á. and Carro, L., “Fault Tolerance AgainstMultiple SEUs using Memory-Based Circuits to Improve the ArchitecturalVulnerability Factor”, in Informal Digest of Papers of the 11th IEEE EuropeanTest Symposium - ETS 2006, pp. 229-234, IEEE Computer Society, NewYork, May 2006.

• Michels, Á., Petroli, L., Lisbôa, C. A. L., Kastensmidt, F. and Carro, L. “SETFault Tolerant Combinational Circuits Based on Majority Logic”, inProceedings of the 21st IEEE International Symposium on Defect and FaultTolerance in VLSI Systems - DFT 2006, pp. 345-352, IEEE Computer Society,Los Alamitos, CA, October 2006..

• Lisbôa, C. A. L., Carro, L., Sonza Reorda, M., and Violante, M. “OnlineHardening of Programs against SEUs and SETs”, in Proceedings of the 21stIEEE International Symposium on Defect and Fault Tolerance in VLSISystems - DFT 2006, pp. 280-288, IEEE Computer Society, Los Alamitos, CA,October 2006.

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 54

Publications - already published

• Lisbôa, C. A. L.; Erigson, M. I.; and Carro, L., “A Low Cost Checker for MatrixMultiplication”, in Informal Proceedings of the 8th IEEE Latin-American TestWorkshop - LATW 2007, Session 10, IEEE Computer Society TestTechnology Technical Council - TTTC, Cuzco, Peru, March 2007.

• Rhod, E. L.; Lisbôa, C. A. L.; Carro, L.; Violante, M.; and Sonza Reorda, M.,“A non-intrusive on-line control flow error detection technique for SoCs”, inInformal Proceedings of the 8th IEEE Latin-American Test Workshop - LATW2007, Sess ion 10, IEEE Computer Society Test Technology TechnicalCouncil - TTTC, Cuzco, Peru, March 2007.

• Rhod, E. L.; Lisbôa, C. A. L.; and Carro, L., “A Low-SER Efficient ProcessorArchitecture for Future Technologies”, in Proceedings of the Design,Automation and Test in Europe 2007 Conference - DATE 2007, IEEEComputer Society, Los Alamitos, CA, April 16-20, 2007, pp 1448-1453, ISBN978-3-9810801-2-4.

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 55

Publications - accepted

• Lisbôa, C. A. L. and Carro, L., “System Level Approaches for Mitigation ofLong Duration Transient Faults in Future Technologies”, accepted forpresentation at the 12th European Test Symposium – ETS 2007, in Freiburg,Germany, May 20-24, 2007.

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 56

Publications - under revision

• Petroli, L.; Lisboa, C. A. L.; Kastensmidt, F.; and Carro, L., “Majority LogicMapping for Soft Error Dependability”, accepted for publication in the specialissue for DFT 2006 of the Journal of Electronic Testing: Theory andApplications, subject to revisions, already re-submitted to the editor.

• Rhod, E.; Lisboa, C. A. L.; Carro, L., Violante, M.; and Sonza Reorda, M.,“Hardware and software transparency in the protection of programs againstSEUs and SETs”, accepted for publication in the special issue for DFT 2006of the Journal of Electronic Testing: Theory and Applications, subject torevisions, already re-submitted to the editor.

• Lisboa, C. A. L.; Kastensmidt, F.; Henes Neto, E.; Wirth, G.; and Carro, L.,“Using Built-in Sensors to Cope with Long Duration Transient Faults in FutureTechnologies”, submitted for the International Test Conference 2007, underreview.

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 57

Questions ?

Contact: [email protected]

Thank You !

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 58

Fault model: single and multiple faults

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 59

Area comparison(*)

(thousands of gates)

Matrix width (n) 3 8 16 32

Size of multiplier 10.8 218.7 1,850.1 16,192.5

Size of checker 5.4 61.8 386.3 2,640.1

Checker areaoverhead (%) 50.0 28.3 20.9 16.3

☺DWC> 100%

TMR> 200%

(*) fully combinational circuit, using Quartus II tool

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 60

Time comparison in µs(*)

Matrix width (n) 3 8 16 32

Multiplication 1.94 25.94 184.98 1,350

Verification 0.50 2.90 10.90 40

Overhead (%) 25.8% 11.2% 5.9% 3.0%

(*) sequential circuit, 1120 lines of VHDL code, parameterized by n

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 61

Sequential circuit implementation

• implemented using VHDL• 1,120 lines of code• parameterized by n

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 62

Area comparison - FPGA usage(*)

Type ofcomponent

LogicElements

Registers

Multiplication 409 305

Verification 119 143

Overhead (%) 29.1% 46.9%

(*) sequential circuit, 1120 lines of VHDL code, parameterized by n

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 63

• when k = 2, if C ≠ A x B

the probability of Cr = ABr is ≤ ¼

• r: is a vector of randomly selected 1s and 0s

e.g.: r = [ 1 0 0 1 0 ... 0 1 1 1 0]

• rc: a vector in which rc[ i ] = 1 - r[ i ] (1s’ complement)

e.g.: rc = [ 0 1 1 0 1 ... 1 0 0 0 1]

Going beyond Freivalds’ technique

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 64

Σ,× ⇒C11 C12 C1n

C21 C22 C2n

Cn1 Cn2 Cnn

0

0

0

Crc 1

Crc 2

Crc n

... ... ...

...

...

...

...

... ...

= 0

Crc = C x rc(also for Brc)

rc

ABr ≠ Cr OR ABrc ≠ Crc

Crci = Ci1.rc1 + Ci2.rc2 + ... + Cin.rcn ⇒ Crci = 0because all rci = 0

Further optimizing the technique

Σ,× ⇒C11 C12 C1n

C21 C22 C2n

Cn1 Cn2 Cnn

1

1

1

Cr1

Cr2

Crn

... ... ...

...

...

...

...

... ...

= ABr1

Cr = C x r(also for Br)

r

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 65

Error correction - main issues

• Recomputation ⇒ small control overhead

• Must save initial state when start computation

• Power-off control circuit when not in use ?

• Circular buffer to cope with hard deadlines in

real-time systems ?

buffer out

buffercontrol

errordetector

maincircuit

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 66

Time comparison - software(number of * and + operations)

Matrix width (n) 3 8 16 32

Multiplication 45 960 7,936 64,512

Verification 27 232 976 4,000

Overhead (%) 60% 24% 12% 6%

Carlos A. L. Lisbôa Thesis Proposal - May 16, 2007 67

Time comparison - software(number of * and + operations)

Matrix width (n) 3 8 16 32

Multiplication 54 1,024 8,192 64,512

Verification 36 256 1024 4,000

Overhead (%) 66.7% 25% 12.5% 6.25%