1. ED 4 I: Error Detection by Diverse Data and Duplicated Instructions Greg Bronevetsky

2. ED 4 I Background A code transformation system developed at the Stanford Center for Reliable Computing. Authors: Nahmsuk Oh, Subhasish Mitra, Edward J. McCluskey ED 4 I allows us to run a program on two slightly different inputs and still be able to compare results at the end.

3. Motivation The simplest way to detect Byzantine Faults is to run the same program on multiple processors and compare results. ED 4 I is Byzantine Fault detection for uniprocessors. Must take into account both temporary and and permanent faults.

4. Definitions Temporary Faults – any fault that temporarily affects a processor, long enough to execute several instructions. Ex: Radiation hitting wires, frayed wires. Permanent Faults – a fault that affects a processor for a long period of time. Ex: Spilling Coke on the chip, cut wires.

5. Problem Statement We can detect Byzantine Failures by running each program or procedure twice and comparing the results. However, this does not guard against permanent faults since the results of both runs will be the same. Need to make the two runs different so that the same fault will affect the results differently. Overhead = 100%.

6. Key Idea Lets feed into the program two different sets of data and then compare the results. Key Insight: If the program only uses arithmetic operations, we can alter the input by multiplying all input numbers by a constant. Then the modified output will be the (real output) * (the constant). Thus, you can verify that the two computations succeeded AND the two computations will be affected by errors differently.

7. New Program If we alter the input to the program, we must alter the program to work with this modified input. The transformation is given the constant k (called the “diversity factor”) and it creates the “k-factor diverse program”. The new program will have the same control flow graph as the old program but all the variables will be k-multiples of the of original ones.

8. Transformations If k ↔ <, ≥ ↔ ≤) All constants in code get multiplied by k. Addition and Subtraction of variables unchanged. Multiplication: v 1 *v 2 *....*v n → (v 1 *v 2 *....*v n )/k n-1 Division: v 1 /v 2 → (v 1 /v 2 )*k

9. Fault Detection Probability For functional unit h i (such as the adder), fault f and diversity factor k: X i = is the set of inputs to h i E i = subset of X containing the inputs that will result in erroneous output due to the fault. E' i = subset of E i that will escape detection C i (k) = Probability of catching an error in h i.

10. Data Integrity Probability For functional unit h i, fault f and diversity factor k: X i = is the set of inputs to h i E i = subset of X containing the inputs that will result in erroneous output due to the fault. E' i = subset of E i that will escape detection D i (k) = Probability of missing no errors in h i.

11. Choosing the value of k For some functional units we can derive C i (k) and D i (k) analytically for each k. This is too hard in general so we resort to trying out a range of k's empirically to determine C i (k) and D i (k).

12. Bus Signal Line Bus wire stuck at either 0 or 1. Derived results for a 12-bit bus:

13. Adder Experimental results for a 12-bit ripple carry adder: Experimental results for a 12-bit carry look- ahead adder:

14. Multiplier & Divider Experimental Results for 12-bit array multiplier 8-bit Wallace Tree multiplier SRT divider

15. Shifter Experimental Results for 16-bit multiplexer- based shifter:

16. Using Benchmarks to pick k Need to determine how much each functional unit is used in the average program. Add, sub, mult and shift use the obvious functional units. “memory access” uses the memory bus “branch” uses a carry-lookahead adder

17. Benchmarked Data Integrity Calculated Data Integrity=D i (k) given above usage statistics. (high D i (k) top priority) Highlighted columns provide the best data integrity for each benchmark.

18. Benchmarked Detection Probability Calculated Detection Probability=C i (k) given above usage statistics. Highlighted columns provide the best detection probability for each benchmark.

19. Optimum k Optimum k selected: Must maximize the Data Integrity=D i (k). Given maximum D i (k), maximize C i (k). For each program, should get an estimate of how it uses the different functional units and pick k accordingly.

20. Dealing with Overflow By multiplying all variables by k, we may cause them to overflow. Can scale variables up to next largest type. Scale down variables by dividing by k. Must only check higher order bits when comparing new results to results of original program. Can use compile-time range checking to determine vulnerability to overflow and pick k accordingly

21. Floating Point Numbers Above technique fails for floating point numbers. IEEE 754 format: K=-2 will only change the sign bit and some bits in the exponent. Solution: pick separate k's for the exponent and the mantissa and run the program once with each k. Overhead = 200%.

22. Picking k for the mantissa To find errors in mantissa, pick k to be 3/2. A stuck-at-1 fault: In original program, variable x's value corrupted to: In transformed program, Since However, the mantissa must be <2, so if the mantissa is right shifted by 1 and normalized.

23. Transformed variables So now, the value in transformed program is: Value in original program is:

24. Fault Detection in Mantissa If there is a stuck-at-1 fault Value in transformed program: Value in original program * k (for checking):

25. We can detect Mantissa errors! Note that the error values for the original and the transformed programs are different! We actually use k= in order to flip the sign bit for improved detection capability

26. k for exponents In order to flip all the bits of the exponent, need to transform program to use k= and k= If a fault invalidates a bit of the exponent, the fault will be detected by comparing to the exponents of one of the two transformed programs.

27. Effectiveness for Mantissa Effectiveness of k= (for IEEE 754 single precision)

28. Effectiveness for Exponent Effectiveness of k= (for IEEE 754 single precision)

29. Summary ED 4 I effectively detects Byzantine Failures in numerical applications on uniprocessors. Purely software solution using Data Diversity. Detects permanent and temporary faults. Works with fixed-point and floating point numbers. Compatible with arithmetic and logical operations (probably with any bitwise logical operation if it can be recast into arithmetic) High overhead: 100% or 200%.