Statistical fault injection: Quantified error and confidence

May 13, 2026 | 541 words | 3min read

Paper Title: Statistical fault injection: Quantified error and confidence

Link to Paper: https://ieeexplore.ieee.org/document/5090716/

Date: 20. April 2009

Paper Type: dependability analysis, statistical fault injection

Short Abstract: The paper addresses a key limitation of statistical fault injection: while only a random subset of possible faults is usually tested in complex circuits, previous work rarely quantified the reliability of the results. It introduces a statistical framework to calculate confidence intervals, margins of error, and the number of fault injections required to achieve a desired level of accuracy and confidence.

Summary

Core Problem

Modern integrated circuits are vulnerable to soft errors caused by:

• radiation (alpha particles, neutrons),
• electromagnetic interference,
• or deliberate fault attacks (e.g., lasers targeting cryptographic chips).

Testing every possible fault in a complex circuit is computationally infeasible because the number of possible:

• fault locations,
• timings,
• and bit combinations

grows explosively.

As a result, researchers commonly use Statistical Fault Injection, where only a random subset of faults is injected.

The authors argue that previous work often claimed:

“a large number of faults were injected”

without rigorously quantifying:

• how accurate the reported results are,
• or how confident we should be in them.

The paper’s main goal is to provide a formal statistical framework for quantifying:

1. margin of error, and
2. confidence level

in SFI experiments.

Method

The paper adapts classical statistical sampling theory (similar to polling/surveys) to fault injection campaigns.

The framework allows researchers to determine:

• how many faults must be injected to achieve a desired accuracy,
• or, conversely,
• what confidence/error bounds result from a given sample size.

Statistical Model

The authors model all possible faults as a finite population of size (N).

A random sample of size (n) is selected uniformly.

The method assumes:

• normal distribution behavior,
• unbiased random sampling,
• finite population sampling without replacement.

The sample size formula is:

n=\frac{N t^2 p(1-p)}{e^2(N-1)+t^2 p(1-p)}

Where:

• (N): total number of possible faults,
• (p): estimated proportion of faults with a given property,
• (e): desired margin of error,
• (t): critical value corresponding to the confidence level.

The paper recommends using:

p=0.5

because it produces the most conservative (largest) required sample size.

Key Insights

1. Very Small Samples Can Be Sufficient

One of the most important findings is that only a tiny fraction of all possible faults may be needed to obtain statistically reliable results.

Example:

• Total fault population: [ N = 4{,}063{,}170 ]
• For:
  • 95% confidence
  • 5% margin of error

only: [ n = 385 ]

fault injections are required.

That is only: [ 0.009% ]

of the total fault space.

This is a major practical result because exhaustive fault injection is usually impossible.

2. Margin of Error Matters More Than Confidence

The paper shows:

• Increasing confidence (e.g., 95% → 99%) only moderately increases required sample size.
• Reducing margin of error (e.g., 5% → 1%) dramatically increases required experiments.

For example:

So precision is much more expensive than confidence.

Conclusion

The paper demonstrates that Statistical Fault Injection can be made scientifically rigorous by applying classical sampling theory.

Its central message is:

Dependability evaluation does not require exhaustive fault injection if sampling error and confidence are quantified correctly.

This significantly reduces experimental cost while preserving measurable reliability in the results.

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