constant-time-testing

# Constant-Time Testing

Timing attacks exploit variations in execution time to extract secret information from cryptographic implementations. Unlike cryptanalysis that targets theoretical weaknesses, timing attacks leverage implementation flaws - and they can affect any cryptographic code.

## Background

Timing attacks were introduced by [Kocher](https://paulkocher.com/doc/TimingAttacks.pdf) in 1996. Since then, researchers have demonstrated practical attacks on RSA ([Schindler](https://link.springer.com/content/pdf/10.1007/3-540-44499-8_8.pdf)), OpenSSL ([Brumley and Boneh](https://crypto.stanford.edu/~dabo/papers/ssl-timing.pdf)), AES implementations, and even post-quantum algorithms like [Kyber](https://eprint.iacr.org/2024/1049.pdf).

### Key Concepts

| Concept | Description |
|---------|-------------|
| Constant-time | Code path and memory accesses independent of secret data |
| Timing leakage | Observable execution time differences correlated with secrets |
| Side channel | Information extracted from implementation rather than algorithm |
| Microarchitecture | CPU-level timing differences (cache, division, shifts) |

### Why This Matters

Timing vulnerabilities can:
- **Expose private keys** - Extract secret exponents in RSA/ECDH
- **Enable remote attacks** - Network-observable timing differences
- **Bypass cryptographic security** - Undermine theoretical guarantees
- **Persist silently** - Often undetected without specialized analysis

Two prerequisites enable exploitation:
1. **Access to oracle** - Sufficient queries to the vulnerable implementation
2. **Timing dependency** - Correlation between execution time and secret data

### Common Constant-Time Violation Patterns

Four patterns account for most timing vulnerabilities:

```c
// 1. Conditional jumps - most severe timing differences
if(secret == 1) { ... }
while(secret > 0) { ... }

// 2. Array access - cache-timing attacks
lookup_table[secret];

// 3. Integer division (processor dependent)
data = secret / m;

// 4. Shift operation (processor dependent)
data = a << secret;
```

**Conditional jumps** cause different code paths, leading to vast timing differences.

**Array access** dependent on secrets enables cache-timing attacks, as shown in [AES cache-timing research](https://cr.yp.to/antiforgery/cachetiming-20050414.pdf).

**Integer division and shift operations** leak secrets on certain CPU architectures and compiler configurations.

When patterns cannot be avoided, employ [masking techniques](https://link.springer.com/chapter/10.1007/978-3-642-38348-9_9) to remove correlation between timing and secrets.

### Example: Modular Exponentiation Timing Attacks

Modular exponentiation (used in RSA and Diffie-Hellman) is susceptible to timing attacks. RSA decryption computes:

$$ct^{d} \mod{N}$$

where $d$ is the secret exponent. The *exponentiation by squaring* optimization reduces multiplications to $\log{d}$:

$$
\begin{align*}
& \textbf{Input: } \text{base }y,\text{exponent } d=\{d_n,\cdots,d_0\}_2,\text{modulus } N \\
& r = 1 \\
& \textbf{for } i=|n| \text{ downto } 0: \\
& \quad\textbf{if } d_i == 1: \\
& \quad\quad r = r * y \mod{N} \\
& \quad y = y * y \mod{N} \\
& \textbf{return }r
\end{align*}
$$

The code branches on exponent bit $d_i$, violating constant-time principles. When $d_i = 1$, an additional multiplication occurs, increasing execution time and leaking bit information.

Montgomery multiplication (commonly used for modular arithmetic) also leaks timing: when intermediate values exceed modulus $N$, an additional reduction step is required. An attacker constructs inputs $y$ and $y'$ such that:

$$
\begin{align*}
y^2 < y^3 < N \\
y'^2 < N \leq y'^3
\end{align*}
$$

For $y$, both multiplications take time $t_1+t_1$. For $y'$, the second multiplication requires reduction, taking time $t_1+t_2$. This timing difference reveals whether $d_i$ is 0 or 1.

## When to Use

**Apply constant-time analysis when:**
- Auditing cryptographic implementations (primitives, protocols)
- Code handles secret keys, passwords, or sensitive cryptographic material
- Implementing crypto algorithms from scratch
- Reviewing PRs that touch crypto code
- Investigating potential timing vulnerabilities

**Consider alternatives when:**
- Code does not process secret data
- Public algorithms with no secret inputs
- Non-cryptographic timing requirements (performance optimization)

## Quick Reference

| Scenario | Recommended Approach | Skill |
|----------|---------------------|-------|
| Prove absence of leaks | Formal verification | SideTrail, ct-verif, FaCT |
| Detect statistical timing differences | Statistical testing | **dudect** |
| Track secret data flow at runtime | Dynamic analysis | **timecop** |
| Find cache-timing vulnerabilities | Symbolic execution | Binsec, pitchfork |

## Constant-Time Tooling Categories

The cryptographic community has developed four categories of timing analysis tools:

| Category | Approach | Pros | Cons |
|----------|----------|------|------|
| **Formal** | Mathematical proof on model | Guarantees absence of leaks | Complexity, modeling assumptions |
| **Symbolic** | Symbolic execution paths | Concrete counterexamples | Time-intensive path exploration |
| **Dynamic** | Runtime tracing with marked secrets | Granular, flexible | Limited coverage to executed paths |
| **Statistical** | Measure real execution timing | Practical, simple setup | No root cause, noise sensitivity |

### 1. Formal Tools

Formal verification mathematically proves timing properties on an abstraction (model) of code. Tools create a model from source/binary and verify it satisfies specified properties (e.g., variables annotated as secret).

**Popular tools:**
- [SideTrail](https://github.com/aws/s2n-tls/tree/main/tests/sidetrail)
- [ct-verif](https://github.com/imdea-software/verifying-constant-time)
- [FaCT](https://github.com/plsyssec/fact)

**Strengths:** Proof of absence, language-agnostic (LLVM bytecode)
**Weaknesses:** Requires expertise, modeling assumptions may miss real-world issues

### 2. Symbolic Tools

Symbolic execution analyzes how paths and memory accesses depend on symbolic variables (secrets). Provides concrete counterexamples. Focus on cache-timing attacks.

**Popular tools:**
- [Binsec](https://github.com/binsec/binsec)
- [pitchfork](https://github.com/PLSysSec/haybale-pitchfork)

**Strengths:** Concrete counterexamples aid debugging
**Weaknesses:** Path explosion leads to long execution times

### 3. Dynamic Tools

Dynamic analysis marks sensitive memory regions and traces execution to detect timing-dependent operations.

**Popular tools:**
- [Memsan](https://clang.llvm.org/docs/MemorySanitizer.html): [Tutorial](https://crocs-muni.github.io/ct-tools/tutorials/memsan)
- **Timecop** (see below)

**Strengths:** Granular control, targeted analysis
**Weaknesses:** Coverage limited to executed paths

> **Detailed Guidance:** See the **timecop** skill for setup and usage.

### 4. Statistical Tools

Execute code with various inputs, measure elapsed time, and detect inconsistencies. Tests actual implementation including compiler optimizations and architecture.

**Popular tools:**
- **dudect** (see below)
- [tlsfuzzer](https://github.com/tlsfuzzer/tlsfuzzer)

**Strengths:** Simple setup, practical real-world results
**Weaknesses:** No root cause info, noise obscures weak signals

> **Detailed Guidance:** See the **dudect** skill for setup and usage.

## Testing Workflow

```
Phase 1: Static Analysis        Phase 2: Statistical Testing
┌─────────────────┐            ┌─────────────────┐
│ Identify secret │      →     │ Detect timing   │
│ data flow       │            │ differences     │
│ Tool: ct-verif  │            │ Tool: dudect    │
└─────────────────┘            └─────────────────┘
         ↓                              ↓
Phase 4: Root Cause             Phase 3: Dynamic Tracing
┌─────────────────┐            ┌─────────────────┐
│ Pinpoint leak   │      ←     │ Track secret    │
│ location        │            │ propagation     │
│ Tool: Timecop   │            │ Tool: Timecop   │
└─────────────────┘            └─────────────────┘
```

**Recommended approach:**
1. **Start with dudect** - Quick statistical check for timing differences
2. **If leaks found** - Use Timecop to pinpoint root cause
3. **For high-assurance** - Apply formal verification (ct-verif, SideTrail)
4. **Continuous monitoring** - Integrate dudect into CI pipeline

## Tools and Approaches

### Dudect - Statistical Analysis

[Dudect](https://github.com/oreparaz/dudect/) measures execution time for two input classes (fixed vs random) and uses Welch's t-test to detect statistically significant differences.

> **Detailed Guidance:** See the **dudect** skill for complete setup, usage patterns, and CI integration.

#### Quick Start for Constant-Time Analysis

```c
#define DUDECT_IMPLEMENTATION
#include "dudect.h"

uint8_t do_one_computation(uint8_t *data) {
    // Code to measure goes here
}

void prepare_inputs(dudect_config_t *c, uint8_t *input_data, uint8_t *classes) {
    for (size_t i = 0; i < c->number_measurements; i++) {
        classes[i] = randombit();
        uint8_t *input = input_data + (size_t)i * c->chunk_size;
        if (classes[i] == 0) {
            // Fixed input class
        } else {
            // Random input class
        }
    }
}
```

**Key advantages:**
- Simple C header-only integration
- Statistical rigor via Welch's t-test
- Works with compiled binaries (real-world conditions)

**Key limitations:**
- No root cause information when leak detected
- Sensitive to measurement noise
- Cannot guarantee absence of leaks (statistical confidence only)

### Timecop - Dynamic Tracing

[Timecop](https://post-apocalyptic-crypto.org/timecop/) wraps Valgrind to detect runtime operations dependent on secret memory regions.

> **Detailed Guidance:** See the **timecop** skill for installation, examples, and debugging.

#### Quick Start for Constant-Time Analysis

```c
#include "valgrind/memcheck.h"

#define poison(addr, len) VALGRIND_MAKE_MEM_UNDEFINED(addr, len)
#define unpoison(addr, len) VALGRIND_MAKE_MEM_DEFINED(addr, len)

int main() {
    unsigned long long secret_key = 0x12345678;

    // Mark secret as poisoned
    poison(&secret_key, sizeof(secret_key));

    // Any branching or memory access dependent on secret_key
    // will be reported by Valgrind
    crypto_operation(secret_key);

    unpoison(&secret_key, sizeof(secret_key));
}
```

Run with Valgrind:
```bash
valgrind --leak-check=full --track-origins=yes ./binary
```

**Key advantages:**
- Pinpoints exact line of timing leak
- No code instrumentation required
- Tracks secret propagation through execution

**Key limitations:**
- Cannot detect microarchitecture timing differences
- Coverage limited to executed paths
- Performance overhead (runs on synthetic CPU)

## Implementation Guide

### Phase 1: Initial Assessment

**Identify cryptographic code handling secrets:**
- Private keys, exponents, nonces
- Password hashes, authentication tokens
- Encryption/decryption operations

**Quick statistical check:**
1. Write dudect harness for the crypto function
2. Run for 5-10 minutes with `timeout 600 ./ct_test`
3. Monitor t-value: high absolute values indicate leakage

**Tools:** dudect
**Expected time:** 1-2 hours (harness writing + initial run)

### Phase 2: Detailed Analysis

If dudect detects leakage:

**Root cause investigation:**
1. Mark secret variables with Timecop `poison()`
2. Run under Valgrind to identify exact line
3. Review the four common violation patterns
4. Check assembly output for conditional branches

**Tools:** Timecop, compiler output (`objdump -d`)

### Phase 3: Remediation

**Fix the timing leak:**
- Replace conditional branches with constant-time selection (bitwise operations)
- Use constant-time comparison functions
- Replace array lookups with constant-time alternatives or masking
- Verify compiler doesn't optimize away constant-time code

**Re-verify:**
1. Run dudect again for extended period (30+ minutes)
2. Test across different compilers and optimization levels
3. Test on different CPU architectures

### Phase 4: Continuous Monitoring

**Integrate into CI:**
- Add dudect tests to test suite
- Run for fixed duration (5-10 minutes in CI)
- Fail build if leakage detected

See the **dudect** skill for CI integration examples.

## Common Vulnerabilities

| Vulnerability | Description | Detection | Severity |
|---------------|-------------|-----------|----------|
| Secret-dependent branch | `if (secret_bit) { ... }` | dudect, Timecop | CRITICAL |
| Secret-dependent array access | `table[secret_index]` | Timecop, Binsec | HIGH |
| Variable-time division | `result = x / secret` | Timecop | MEDIUM |
| Variable-time shift | `result = x << secret` | Timecop | MEDIUM |
| Montgomery reduction leak | Extra reduction when intermediate > N | dudect | HIGH |

### Secret-Dependent Branch: Deep Dive

**The vulnerability:**
Execution time differs based on whether branch is taken. Common in optimized modular exponentiation (square-and-multiply).

**How to detect with dudect:**
```c
uint8_t do_one_computation(uint8_t *data) {
    uint64_t base = ((uint64_t*)data)[0];
    uint64_t exponent = ((uint64_t*)data)[1]; // Secret!
    return mod_exp(base, exponent, MODULUS);
}

void prepare_inputs(dudect_config_t *c, uint8_t *input_data, uint8_t *classes) {
    for (size_t i = 0; i < c->number_measurements; i++) {
        classes[i] = randombit();
        uint64_t *input = (uint64_t*)(input_data + i * c->chunk_size);
        input[0] = rand(); // Random base
        input[1] = (classes[i] == 0) ? FIXED_EXPONENT : rand(); // Fixed vs random
    }
}
```

**How to detect with Timecop:**
```c
poison(&exponent, sizeof(exponent));
result = mod_exp(base, exponent, modulus);
unpoison(&exponent, sizeof(exponent));
```

Valgrind will report:
```
Conditional jump or move depends on uninitialised value(s)
  at 0x40115D: mod_exp (example.c:14)
```

**Related skill:** **dudect**, **timecop**

## Case Studies

### Case Study: OpenSSL RSA Timing Attack

Brumley and Boneh (2005) extracted RSA private keys from OpenSSL over a network. The vulnerability exploited Montgomery multiplication's variable-time reduction step.

**Attack vector:** Timing differences in modular exponentiation
**Detection approach:** Statistical analysis (precursor to dudect)
**Impact:** Remote key extraction

**Tools used:** Custom timing measurement
**Techniques applied:** Statistical analysis, chosen-ciphertext queries

### Case Study: KyberSlash

Post-quantum algorithm Kyber's reference implementation contained timing vulnerabilities in polynomial operations. Division operations leaked secret coefficients.

**Attack vector:** Secret-dependent division timing
**Detection approach:** Dynamic analysis and statistical testing
**Impact:** Secret key recovery in post-quantum cryptography

**Tools used:** Timing measurement tools
**Techniques applied:** Differential timing analysis

## Advanced Usage

### Tips and Tricks

| Tip | Why It Helps |
|-----|--------------|
| Pin dudect to isolated CPU core (`taskset -c 2`) | Reduces OS noise, improves signal detection |
| Test multiple compilers (gcc, clang, MSVC) | Optimizations may introduce or remove leaks |
| Run dudect for extended periods (hours) | Increases statistical confidence |
| Minimize non-crypto code in harness | Reduces noise that masks weak signals |
| Check assembly output (`objdump -d`) | Verify compiler didn't introduce branches |
| Use `-O3 -march=native` in testing | Matches production optimization levels |

### Common Mistakes

| Mistake | Why It's Wrong | Correct Approach |
|---------|----------------|------------------|
| Only testing one input distribution | May miss leaks visible with other patterns | Test fixed-vs-random, fixed-vs-fixed-different, etc. |
| Short dudect runs (< 1 minute) | Insufficient measurements for weak signals | Run 5-10+ minutes, longer for high assurance |
| Ignoring compiler optimization levels | `-O0` may hide leaks present in `-O3` | Test at production optimization level |
| Not testing on target architecture | x86 vs ARM have different timing characteristics | Test on deployment platform |
| Marking too much as secret in Timecop | False positives, unclear results | Mark only true secrets (keys, not public data) |

## Related Skills

### Tool Skills

| Skill | Primary Use in Constant-Time Analysis |
|-------|---------------------------------------|
| **dudect** | Statistical detection of timing differences via Welch's t-test |
| **timecop** | Dynamic tracing to pinpoint exact location of timing leaks |

### Technique Skills

| Skill | When to Apply |
|-------|---------------|
| **coverage-analysis** | Ensure test inputs exercise all code paths in crypto function |
| **ci-integration** | Automate constant-time testing in continuous integration pipeline |

### Related Domain Skills

| Skill | Relationship |
|-------|--------------|
| **crypto-testing** | Constant-time analysis is essential component of cryptographic testing |
| **fuzzing** | Fuzzing crypto code may trigger timing-dependent paths |

## Skill Dependency Map

```
                    ┌─────────────────────────┐
                    │  constant-time-analysis │
                    │     (this skill)        │
                    └───────────┬─────────────┘
                                │
                ┌───────────────┴───────────────┐
                │                               │
                ▼                               ▼
    ┌───────────────────┐           ┌───────────────────┐
    │      dudect       │           │     timecop       │
    │  (statistical)    │           │    (dynamic)      │
    └────────┬──────────┘           └────────┬──────────┘
             │                               │
             └───────────────┬───────────────┘
                             │
                             ▼
              ┌──────────────────────────────┐
              │   Supporting Techniques      │
              │ coverage, CI integration     │
              └──────────────────────────────┘
```

## Resources

### Key External Resources

**[These results must be false: A usability evaluation of constant-time analysis tools](https://www.usenix.org/system/files/sec24fall-prepub-760-fourne.pdf)**
Comprehensive usability study of constant-time analysis tools. Key findings: developers struggle with false positives, need better error messages, and benefit from tool integration. Evaluates FaCT, ct-verif, dudect, and Memsan across multiple cryptographic implementations. Recommends improved tooling UX and better documentation.

**[List of constant-time tools - CROCS](https://crocs-muni.github.io/ct-tools/)**
Curated catalog of constant-time analysis tools with tutorials. Covers formal tools (ct-verif, FaCT), dynamic tools (Memsan, Timecop), symbolic tools (Binsec), and statistical tools (dudect). Includes practical tutorials for setup and usage.

**[Paul Kocher: Timing Attacks on Implementations of Diffie-Hellman, RSA, DSS, and Other Systems](https://paulkocher.com/doc/TimingAttacks.pdf)**
Original 1996 paper introducing timing attacks. Demonstrates attacks on modular exponentiation in RSA and Diffie-Hellman. Essential historical context for understanding timing vulnerabilities.

**[Remote Timing Attacks are Practical (Brumley & Boneh)](https://crypto.stanford.edu/~dabo/papers/ssl-timing.pdf)**
Demonstrates practical remote timing attacks against OpenSSL. Shows network-level timing differences are sufficient to extract RSA keys. Proves timing attacks work in realistic network conditions.

**[Cache-timing attacks on AES](https://cr.yp.to/antiforgery/cachetiming-20050414.pdf)**
Shows AES implementations using lookup tables are vulnerable to cache-timing attacks. Demonstrates practical attacks extracting AES keys via cache timing side channels.

**[KyberSlash: Division Timings Leak Secrets](https://eprint.iacr.org/2024/1049.pdf)**
Recent discovery of timing vulnerabilities in Kyber (NIST post-quantum standard). Shows division operations leak secret coefficients. Highlights that constant-time issues persist even in modern post-quantum cryptography.

### Video Resources

- [Trail of Bits: Constant-Time Programming](https://www.youtube.com/watch?v=vW6wqTzfz5g) - Overview of constant-time programming principles and tools