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 in 1996. Since then, researchers have demonstrated practical attacks on RSA (Schindler), OpenSSL (Brumley and Boneh), AES implementations, and even post-quantum algorithms like Kyber.

Key Concepts

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:

// 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.

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

When patterns cannot be avoided, employ masking techniques 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

Constant-Time Tooling Categories

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

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
• ct-verif
• 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
• 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: Tutorial
• 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

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 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

#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 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

#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:

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