FEAL Linear Cryptanalysis

Overview

This skill provides procedural guidance for attacking the FEAL (Fast Data Encipherment Algorithm) cipher using linear cryptanalysis. FEAL is a well-studied cipher with known vulnerabilities that make it susceptible to linear cryptanalysis attacks, which exploit linear approximations of non-linear components to recover key bits statistically.

Critical Decision Point: Brute Force vs. Linear Cryptanalysis

Before writing any code, determine the attack approach:

Rule of thumb: If the problem mentions "linear attack" or provides many known plaintext-ciphertext pairs (typically 32+), linear cryptanalysis is the intended approach. Brute force is almost never the answer for academic cryptanalysis challenges.

Linear Cryptanalysis Approach

Phase 1: Cipher Analysis

Before implementing any attack:

1. Study the cipher structure

   • Identify the G-function (basic operation)
   • Understand the F-function (round function)
   • Map the key schedule and subkey derivation
   • Count the number of rounds

2. Identify linear approximations

   • Analyze the G-function for linear biases
   • The G-function typically uses rotation and XOR
   • Find input/output bit masks with bias > 0.5
   • Document approximations with their probabilities

3. Chain approximations through rounds

   • Connect single-round approximations
   • Calculate cumulative bias using piling-up lemma: bias = 2^(n-1) * Π(bias_i)
   • Identify which key bits are exposed by each approximation

Phase 2: Statistical Attack Implementation

1. Collect statistics from known pairs

   • For each plaintext-ciphertext pair, evaluate the linear approximation
   • Count how often the approximation holds (equals 0 or 1)
   • The bias indicates the correct key guess

2. Key recovery strategy

   • Attack round keys one at a time, typically starting from the last round
   • For each candidate partial key, count approximation matches
   • The correct key produces the strongest bias deviation from 0.5
   • Use maximum likelihood or chi-squared test for key ranking

3. Recover full key from subkeys

   • Once round keys are known, reverse the key schedule
   • Verify by encrypting a known plaintext

Phase 3: Verification

Before running the full attack:

1. Test with known keys

   • Generate your own key and encrypt test plaintexts
   • Verify your attack recovers the known key
   • This catches implementation bugs early

2. Verify cipher implementation

   • Compare encrypt/decrypt outputs with reference implementations
   • Check that your FEAL implementation matches the provided one exactly

3. Validate approximation bias

   • With random keys, verify your approximations show expected bias
   • If bias is ~0.5, the approximation is wrong

Common Pitfalls

Pitfall 1: Attempting Brute Force Despite Hints

Problem: Ignoring explicit hints about "linear attack" and attempting brute force variations.

Signs of this mistake:

• Writing meet-in-the-middle code
• Creating hash tables for partial key enumeration
• Running out of memory with 2^40+ entries
• Optimizing brute force with "prioritized" key values

Solution: If a problem mentions linear cryptanalysis, implement linear cryptanalysis. No amount of brute force optimization will work for 80-bit keys.

Pitfall 2: Insufficient Cipher Analysis

Problem: Starting to code without understanding the cipher's mathematical properties.

Signs of this mistake:

• Copy-pasting encrypt/decrypt without understanding internals
• Unable to explain what makes FEAL vulnerable
• No identification of specific linear approximations

Solution: Read the reference material on FEAL structure. Understand G(a,b) = ROL2((a+b) mod 256) and how this creates linear biases.

Pitfall 3: Not Verifying Implementation

Problem: Running attacks against unknown keys without first testing on known keys.

Signs of this mistake:

• Attack produces random-looking results
• No verification that cipher implementation matches specification
• Debugging complex attack logic instead of simple test cases

Solution: Always create a test harness:

1. Generate random key K
2. Encrypt known plaintext P -> C
3. Run attack with (P, C) pair
4. Verify attack recovers K

Pitfall 4: Memory-Intensive Approaches

Problem: Using hash tables or sorted arrays for 2^40+ entries.

Signs of this mistake:

• Out-of-memory errors
• Chunked processing that never completes
• Disk-based storage that's too slow

Solution: Linear cryptanalysis requires O(n) memory where n is the number of known pairs, not O(2^k) where k is key bits.

Pitfall 5: Repetitive Failed Attempts

Problem: Trying minor variations of the same failed approach instead of fundamentally changing strategy.

Pattern to recognize:

1. Write brute-force code -> too slow
2. Add meet-in-the-middle -> out of memory
3. Chunk the hash table -> still too slow
4. Prioritize small keys -> doesn't find key
5. Repeat variations...

Solution: After 2-3 failed variations of the same approach, completely abandon it and implement the mathematically correct attack.

Time Management Strategy

1. First 10% of time: Analyze the cipher and understand the attack
2. Next 20% of time: Implement and verify with known test vectors
3. Remaining 70% of time: Run the actual attack

If still attempting brute force after 30% of available time, stop and implement linear cryptanalysis.

Complexity Sanity Check

Before running any attack, calculate:

• Total operations: Should be polynomial in the number of pairs, not exponential in key size
• Memory usage: Should be O(pairs), not O(2^key_bits)
• Expected runtime: Should complete in minutes, not hours

If calculations show infeasibility, the approach is wrong.

Resources

For detailed technical information on FEAL structure and linear approximation derivation, refer to:

• references/linear_cryptanalysis_guide.md - Detailed guide on linear cryptanalysis techniques and FEAL-specific approximations