Feature/lwe encryption

[dilaraacetin]

Fixes #7630

Description

Adds an educational implementation of Learning With Errors (LWE) public-key encryption (Regev, 2005) to the ciphers package. This is the first lattice-based algorithm in the repository and shows the mathematical foundation of ML-KEM (Kyber, FIPS 203).

How it works

• Key generation: a secret s ∈ Z_q^n, a random matrix A ∈ Z_q^{m×n} and a small error e ∈ [-B, B]^m are sampled. The public key is (A, b = A·s + e mod q), and the private key is s.
• Encryption of a bit: a random r ∈ {0,1}^m is chosen, then u = rᵀ·A (mod q) and v = r·b + bit·⌊q/2⌋ (mod q) are computed. Encryption is randomized.
• Decryption: d = v − u·s (mod q) equals r·e + bit·⌊q/2⌋. The bit is 1 if q/4 ≤ d < 3q/4, otherwise 0.
• Messages: byte[] messages are encrypted bit by bit, MSB first.

Changes

• src/main/java/com/thealgorithms/ciphers/LWEEncryption.java: implementation with a nested immutable Ciphertext class
• src/test/java/com/thealgorithms/ciphers/LWEEncryptionTest.java: JUnit 5 tests

No other files are modified.

Design notes

• Toy parameters: n = 32, q = 3329 (the ML-KEM modulus), m = 128, B = 3.
• Correctness is guaranteed: the accumulated noise is at most m·B = 384 < q/4 ≈ 832, so decryption never fails. This condition is documented and checked in the code.
• Math.floorMod is used for all modular reductions, and inner products are accumulated in long to avoid overflow.
• SecureRandom is used for all sampling.
• Ciphertext is immutable; arrays are defensively copied.
• For simplicity, one instance holds both the public and private key; the Javadoc notes that these are separate in real usage.
• Educational only: the parameters are not secure. Real schemes use discrete Gaussian noise and much larger dimensions, as stated in the Javadoc.

Tests

• Bit roundtrip: bits 0 and 1, 100 repetitions each, all correct.
• Byte roundtrip: "Hello, PQC!" and a random 1 KB message.
• Randomized encryption: encrypting the same bit twice gives different ciphertexts.
• Wrong key: decrypting an 8+ byte message with another instance does not recover the original.
• Bit ordering, including negative byte values such as 0xFF and 0x80.
• Input validation (invalid bits, null/empty inputs, malformed ciphertexts) and Ciphertext immutability.

mvn clean verify passes locally, and all files are formatted with clang-format -i --style=file.

References

• https://en.wikipedia.org/wiki/Learning_with_errors

• O. Regev, "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography", STOC 2005

• FIPS 203 (ML-KEM): https://csrc.nist.gov/pubs/fips/203/final

• I have read CONTRIBUTING.md.

• This pull request is all my own work -- I have not plagiarized it.

• All filenames are in PascalCase.

• All functions and variable names follow Java naming conventions.

• All new algorithms have a URL in their comments that points to Wikipedia or other similar explanations.

• All new algorithms include a corresponding test class that validates their functionality.

• All new code is formatted with clang-format -i --style=file path/to/your/file.java

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 81.74%. Comparing base (922c4d3) to head (2eb9fbd).

Additional details and impacted files

@@             Coverage Diff              @@
##             master    #7631      +/-   ##
============================================
+ Coverage     81.68%   81.74%   +0.05%     
- Complexity     8192     8223      +31     
============================================
  Files           838      839       +1     
  Lines         25734    25799      +65     
  Branches       5027     5042      +15     
============================================
+ Hits          21022    21089      +67     
+ Misses         3924     3923       -1     
+ Partials        788      787       -1     

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[alxkm]

Looks good. Thank you for the contribution.