This paper presents JAL CIPHER, a symmetric product cipher combining two complementary layers motivated by the mechanics of jigsaw puzzles. The first layer, JIGSAWCIPHER, is a key-driven transposition cipher: plaintext is cut into variable-length pieces whose boundaries are secret, and the pieces are assembled in a key-derived order. The second layer, the Jigsaw Affine Layer (JAL), applies a distinct affine cipher S(x) = (aᵢ·x + bᵢ) mod m to every piece Pᵢ. The two layers are computationally independent: breaking the cipher requires defeating both the transposition search space Ω(L) and the substitution space |AGL(1, ℤₘ)|ᵏ simultaneously. All mathematical foundations — modular arithmetic, the Extended Euclidean Algorithm, Euler's totient function, the affine group AGL(1, ℤₘ), and the binomial coefficient — are developed from first principles in journal display-equation format, with full step-by-step derivations for all six modular inverses and every encryption and decryption computation. The combined cipher's parameter space corresponds to 99.41 bits of keyspace at L=17 and exceeds 200 bits at L=32, while requiring only O(L log L) computation. An empirical cryptanalysis against statistically-informed adversaries is further reported: per-piece affine recovery using English letter statistics, a crib attack, and a nonce-reuse demonstration, which together show that realized security against natural-language plaintext is substantially below the nominal keyspace size, particularly for short pieces.

Affine Cipher classical cryptography fragmentation cipher jigsaw puzzle key-driven permutation product cipher transposition cipher