NTT: Fast Polynomial Multiplication for Lattice Cryptography (2D Butterfly View)

Every encryption and signature operation in post-quantum standards like CRYSTALS-Kyber and CRYSTALS-Dilithium boils down to multiplying polynomials inside a negacyclic ring ℤ_q[x]/(xⁿ+1). Done the naive way that costs O(n²) modular multiplications; done via the Number Theoretic Transform — the modular-arithmetic cousin of the FFT — it costs O(n log n). This 2D companion draws the real Cooley-Tukey butterfly network stage by stage on canvas (bit-reversal, twiddle-factor butterflies, pointwise multiply, inverse transform), cross-checks every result against a direct O(n²) convolution, and benchmarks real wall-clock timing of both methods as the polynomial degree grows.