Deterministic Cofactor Recovery, NC-Class ASIC/FPGA Architecture, and the Super-Deterministic Communication Protocol (SDCP)

Abstract This paper introduces the Generalized FB-Direct Theorem, a deterministic method for recovering cofactor components in $\mathbb{Z}[\sqrt{-D}]$ without modular exponentiation or iterative modular square roots. By reducing the computational complexity from $O(\log^3 p)$ to fundamental arithmetic operations $O(M(n))$, this work establishes the theoretical foundation for two novel engineering applications disclosed herein: a shallow-depth NC-Class ASIC/FPGA Architecture and the Super-Deterministic Communication Protocol (SDCP). Key Technical Disclosures This publication explicitly places the following methods and designs into the public domain to serve as prior art: Hardware Architecture (ASIC/FPGA): A feed-forward, non-iterative circuit design for cofactor recovery. Utilization of Wallace Tree or Dadda multipliers combined with Carry-Lookahead adders to achieve a critical path depth of approximately 25 gate delays for 1024-bit inputs. Replacement of general division logic with Deterministic Exact Division Scaling, bypassing the need for complex restoration algorithms. Communication Protocol (SDCP): A physical layer (PHY) security mechanism defined as "Algebraic Armor". Zero-Latency Error Detection: Utilization of quadratic form alignment ($N=X^2+DY^2$) to detect signal noise deterministically. A method for rejecting noise-corrupted frames based on integer divisibility failure, eliminating the need for probabilistic checksums (CRC) or Reed-Solomon codes. Mathematical Algorithm: Deterministic recovery of cofactor components $(c, d)$ using the Generalized Brahmagupta-Fibonacci identity. Proof of complexity reduction to $O(M(n))$, enabling real-time processing suitable for optical and photonic computing architectures.