Derivative works — US Patent 8666062

Defensive Disclosure: Method and Apparatus for Performing Finite Field Calculations

Publication Date: May 10, 2026

Subject Matter: This document discloses novel extensions, applications, and implementations derived from the architectural principles of U.S. Patent 8,666,062. The purpose of this disclosure is to place these concepts into the public domain, thereby establishing prior art against future patent applications on these and similar incremental innovations. The core concept of the '062 patent, a two-stage process involving a generalized "wordsized" operation followed by a specific modular reduction, is expanded upon herein.

Claim 1 & 15 Derivative: Method and Cryptographic Engine

Axis 1: Material & Component Substitution

1. FPGA-Based Reconfigurable Crypto-Processor

Enabling Description: A cryptographic engine is implemented on a Field-Programmable Gate Array (FPGA). The "first set of instructions" (wordsized operations like multiplication, addition) is realized as a permanent, optimized, and generic logic block synthesized from a hardware description language (e.g., Verilog or VHDL). This block accepts operands of a fixed maximum width (e.g., 512 bits). The "second set of instructions" (modular reduction) is not fixed logic. Instead, it is a partial reconfiguration bitstream, specific to a given finite field modulus (e.g., NIST P-256 prime). Upon initialization of a cryptographic protocol, a host processor loads the appropriate partial bitstream into a designated reconfigurable region of the FPGA. This dynamically programs the reduction logic, which then receives the unreduced output from the fixed wordsized block. This allows for field-agile cryptography in hardware without requiring a full re-synthesis of the FPGA.

Mermaid Diagram:

graph TD
    subgraph FPGA Fabric
        subgraph Static Region
            A[Input A Register] --> WordOp
            B[Input B Register] --> WordOp
            WordOp{Wordsized Operator<br>(e.g., Full Multiplier)} --> UnreducedResult[Unreduced Result Bus]
        end
        subgraph Dynamic Reconfigurable Region
            UnreducedResult --> ModReducer
            ModReducer{Modular Reducer<br>(Logic loaded from bitstream)} --> ReducedResult[Reduced Result Register]
        end
    end

    HostCPU[Host CPU] -- "Load Reduction Bitstream (e.g., for P-256)" --> ModReducer
    HostCPU -- "Provide Operands" --> A & B
    ReducedResult --> HostCPU

    style ModReducer fill:#f9f,stroke:#333,stroke-width:2px

2. In-Memory Computing with Resistive RAM (ReRAM)

Enabling Description: Finite field operations are performed directly within a ReRAM crossbar array, eliminating the CPU-memory bus bottleneck. Field elements are stored as resistance levels in ReRAM cells. The "first set of instructions" is a sequence of voltage pulses applied to wordlines and bitlines, performing analog matrix-vector multiplications that result in an unreduced product, accumulated as charge on the bitlines. This unreduced analog value is then processed by the "second set of instructions," which comprises a digital circuit (ADC, control logic, and DAC) integrated at the periphery of the memory array. This peripheral logic reads the analog result, performs a digital modular reduction specific to the finite field, and writes the final reduced value back into the ReRAM array by applying programming pulses. The reduction logic can be re-programmed for different fields.

Mermaid Diagram:

graph TD
    subgraph ReRAM Chip
        A[ReRAM Array<br>Stores Operands] -- Voltage Pulses --> B{Crossbar Array<br>Analog Multiplication}
        B -- Accumulated Charge --> C[Peripheral Sense Amps / ADCs]
        C -- Digital Unreduced Value --> D{Modular Reduction Unit<br>(Programmable Logic)}
        D -- Reduced Digital Value --> E[Peripheral Drivers / DACs]
        E -- Programming Pulses --> A
    end
    Controller[External Controller] -- "Set Field (p)" --> D
    Controller -- "Initiate Op(A, B)" --> A

3. GPU-Accelerated Batch Cryptography

Enabling Description: A method for performing bulk cryptographic operations on a Graphics Processing Unit (GPU). A large number of element pairs are loaded into the GPU's global memory. A first CUDA or OpenCL kernel implements the "wordsized" multiplication, where each thread in a block computes a partial product. These are aggregated into an unreduced result, twice the bit-length of the operands. This first kernel is generic for the word size. A second, separate kernel is then launched. This second kernel is selected from a library of pre-compiled reduction kernels, each one optimized for a specific, commonly used cryptographic prime (e.g., secp256k1, Curve25519). This reduction kernel reads the unreduced results from global memory, performs the modular reduction in parallel, and writes the final results back. This two-kernel pipeline maximizes GPU occupancy and leverages specialized instruction sets (like integer multiply-add) for both stages.

Mermaid Diagram:

sequenceDiagram
    participant CPU
    participant GPU

    CPU->>GPU: 1. Transfer Operands (A[], B[]) to Global Memory
    CPU->>GPU: 2. Launch WordsizedMultiply_Kernel(A[], B[], Unreduced_C[])
    activate GPU
    Note right of GPU: Each thread computes C[i] = A[i] * B[i] (unreduced)
    GPU-->>CPU: Kernel 1 Complete
    deactivate GPU
    CPU->>GPU: 3. Launch Reduce_secp256k1_Kernel(Unreduced_C[], Reduced_C[])
    activate GPU
    Note right of GPU: Each thread computes Reduced_C[i] = Unreduced_C[i] mod p
    GPU-->>CPU: Kernel 2 Complete
    deactivate GPU
    CPU->>GPU: 4. Read back Reduced_C[] from Global Memory

Axis 2: Operational Parameter Expansion

4. Cryptography for Deep-Space Radiation-Hardened Systems

Enabling Description: A cryptographic engine for spacecraft operating in high-radiation environments. The "wordsized" arithmetic unit is implemented using Triple Modular Redundancy (TMR) in a radiation-hardened-by-design (RHBD) ASIC. This core logic is simple, robust, and performs basic operations on a fixed word size (e.g., 256-bit operands yielding a 512-bit result). To allow for in-flight updates to cryptographic standards (e.g., moving to a new post-quantum standard), the "modular reduction" logic is stored in reprogrammable, radiation-tolerant MRAM (Magnetoresistive RAM). An uplinked command from ground control can overwrite the MRAM with a new set of reduction micro-instructions, adapting the system to new security protocols without requiring a full software patch of the flight computer, which is a high-risk operation.

Mermaid Diagram:

graph TD
    subgraph Rad-Hard ASIC
        subgraph TMR_Core [TMR Wordsized Core]
            Op1(Operand 1) --> ProcA
            Op2(Operand 2) --> ProcA
            Op1 --> ProcB
            Op2 --> ProcB
            Op1 --> ProcC
            Op2 --> ProcC
            ProcA --> Voter
            ProcB --> Voter
            ProcC --> Voter
        end
        Voter -- "Unreduced Result" --> Reducer
        MRAM[Rad-Tolerant MRAM<br>Stores Reduction Microcode] -- "Instructions" --> Reducer{Microcoded Reduction Unit}
        Reducer -- "Final Result" --> OutputBus
    end
    GroundControl[Ground Control Uplink] -- "Update Microcode" --> MRAM

5. Real-Time Cryptographic Engine for Terahertz (THz) Communications

Enabling Description: For future 6G and beyond communication systems operating in the 100-300 GHz range, data rates will demand cryptographic processing with latencies in the nanosecond range. This method is implemented in a Gallium Nitride (GaN) or Indium Phosphide (InP) integrated circuit. The "wordsized" multiplier is a massively parallel, pipelined Karatsuba multiplier designed for extreme clock speeds. The unreduced output is fed directly into a bank of selectable reduction circuits. The "second set of instructions" is not software, but a hardware multiplexer that routes the unreduced result to one of several hard-wired reduction circuits, each optimized for a specific standard (e.g., one for AES-GCM in GF(2^128), another for an ECC curve). The selection is controlled by a low-latency control signal from the baseband processor, allowing for sub-nanosecond switching between cryptographic schemes.

Mermaid Diagram:

graph TD
    DataIn[High-Speed Data In] --> BasebandProc[Baseband Processor]
    BasebandProc -- "Operands" --> GaN_ASIC
    BasebandProc -- "Select 'AES' or 'ECC'" --> MUX_Control
    
    subgraph GaN_ASIC
        WordsizedMultiplier[Pipelined Karatsuba Multiplier] --> UnreducedBus
        UnreducedBus --> MUX{Multiplexer}
        MUX --> ReducerAES[Hard-wired AES-GCM Reducer]
        MUX --> ReducerECC[Hard-wired ECC Reducer]
        ReducerAES --> EncryptedDataOut
        ReducerECC --> EncryptedDataOut
    end
    
    MUX_Control[Control Signal] --> MUX

Axis 3: Cross-Domain Application

6. Finite Field Engine for Error Correction in Genomic Data Storage

Enabling Description: DNA-based data storage encodes binary data into nucleotide sequences (A, T, C, G). This process is error-prone during synthesis and sequencing. This method is used to implement Reed-Solomon error correction codes over GF(2^8) or GF(2^16). The "wordsized" engine performs the generic polynomial multiplication and division required for syndrome calculation and Chien search. The "modular reduction" instruction set is specific to the generator polynomial of the Reed-Solomon code being used, which can be changed depending on the desired error correction capability (e.g., more redundancy for long-term archival). This allows a single hardware accelerator to be used for different coding schemes optimized for various DNA storage applications.

Mermaid Diagram:

flowchart TD
    A[Genomic Data Chunk] --> B(Encode as Polynomial)
    B --> C{Syndrome Calculation Engine}
    C --> D{Error Locator Polynomial Calc}
    D --> E{Error Value Calculation}
    E --> F(Corrected Polynomial) --> G[Corrected Genomic Data]

    subgraph Finite Field Accelerator
        C -- "Generic Poly Multiply" --> WordsizedEngine
        D -- "Generic Poly Multiply/Divide" --> WordsizedEngine
        E -- "Generic Poly Evaluation" --> WordsizedEngine
        WordsizedEngine -- Unreduced Result --> ReductionEngine
        ReductionEngine -- Reduced Result --> C & D & E
    end

    Control[Storage Controller] -- "Load RS-Code Generator Polynomial" --> ReductionEngine

7. Dynamic Simulation of Crystalline Structures

Enabling Description: In computational materials science, particularly crystallography, operations within finite groups and fields are used to model lattice symmetries. This method is applied to accelerate these simulations. A generalized "wordsized" engine computes group operations (represented as matrix multiplications) in a large, encompassing field. The "second set of instructions" implements a modular reduction specific to the symmetry group (e.g., one of the 230 space groups) of the crystal being simulated. This allows researchers to use the same core computational hardware to simulate different materials (e.g., silicon, quartz, perovskites) by simply loading a different, compact reduction module for each material's crystal structure.

Mermaid Diagram:

graph LR
    SimConfig[Simulation Config: Material='Quartz'] -->|Selects Space Group P3121| Controller
    Controller -->|Loads 'P3121' Reduction Module| ReductionUnit
    
    subgraph Physics_Core
        StateA[Atom Positions Vector] --> Op
        Transform[Symmetry Transform Matrix] --> Op{Wordsized Matrix Multiply}
        Op --> UnreducedState[Unreduced State Vector]
        UnreducedState --> ReductionUnit{Reduction Unit}
        ReductionUnit --> NewState[New Atom Positions]
    end

    NewState --> NextIteration[Next Simulation Step]

Axis 4: Integration with Emerging Tech

8. AI-Driven Adaptive Cryptography for IoT Networks

Enabling Description: An AI-based network security orchestrator monitors an IoT network for threats and computational constraints (e.g., device power levels, network latency). Based on this real-time analysis, it determines the optimal cryptographic curve and parameters for different segments of the network. For a high-power gateway, it might select a secure 521-bit curve. For a battery-powered sensor, it might select a more efficient 163-bit curve. The orchestrator generates the specific "modular reduction" instructions for the chosen curve and securely distributes them to the IoT devices. The devices, all equipped with the same generic "wordsized" engine (the first instruction set), load this new reduction module to seamlessly switch cryptographic schemes without requiring a full firmware update. This creates a self-optimizing, agile cryptographic infrastructure.

Mermaid Diagram:

sequenceDiagram
    participant AI_Orchestrator
    participant IoT_Gateway
    participant IoT_Sensor

    AI_Orchestrator->>IoT_Sensor: Monitor(Energy_Level, Latency)
    Note over AI_Orchestrator: Energy is low. Select efficient curve.
    AI_Orchestrator->>AI_Orchestrator: Generate 'K-163' Reduction Module
    AI_Orchestrator->>IoT_Sensor: Deploy(ReductionModule_K163)
    IoT_Sensor->>IoT_Sensor: Load K-163 into FF Engine

    AI_Orchestrator->>IoT_Gateway: Monitor(Threat_Level)
    Note over AI_Orchestrator: High threat detected. Select robust curve.
    AI_Orchestrator->>AI_Orchestrator: Generate 'P-521' Reduction Module
    AI_Orchestrator->>IoT_Gateway: Deploy(ReductionModule_P521)
    IoT_Gateway->>IoT_Gateway: Load P-521 into FF Engine

Axis 5: The "Inverse" or Failure Mode

9. Cryptographic Watchdog with Graceful Degradation

Enabling Description: A cryptographic engine designed for high-availability systems where failure is not an option. The engine operates in three modes.
- Mode 1 (Normal): Executes both the "wordsized" operation and the specific "modular reduction" for full cryptographic security.
- Mode 2 (Degraded/Low-Power): If the processor detects a fault in the reduction unit or a low-power directive, it bypasses the second stage. It uses only the "wordsized" engine to produce an unreduced result. This result is then used as a non-cryptographic hash (e.g., for a checksum) to ensure data integrity, though not confidentiality.
- Mode 3 (Fail-Safe): If the "wordsized" engine itself reports an error (e.g., via internal hardware checks), the entire engine is disabled, and a "zeroize" command is triggered to clear sensitive key material from memory. This prevents the leakage of corrupted or insecure cryptographic outputs.

Mermaid Diagram:

stateDiagram-v2
    [*] --> Normal
    Normal: Full Crypto (Wordsized + Reduction)
    Degraded: Integrity Checksum (Wordsized Only)
    FailSafe: Zeroize Keys

    Normal --> Degraded: Low Power Signal / Fault in Reducer
    Degraded --> Normal: Power Restored / Fault Cleared
    Normal --> FailSafe: Wordsized Unit Fault
    Degraded --> FailSafe: Wordsized Unit Fault

10. One-Time Reduction Module for Perfect Forward Secrecy

Enabling Description: In a key exchange protocol like ECDH, this method is used to enforce perfect forward secrecy at the firmware level. The "wordsized" engine is part of the standard system firmware. The "modular reduction" instructions, however, are generated dynamically for each session as part of the ephemeral key generation process. This session-specific reduction module is loaded into a secure memory enclave (e.g., Intel SGX or ARM TrustZone), used exactly once to perform the scalar multiplication for the key exchange, and then immediately purged from memory. Any attempt to re-use the reduction module or access it after the operation will result in a hardware fault. This ensures that even if the device's long-term keys are compromised, the ephemeral session key cannot be recreated, as the specific reduction code used to compute it is gone forever.

Mermaid Diagram:

sequenceDiagram
    participant Client
    participant Server

    Client->>Client: Generate Ephemeral Keypair (k_c, P_c)
    Client->>Client: Dynamically Generate Reduction Module (M_c) for curve
    Client->>Server: Send Public Key P_c
    
    Server->>Server: Generate Ephemeral Keypair (k_s, P_s)
    Server->>Server: Dynamically Generate Reduction Module (M_s) for curve
    Server->>Client: Send Public Key P_s

    Client->>Client: Load M_c into Secure Enclave
    Client->>Client: Compute Shared Secret = k_c * P_s (using Wordsized Engine + M_c)
    Client->>Client: **Purge M_c from Enclave**

    Server->>Server: Load M_s into Secure Enclave
    Server->>Server: Compute Shared Secret = k_s * P_c (using Wordsized Engine + M_s)
    Server->>Server: **Purge M_s from Enclave**

Combination Prior Art Scenarios

1. Integration with RISC-V Cryptography Extension ("Scalar" Profile)

Enabling Description: The '062 patent's method is implemented as part of the open-source RISC-V ISA. A set of custom instructions is defined. p.mul.w rD, rA, rB performs a "wordsized" polynomial multiplication on the registers rA and rB, storing the 2n-bit unreduced result in the register pair rD:rD+1. A separate configuration register, fcr (field control register), is loaded with a pointer to a memory region containing the irreducible polynomial for the specific field. A p.reduce rD instruction then executes the modular reduction of the wide register rD:rD+1 using the polynomial defined by fcr. This directly maps the two-stage process onto an open-standard CPU architecture, making the combination obvious to a person skilled in the art of processor design. The implementation can be prototyped using the open-source Spike RISC-V simulator and Rocket Chip generator.

2. Implementation as a WebAssembly (WASM) System Interface

Enabling Description: The '062 method is provided as a high-performance cryptographic backend for web applications. The "wordsized" arithmetic routines are compiled into a core crypto.wasm module. This module is sandboxed and highly optimized but field-agnostic. The Web Cryptography API in the browser is extended with a new function: crypto.subtle.defineField(name, algorithm, modulus). When a web application calls this, the browser's native C++ code JIT-compiles a highly optimized "modular reduction" function specific to that modulus. It then passes a function pointer for this JIT-compiled code into the crypto.wasm module's memory space. Subsequent calls to crypto.subtle.encrypt within the WASM module will call out to this browser-provided, field-specific reduction function pointer after performing the wordsized multiplication internally. This combines the patent's method with the open standards of WebAssembly and the Web Crypto API.

3. Integration into the Linux Kernel Crypto API

Enabling Description: The '062 patent's method is integrated into the Linux kernel's cryptographic framework. A new algorithm type, akcipher_ws (wordsized asymmetric cipher), is created. Drivers for cryptographic hardware accelerators register two separate function pointers with the kernel: one for the wordsized operation (op_wordsized) and a table of pointers for supported reductions (op_reduce_modN). When a user-space application (like OpenSSL) requests a cryptographic operation, the kernel first calls the generic op_wordsized function. It then looks up the required modulus in the hardware driver's table and calls the corresponding op_reduce_modN function on the intermediate result. This provides a standardized kernel interface that directly reflects the patent's two-stage architecture, making it an obvious software design pattern for integrating field-agile crypto accelerators into any Linux-based system.