Multi-Layer Residue Number System for Large Dynamical Range

The patent divides the RNS representation into multiple layers, where each layer uses a separate set of moduli. The upper layer handles large integers with large moduli, while the lower layer handles smaller integers with small moduli. This segmentation allows the system to achieve a large dynamical range without requiring a single large table, as each layer can use its own optimized table size appropriate to its modulus size.

The patent transitions from a single-layer RNS to a multi-layer RNS structure, adding a dimensional aspect to the representation. Instead of using one set of moduli to achieve large range, the system stacks multiple RNS layers vertically, where each layer contributes to the overall dynamical range. This dimensional change allows the system to achieve large computational range while keeping individual table sizes small and manageable.

The patent segments the computational task across multiple layers, each handling different ranges of values with appropriately sized moduli. This segmentation allows the system to use small moduli in lower layers (requiring fewer and smaller tables) while still achieving large overall dynamical range through the combination of layers. The upper layer uses larger moduli but fewer of them, reducing the total number of tables required.

The patent changes the parameters of the RNS representation by introducing a hierarchical structure with varying modulus sizes across layers. Instead of uniformly using large moduli throughout, the system uses small moduli in lower layers and large moduli in upper layers. This parameter change optimizes the balance between computational range and the number of tables required, as small moduli require fewer table entries.

The patent segments the computation into multiple layers, each optimized for specific ranges of values. This segmentation allows parallel processing within each layer, reducing overall computation time. Each layer can independently perform computations on its residues without waiting for other layers, and the results are combined through the Chinese Remainder Theorem, achieving large dynamical range without proportional increase in computation time.

The patent performs preliminary decomposition of large integer operations into smaller operations on each layer before combining results. By pre-organizing the computation into layered residues and using pre-computed tables for each layer, the system avoids the need for complex single-step operations on very large integers, thereby reducing computation time while maintaining large dynamical range capability.