Packed Ciphertext Reshaping With Identity Elements
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Solution Overview
Problem
Homomorphic encryption operations face challenges in efficiently reshaping tile tensors due to the complexity of transpose or reshape operations, which often require n^2 rotations and masking operations, making them inefficient and cumbersome.
Innovation Solution
The system employs multiplication by an identity element to efficiently reshape data structures, allowing for efficient execution of homomorphic computations on encrypted packed plaintexts, including the use of identity matrices and permutations to achieve target shapes without decrypting the data.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If traditional transpose or reshape operations are used on tile tensors under HE, then the data structure can be transformed to different shapes, but the operation complexity increases dramatically requiring n^2 rotations and masking operations
Solution Approach 1:
The patent introduces an intermediary identity element (identity matrix or identity scalar) that mediates the reshape operation. Instead of directly performing complex n^2 rotations and masking operations, the system multiplies the tile tensor by this identity element, which naturally repositions elements into the target shape through the algebraic properties of identity elements in the underlying ring structure.
Solution Approach 2:
The patent changes the operational parameter from complex geometric transformations (rotations and masking) to simple algebraic multiplication. By changing the reshape operation from a spatial manipulation task to an algebraic operation with identity elements, the complexity is reduced from O(n^2) to a constant number of multiplications regardless of tensor size.
2Adaptability or versatility
If traditional reshape operations are used on encrypted data, then the data structure can be transformed, but computational overhead and latency increase significantly
Solution Approach 1:
The patent replaces the mechanical system of n^2 individual rotation and masking operations with a single algebraic multiplication operation. This substitution leverages the mathematical properties of identity elements to achieve the same reshaping effect in constant time, independent of the tensor dimensions, thereby dramatically reducing computational latency.
3Ease of operation
If tile tensor dimensions are transposed or reshaped under HE, then operations can be performed on encrypted data, but the operation becomes cumbersome and inefficient
Solution Approach 1:
The patent makes the identity element multiplication a universal operation that can handle any reshape or transpose operation on tile tensors. A single multiplication by an appropriately constructed identity element can achieve dimension permutation, transposition, and reshaping, making the operation both easy to perform and universally applicable to any tile tensor configuration.
Data Source
AI summary
An example system includes a processor to receive a ciphertext including an encrypted packed plaintext. The processor can, in response to detecting that a reshape is to be executed for a data structure including the packed ciphertext, identify a target shape for the data structure and execute the reshape based on the identified target shape. The reshape is executed using multiplication by an identity element. The processor can then execute a homomorphic computation using the reshaped data structure.