BITWISE HOMOMORPHIC OPERATIONS ON FLOATING POINT NUMBERS

Abstract

Homomorphic cryptography is a special kind of cryptography that allows you to perform operations on encrypted data without first decrypting it. Due these features, homomorphic cryptography can be effectively used to perform secure cloud computing. To solve various applied problems, support for all mathematical operations is required, as well as support for rational numbers in order to effectively implement the division operation and reduce the loss of accuracy during rounding of the result. Also, to improve the accuracy of calculations, it is necessary to use numbers in floating point format, but this topic has not been sufficiently researched. Support for all arithmetic and logical operations within a single homomorphic encryption scheme will allow us to perform a homomorphic implementation of almost any data processing algorithm, and the representation of numbers in floating point format will improve the accuracy of calculations and the maximum dimension of the processed data with the same amount of memory consumed, when compared with bitwise homomorphic algorithm over integers. For example, to solve SLAE by the Gaussian method, it is necessary to support the operations of difference, multiplication, division, and comparison of numbers, and it is also necessary to represent numbers in floating point format, otherwise, during the back substitution after each division operation, rounding of the result will occur, and the error will accumulate. This article discusses the possibility of performing homomorphic bitwise operations on numbers in floating point format. The most common floating-point representation format, IEEE 754, is considered. Alternative solutions for homomorphic processing of rational numbers are considered. An analysis is made of the possibility of implementing bitwise homomorphic arithmetic operations - addition, difference, multiplication and division, over homomorphically encrypted numbers in floating point format. Difficulties arising in the implementation of homomorphic arithmetic operations are analyzed, methods for solving them are considered, and the resulting algorithms for homomorphically encrypted data are presented. The analysis of the results obtained is carried out and recommendations are given regarding the choice of a method for representing homomorphically encrypted data, depending on the problem being solved.