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• THE LIBRARY OF FULLY HOMOMORPHIC ENCRYPTION OVER THE INTEGERS

  L.K. Babenko, I.D. Rusalovsky

  2020-07-20

  Abstract ▼

  The article discusses one of the new directions of cryptography, a homomorphic cryptography.
  Its distinctive feature is that this type of cryptography allows you to process encrypted data
  without first decrypting it in such a way that the result of operations on encrypted data is equivalent
  after decryption to the result of operations on open data. The paper describes the main areas
  of application of homomorphic encryption. The analysis of existing developments in the field of
  homomorphic encryption is performed. The analysis showed that existing library implementations
  only allow processing of bits or arrays of bits and do not support the division operation. However,
  to solve applied problems, support for performing integer operations is necessary. The analysis
  revealed the need to implement the operation of homomorphic division, as well as the relevance of
  developing your own implementation of a library of homomorphic encryption over integers. The
  ability to perform four operations (addition, difference, multiplication and division) on encrypted
  data will expand the field of application of homomorphic encryption. A method of homomorphic
  division is proposed, which allows performing the division operation on homomorphically encrypted
  data. A library architecture of completely homomorphic operations on integers is proposed.
  The library supports the basic homomorphic operations on integers, as well as the division
  operation, thanks to the method of homomorphic division. Based on the proposed method of
  homomorphic division and library architecture, a library of homomorphic operations on integers
  was implemented. The article also provides measurements of the time required to perform certain
  operations on encrypted data and analyzes the effectiveness of the developed library implementation.
  Conclusions and possible ways of further development are given.

• METHOD OF IMPLEMENTING HOMOMORPHIC DIVISION

  L. K. Babenko, I. D. Rusalovsky

  2020-11-22

  Abstract ▼

  The article deals with the problems of homomorphic cryptography. Homomorphic cryptography
  is one of the young directions of cryptography. Its peculiarity lies in the fact that it is possible
  to process encrypted data without preliminary decryption in such a way that the result of operations
  on encrypted data is equivalent, after decryption, to the result of operations on open data.
  The article provides a brief overview of the areas of application of homomorphic encryption. To
  solve various applied problems, support for all mathematical operations is required, including the
  division operation, and the ability to perform this operation homomorphically will expand the
  possibilities of using homomorphic encryption. The paper proposes a method of homomorphic
  division based on an abstract representation of the ciphertext in the form of an ordinary fraction.
  The paper describes in detail the proposed method. In addition, the article contains an example of
  the practical implementation of the proposed method. It is proposed to divide the levels of data
  processing into 2 levels – cryptographic and mathematical. At the cryptographic level, a completely homomorphic encryption algorithm is used and the basic homomorphic mathematical operations
  are performed – addition, multiplication and difference. The mathematical level is a superstructure
  on top of the cryptographic level and expands its capabilities. At the mathematical level,
  the ciphertext is represented as a simple fraction and it becomes possible to perform the
  homomorphic division operation. The paper also provides a practical example of applying the
  homomorphic division method based on the Gentry algorithm for integers. Conclusions and possible
  ways of further development are given.

• DEVELOPMENT OF HOMOMORPHIC DIVISION METHODS

  I.D. Rusalovsky, L.K. Babenko, О.B. Makarevich

  2022-11-01

  Abstract ▼

  The article deals with the problems of homomorphic cryptography. Homomorphic cryptography
  is one of the young areas of cryptography. Its distinguishing feature is that it is possible to
  process encrypted data without decrypting it first, so that the result of operations on encrypted
  data is equivalent to the result of operations on open data after decryption. Homomorphic encryption
  can be effectively used to implement secure cloud computing. To solve various applied problems,
  support for all mathematical operations, including the division operation, is required, but
  this topic has not been sufficiently developed. The ability to perform the division operation
  homomorphically will expand the application possibilities of homomorphic encryption and will
  allow performing a homomorphic implementation of many algorithms. The paper considers the
  existing homomorphic algorithms and the possibility of implementing the division operation within
  the framework of these algorithms. The paper also proposes two methods of homomorphic division.
  The first method is based on the representation of ciphertexts as simple fractions and the
  expression of the division operation through the multiplication operation. As part of the second
  method, it is proposed to represent ciphertexts as an array of homomorphically encrypted bits, and
  all operations, including the division operation considered in this article, are implemented
  through binary homomorphic operations. Possible approaches to the implementation of division
  through binary operations are considered and an approach is chosen that is most suitable for a
  homomorphic implementation. The proposed methods are analyzed and their advantages and disadvantages
  are indicated.

• BITWISE HOMOMORPHIC OPERATIONS ON FLOATING POINT NUMBERS

  L.К., I.D. Rusalovsky

  2023-10-23

  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.