Boris S. Verkhovsky

Pseudo-Random Pixel Rearrangement Algorithm Based on Gaussian Integers for Image Watermarking

This paper presents a pseudo-random pixel rearrangement algorithm to improve the security of most image watermarking techniques. Many published watermarking algorithms rely on methods of rearranging pixels. They often use chaotic maps as a part of the watermarking procedure. In this paper, we propose a new method of rearranging image pixels based on the properties of Gaussian integers. It results in a more randomlooking image transformation that, in turn, significantly improves the security of the embedded watermark. The computation time is much better than the computation time of Arnold cat map chaotic transformation algorithm, used in methods previously published.

Cubic Root Extractors of Gaussian Integers and Their Application in Fast Encryption for Time-Constrained Secure Communication

There are settings where encryption must be performed by a sender under a time constraint. This paper de-scribes an encryption/decryption algorithm based on modular arithmetic of complex integers called Gaus-sians. It is shown how cubic extractors operate and how to find all cubic roots of the Gaussian. All validations (proofs) are provided in the Appendix. Detailed numeric illustrations explain how to use the method of digital isotopes to avoid ambiguity in recovery of the original plaintext by the receiver.

Cryptosystem Based on Extraction of Square Roots of Complex Integers

This paper introduces an extension of Rabin cryptosystem into the domain of complex integers. The extended cryptosystem employs a new square root algorithm for complex integers that is presented. The extended Rabin cryptosystem is efficient, provably secure and has certain advantages over the real- integer Rabin cryptosystem.

Enhanced Euclid Algorithm for Modular Multiplicative Inverse and Its Application in Cryptographic Protocols

Numerous cryptographic algorithms (ElGamal, Rabin, RSA, NTRU etc) require multiple computations of modulo multiplicative inverses. This paper describes and validates a new algorithm, called the Enhanced Euclid Algorithm, for modular multiplicative inverse (MMI). Analysis of the proposed algorithm shows that it is more efficient than the Extended Euclid algorithm (XEA). In addition, if a MMI does not exist, then it is not necessary to use the Backtracking procedure in the proposed algorithm; this case requires fewer operations on every step (divisions, multiplications, additions, assignments and push operations on stack), than the XEA. Overall, XEA uses more multiplications, additions, assignments and twice as many variables than the proposed algorithm.

Integer Factorization of Semi-Primes Based on Analysis of a Sequence of Modular Elliptic Equations *

In this paper is demonstrated a method for reduction of integer factorization problem to an analysis of a sequence of modular elliptic equations. As a result, the paper provides a non-deterministic algorithm that computes a factor of a semi-prime integer n=pq, where prime factors p and q are unknown. The proposed algorithm is based on counting points on a sequence of at least four elliptic curves y2=x(x2+b2)(modn) , where b is a control parameter. Although in the worst case, for some n the number of required values of parameter b that must be considered (the number of basic steps of the algorithm) substantially exceeds four, hundreds of computer experiments indicate that the average number of the basic steps does not exceed six. These experiments also confirm all important facts discussed in this paper.

Double-Moduli Gaussian Encryption/Decryption with Primary Residues and Secret Controls

In this paper an encryption-decryption algorithm based on two moduli is described: one in the real field of integers and another in the field of complex integers. Also the proper selection of cryptographic system parameters is described. Several numeric illustrations explain step-by-step how to precondition a plaintext, how to select secret control parameters, how to ensure feasibility of all private keys and how to avoid ambiguity in the process of information recovery. The proposed cryptographic system is faster than most of known public key cryptosystems, since it requires a small number of multiplications and additions, and does not require exponentiations for its implementation.

Space Complexity of Algorithm for Modular Multiplicative Inverse

In certain computational systems the amount of space required to execute an algorithm is even more restrictive than the corresponding time necessary for solution of a problem. In this paper an algorithm for modular multiplicative inverse is introduced and its computational space complexity is analyzed. A tight upper bound for bit storage required for execution of the algorithm is provided. It is demonstrated that for range of numbers used in public-key encryption systems, the size of bit storage does not exceed a 2K-bit threshold in the worst-case. This feature of the Enhanced-Euclid algorithm allows designing special-purpose hardware for its implementation as a subroutine in communication-secure wireless devices.

Information Protection Based on Extraction of Square Roots of Gaussian Integers

A cryptosystem based on computation of square roots of complex integers modulo composite n is described in this paper. This paper provides an algorithm extracting a square root of Gaussian integer. Various properties of square roots and a method for finding Gaussian generators are demonstrated. The generators can be instrumental in constructing other cryptosystems. It is shown how to significantly reduce average complexity of decryption per each block of ciphertext.

Analysis of RSA over Gaussian Integers Algorithm

In this paper we analyze the extended RSA algorithm into the field of Gaussian integers. We examine in depth the perceived advantages of this extension, such as security and efficiency. We found that the extended RSA is slightly less efficient and could be more secure only if RSA is not as strong as factoring (even in this case it is not guaranteed to add security).

Scheme for Secure Communication via Information Hiding Based on Key Exchange and Decomposition Protocols

This paper considers a decomposition framework as a mechanism for information hiding for secure communication via open network channels. Two varieties of this framework are provided: one is based on Gaussian arithmetic with complex modulus and another on an elliptic curve modular equation. The proposed algorithm is illustrated in a numerical example.