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\newenvironment{specHead}[2]% {\vspace{20pt}\hrule\vspace{10pt}% \phantomsection\label{#1}\markright{#2}% \pdfbookmark[2]{#2}{#1}% \hspace{-0.75in}{\bfseries\fontsize{16pt}{18pt}\selectfont#2}% }{} \def\TheFullDate{2021-07-15 (revised: 15 July 2021)} \def\TheID{\makeatother } \def\TheDate{2021-07-15} \title{Optimal Asymmetric Data Encryption Algorithm} \author{}\makeatletter \makeatletter \newcommand*{\cleartoleftpage}{% \clearpage \if@twoside \ifodd\c@page \hbox{}\newpage \if@twocolumn \hbox{}\newpage \fi \fi \fi } \makeatother \makeatletter \thispagestyle{empty} \markright{\@title}\markboth{\@title}{\@author} \renewcommand\small{\@setfontsize\small{9pt}{11pt}\abovedisplayskip 8.5\p@ plus3\p@ minus4\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus2\p@ \belowdisplayshortskip 4\p@ plus2\p@ minus2\p@ \def\@listi{\leftmargin\leftmargini \topsep 2\p@ plus1\p@ minus1\p@ \parsep 2\p@ plus\p@ minus\p@ \itemsep 1pt} } \makeatother 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}{% \vskip8pt \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt } \usepackage[absolute,overlay]{textpos} \makeatother \usepackage{lineno} \linenumbers \begin{document} \author[1]{Kuryazov D.M.} \affil[1]{ National University of Uzbekistan, Uzbekistan} \renewcommand\Authands{ and } \date{\small \em Received: 12 June 2021 Accepted: 4 July 2021 Published: 15 July 2021} \maketitle \begin{abstract} Today, public-key cryptosystems are particularly vulnerable to fetching cipher text and adaptively matched plaintext attacks. To prevent such attacks, in practice, optimal asymmetric algorithms are used, for example, RSA-OAEP and etc. In this article, using the method of encoding messages by points of an elliptic curve, an optimal asymmetric algorithm is proposed for data encryption which is based on elliptic curves. \end{abstract} \keywords{asymmetric algorithms, elliptical curves, encoding and decoding} \begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) \textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark} \end{textblock*} \begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) \textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}} \end{textblock*} \let\tabcellsep& \section[{Introduction}]{Introduction}\par o date, the durability of modern asymmetric algorithms (data encryption and digital signature) is characterized by their properties to withstand all kinds of attacks and the laboriousness of the best known hacking algorithm {\ref [1]}\hyperref[b1]{[2]}\hyperref[b2]{[3]} {\ref [4]} {\ref [5]} {\ref [6]} {\ref [7]} {\ref [8]} {\ref [9]}.\par The standards of asymmetric data encryption algorithms used in practice are based on the problems of factorizing a composite number and discrete logarithm in a finite group of large prime order.\par The main problems in this class of cryptographic transformations are the low speed of such transformations, a significant increase in the size of the cryptogram compared to the size of the original message, and also the decreasing strength due to the development of mathematical methods and cryptanalysis tools.\par In recent years, elliptic cryptography has been intensively developed, discovered independently by N. {\ref Koblitz and V. Miller in 1985}, in which the role of a onesided function is played by scalar multiplication of a point by a constant, implemented on the basis of operations of addition and doubling of points of elliptic curves (EC) in finite fields of various characteristics \hyperref[b8]{[14]}\hyperref[b9]{[15]}.\par In \hyperref[b4]{[11]}, a status of the directional encryption was considered, possibilities of implementing directional encryption in groups of points on the EC were substantiated, in \hyperref[b5]{[12]}, a method of commutative encryption was proposed using computations on the EC, which ensures the exponential strength of the commutative encryption algorithm and its performance increase compared to other algorithms \hyperref[b7]{[13]}.\par For cryptosystems (symmetric and asymmetric), there exist Chosen-plaintext attack (CPA), Chosen-cipher text attack (CCA), and adaptive chosen plaintext attack (CCA-2). The CPA and CCA attacks were originally intended for active cryptanalysis of secret key cryptosystems.\par The purpose of this cryptanalysis is to break the cryptosystem using open and encrypted messages received during the attack \hyperref[b12]{[18]}\hyperref[b13]{[19]}\hyperref[b14]{[20]}. They were then adapted for cryptanalysis of public key cryptosystems.\par The purpose of this work is to propose an optimal asymmetric data encryption algorithm for EC using the method of encoding messages with EC points.\par In the EC encryption algorithm considered below, -bit data block of the message m is encoded by the EC point M, which is then transformed with a secret key. As a result, the cryptogram represents some point C.\par The decryption procedure involves performing inverse transformations over point C, after which point M is restored and decryption is performed, leading to the receipt of message m. \section[{II.}]{II.} \section[{Mainpart}]{Mainpart}\par Let a prime number be given p>3.Then an elliptic curve E defined over a finite prime field Fp is the set of pairs of numbers (x, y), x, y?F p , satisfying the identityy 2 ? x 3 + ax + b (mod ?) , (1)\par where a, b? F p and 4a 3 + 27b 2 is not comparable to zero mod p.\par Analysis shows that public key cryptosystems are especially vulnerable to CCA andCCA-2 \hyperref[b11]{[17]}. Therefore, to prevent such attacks, in practice, optimal asymmetric algorithms are used, for example RSA-OAEP \hyperref[b10]{[16]} and etc.\par .\par An invariant of an elliptic curve is a magnitude J (E) that satisfies the identity) (mod 27 4 4 1728 ) ( 2 3 3 p b a a E J + = ,\textbf{(2)}\par The coefficients a, b of the elliptic curve E, according to the known invariant J (E) are determined as follows? ? ? ? ? ), (mod 2 ) (mod 3 p k b p k a (3) where, p), ( -J(E) J(E) k mod 1728 = J(E) ? 0 or 1728.\par Pairs (x, y) that satisfy identity (1) are called points of the elliptic curve E; x and yare the x-and ycoordinates of the point, respectively.\par The points of the elliptic curve will be denoted by G (x, y) or G. Two points of an elliptic curve are equal if their corresponding x-and y-coordinates are equal.\par On the set of all points of the elliptic curve E we introduce the addition operation, which we will denote by the "+" sign. For two arbitrary points G 1 (x 1 , y 1 )and G 2 (x 2 , y 2 )of the elliptic curve E, we consider several options.\par Let the coordinates of the points G 1 (x 1 , y 1 )and G 2 (x 2 , y 2 ) satisfy the condition x 1 ? x 2 . In this case, their sum will be called the point G 3 (x 3 , y 3 ), the coordinates of which are determined by t he following formula? ? ? ? ? ? ? ? ? ? ? ), (mod ) (\par ), (mod1 3 1 3 2 1 2 3 p y x x y p x x x ? ? (4)\par where ,\par ). (mod1 2 1 2 p x x y y ? ? ? ? If the equalities holdx 1 =x 2 andy 1 = y 2 ? 0 ,then we define the coordinates of the point G 3 , as follows ? ? ? ? ? ? ? ? ? ? ), (mod ) ( ), (mod 2 1 3 1 3 1 2 3 p y x x y p x x ? ? (5) Where, ). (mod 2 3 1 2 1 p y a x + ? ?\par In the case when the conditionx 1 =x 2 andy 1 =-y 2 (mod p) is satisfied sum of the points G 1 and G 2 will be called the zero point 0, without determining its x-and ycoordinates. In this case, the point G 2 is called the negation of the point G 1 . For the zero point 0, the equalities holds.G"+"0=0"+"G=G, (\textbf{6})\par Where G is an arbitrary point of the elliptic curve E.\par On the set of all points of the elliptic curve E, we introduce the subtraction operation which we denote by the sign "-". By the properties of points on elliptic curves, for an arbitrary point G (x, y) of an elliptic curve, the following equality holds:-G(x, y)=G(x, -y) , (\textbf{7}) G 1 (x 1 , y 1 ) -G 2 (x 2 , y 2 )=G 1 (x 1 , y 1 ) +G 2 (x 2 , -y 2 ),\textbf{(8)}\par i.e. a subtraction operation can be converted to an addition operation. With respect to the introduced operation of addition, the set of all points of the elliptic curve E, together with the zero point form a finite abelian (commutative) group of order w, for which the inequality \hyperref[b1]{[2]} holds.p p w p p 2 1 2 1 + + ? ? ? + ,\textbf{(9)}\par A point T is called a point of multiplicity k, or simply a multiple point of an elliptic curve E, if for some point N the equalityN k N N T k ] [ " "..." " = + + = ? ?? ? ?? ? ,\textbf{(10)} \section[{III. Asymmetric Encryption Algorithm Parameters}]{III. Asymmetric Encryption Algorithm Parameters}\par The parameters of the asymmetric data encryption algorithm are:\par 1. Prime number p is the modulus of an elliptic curve satisfying the inequality ?>2 255 . The upper bound of this number should be determined with a specific implementation of the asymmetric algorithm; 2. Elliptic curve E defined by its invariant J (E) or coefficientsa, b?F ? ; 3. Integer w is the order of group points of the elliptic curve E 4. Prime number n is the order of the cyclic subgroup of group points of the elliptic curve E, for which the following conditions are satisfied: The above parameters of the asymmetric encryption algorithm are subject to the following requirements:? ? ? < < ? ? =\par 1. The condition ? i ? 1(mod n)must be fulfilled,for all integersi=1, 2?, B , where ? satisfies the inequality B ? 31; 2. The inequality must be satisfied w ? ?. \section[{Each user of the asymmetric encryption algorithm must have private keys:}]{Each user of the asymmetric encryption algorithm must have private keys:}\par 1. The private key of the asymmetric algorithm d is an integer satisfying the inequality 0