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Introduce a Generalized Public Keys Using a Sequence of Ω- Numbers Built from Beurling's Function of Reals

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Introduce a Generalized Public Keys Using a Sequence of Ω- Numbers Built from Beurling's Function of Reals www.gsjpublications.com Journal of Journal of Global Scientific Research (ISSN: 2523-9376) Global Scientific Research 5 (2020) 596-601 www.gsjpublications.com/jgsr Introduce a Generalized Public Keys Using a Sequence of ω- Numbers Built from Beurling's Function of Reals Sarah Sh. Hasan Department of Mathematics, College of Education, University of Al-Mustansiriyah, Baghdad, Iraq. Email address: [email protected] Received: 14 Aug 2020, Revised: 19 Aug 2020, Accepted: 28 Aug 2020, Online: 30 Aug 2020 Abstract The generalized primes had been introduced during the third decade of the last century by Beurling in which for any infinite sequence of increasing positive real's start with an element sharply greater than one called generalized (or Beurling) primes. The fundamental theorem of arithmetic form the sequence of generalized (or Beurling) integers. So many authors have been working with this subject in order to having a clue of the truth of The Riemann-Hypothesis as a special case. This work gives an algorithm for creating a sequence of weird numbers from any positive real number. This used an arithmetical function 풢(푥) which consists of some filters works together for obtaining the relative weird (or primitive weird) number (or for short ω-number and ωp- number) to the real number 푥. The purpose of obtaining the weird number is to use them (as a special case) for applying them in the Beurling zeta function in order to find a better lower and upper bounds. On the other hand, one could use the sequence of that weird numbers as secret keys for any security algorithm. Keywords: Arithmetical Function, ω-Numbers, ωp-Numbers. 1. Introduction sequence of generalized (or Beurling) integers could be obtained. Therefore, these two sequences During the 3rd decade of the last century, Beurling (of Beurling primes, Beurling integers) called had introduced the generalized sequence of primes Beurling’s system. This work involves Beurling’s where every increasing (positive and infinite) manner to create a concrete filters for building a ∞ sequence of real's {푟푖}푖=1 with 푟푖 > 1 called sequence of ω-numbers as a special sequence Beurling (or generalized) primes. With the sense satisfies Beurling’s condition of generalized of the fundamental theorem of arithmetic, the primes. Hasan, S. S Journal of Global Scientific Research (ISSN: 2523-9376) Vol. 5 2020 597 The main goal of this work are, firstly use the some conditions the positive integer number 푚 mention sequence of ωp-numbers in the upper 푥 = 푐 ∏푖=1 푞푖 is ωp-number with c is a deficient bound of the county number of integers 풩푖(푥) = number such that ∇(푐) ≥ 1 and 푞1, … , 푞푚 are ∑푛∈푁 1 to see the behavior of (풩푖(푥) − 푥) how it primes. They addressed an algorithm to calculate looks like when 푥 takes ω-number values and goes all PA-numbers (see [4]). Define the function to infinity. On the other hand, the arithmetical 풢: 풮 → ℕ, where function 풢(푥) (for 푥 takes a large enough positive real values) prepares a ω-numbers and so one could 풮 = {푥| 푥휖{푥훼}훼≥1, 푤ℎ푒푟푒 1 < 푥1 ≤ 푥2 ≤ 푥3 ≤ ⋯ use them as a keys instead of primes in some and 푥훼 goes to ∞ as 훼 goes to ∞}. security algorithms which make the algorithm as The set of any real numbers belongs to a sequence strong as possible in security cases. 푥훼 satisfies the conditions of Beurling's prime Let 푛 be a natural number belongs to ℕ, and (푛) system (for more details see [5]). Here, 푥 belongs + to be the sum of divisor of 푛. We refer to the to ℝ , where 푥 is greater than a positive number 푎 abundance of 푛 by ∆(푛) which represent the and 풢(푥) is a natural number. difference between (푛) and 2푛, and to its Furthermore, we used a special case of generalized deficiency by ∇(푛) = −∆(푛) which represent the Chebyshev's 휓-function difference between 2푛 and (푛). A natural number푛 is called an abundant if ∆(푛) greater 휓픷(푥) + 1 = 푓푙표푤(푥), than 1, deficient number whenever ∇(푛)greater than 1and perfect number if ∆(푛) = ∇(푛) = 0.A to construct an arithmetic function [6]: number 푛 which is abundant and can be expressed log 푥 as a sum of some or all of its distinct proper ∞ 풢(푥) = ∑푘=1 휇(푘) (휓픷 (푒 푘 ))........................(1) divisors is called 풔풑-number (semiperfect). A ω- number (weird number) is an abundant and not 푠푝- Where the sum is finite (whenever푘 is sharply number.An abundant number 푛 is called 푷푨- log 2 −1 number (primitive abundant) if it is not a multiple greater than ( ) ). Also, 풢 it is a step function log 푥 of smaller abundant number. A ωp-number and increasing with jumps 1. Where this jump clear (primitive weird number) is a ω-number which is only when 푥 ≠ 푎푚, such that 푎, 푚 are integers and not a multiple of any smaller ω-number. greater than 1. In 1972, A ω-numbers was defined by Benkoski Note that, the function 흁(풌) is the Mӧbius [1]. Due to the rarity of these numbers, it is called function and the 풇풍풐풘(풙) is the integer part of 풙. by this name, for instance: There is only seven ω- numbers less than 10000. These numbers are This work shows the using of this step function to infinite. create an algorithm for obtaining a ω-numbers from a positive real's 푥.There are so many usage of In 1980, Pajunen [2] defined ωp-number. This type an infinite sequence of these ω-numbers. One of numbers have been studied by many authors 푟 could use these sequences which are satisfied the which is of the form 2 푞1푞2 where 푞1 and 푞2 are conditions of Beurling’s prime to apply it in the primes (for instance see [3]) and addressed the upper and lower bounds of Beurling zeta function. related properties. The other usage of that sequence is to obtain a Three years ago, a several authors have been secret key for modified algorithm, choosing 푥 to be worked on this and lately they showed that under a large real number (due to the rarity of ω- Hasan, S. S Journal of Global Scientific Research (ISSN: 2523-9376) Vol. 5 2020 598 numbers) and pass it through the filters to find a ω- cryptosystem. It presented in 1977 in the Scientific numbers from it. This work concentrated on American article written by Gardner [11]. It obtaining a sequence of ω-numbers for security invented by its inventors Rivest, Shamir and purposes. Adleman [12]. It has gained considerable importance since it has been used in Now turning our attention is to give a brief communications. It uses in the Transport Layer overview of the information security for needed Security (TLS) protocol and its predecessor, the concepts. Information security follows three basic Secure Sockets Layer (SSL) protocol, and elements, confidentiality, Integrity and availability. symmetric cryptosystem key distribution. Confidentiality prevents unauthorized people from seeing or using data. Integrity keeps data without The public key cryptosystem uses two keys, a change by unauthorized through a tracked. public key and a private key. The public key is free Availability regulars accessible to data when distributed and it is not secured. It generates by authorized users need it [7]. All these goals can sender. It uses by sender in encryption operation achieve using a cryptographic system. and produces a ciphertext. Private key uses by receiver for recovery original text from ciphertext. A cryptographic system is a process of safe data It is share by sender. Hacker is always attacked a from unwanted people for seeing or modification. public key to generate a private key. He benefits It’s a method to convert ordinary text into fromthe generation private key from a public key unintelligible and vice versa. There are three [12]. cryptographic techniques, symmetric key cryptosystem, hash function and public key The RSA has two keys, public key and private key. cryptosystem. In symmetric key cryptosystem, both The public key generates from a multiplication of sender and receiver use the same key for two large prime numbers. The private key number encrypting and decrypting a message so key must also generates from the same two prime numbers. be saved at sender and receiver. Hash function is So the problem of RSA factorizes the large one direction function and it is production fix number, whenever the large number factorizes, length hash value per plain text. It does not use a private key compromises. So that, power of key. Asymmetric key cryptosystem have two keys. encryption rises exponentially when large number The key is known called the public key and key is size is double or triple. secured called the private key. The public key uses to encrypting data and the private key for The main problem in RSA is factorize large decrypting [8]. A cryptographic system has two number. For any large enough real number there is phases encryption and decryption. Encryption is a a weird number less than it, so one can say that method to convert plaintext into ciphertext. there is a hidden large ω-numberin real number. A Decryption is a reverse of encryption. It’s recovery Large ω-number can be generated from a large original plain text. A cryptographic system enough real number using an arithmetical function includes algorithms, keys, and key management of Beurling’s primes: facilities. log 푥 ∞ 풢(푥) = ∑푘=1 휇(푘) (휓픷 (푒 푘 )).........................(1) The term public key cryptography appeared in the mid of 1970’s by Diffie and Hellman [9] and where 휓 is is the generalized (or Beurling) Merkle [10].
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