In block ciphers,the nonlinear components,also known as sub-stitution boxes(S-boxes),are used with the purpose of inducing confusion in cryptosystems.For the last decade,most of the work on designing S-boxes over the ...In block ciphers,the nonlinear components,also known as sub-stitution boxes(S-boxes),are used with the purpose of inducing confusion in cryptosystems.For the last decade,most of the work on designing S-boxes over the points of elliptic curves has been published.The main purpose of these studies is to hide data and improve the security levels of crypto algorithms.In this work,we design pair of nonlinear components of a block cipher over the residue class of Gaussian integers(GI).The fascinating features of this structure provide S-boxes pair at a time by fixing three parameters.But the prime field dependent on the Elliptic curve(EC)provides one S-box at a time by fixing three parameters a,b,and p.The newly designed pair of S-boxes are assessed by various tests like nonlinearity,bit independence criterion,strict avalanche criterion,linear approximation probability,and differential approximation probability.展开更多
In block ciphers,the nonlinear components,also known as substitution boxes(S-boxes),are used with the purpose to induce confusion in cryptosystems.For the last decade,most of the work on designing S-boxes over the poi...In block ciphers,the nonlinear components,also known as substitution boxes(S-boxes),are used with the purpose to induce confusion in cryptosystems.For the last decade,most of the work on designing S-boxes over the points of elliptic curves,chaotic maps,and Gaussian integers has been published.The main purpose of these studies is to hide data and improve the security levels of crypto algorithms.In this work,we design pair of nonlinear components of a block cipher over the residue class of Eisenstein integers(EI).The fascinating features of this structure provide S-boxes pair at a time by fixing three parameters.However,in the same way,by taking three fixed parameters only one S-box is obtained through a prime field-dependent Elliptic curve(EC),chaotic maps,and Gaussian integers.The newly designed pair of S-boxes are assessed by various tests like nonlinearity,bit independence criterion,strict avalanche criterion,linear approximation probability,and differential approximation probability.展开更多
In this paper, the automorphism group is completely determined, of the unitary Cayley graph, the unit graph and the total graph, over the ring of Gaussian integers modulo a prime power.
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 ...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.展开更多
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...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.展开更多
Let m be a positive integer, g（m） be the number of integers t for which 1 ≤ t ≤ m and there does not exist a positive integer n satisfying （ t = t（n） ） t^n＋1≡t（modm）.For a number x≥3, let G（x）=∑m≤tg（...Let m be a positive integer, g（m） be the number of integers t for which 1 ≤ t ≤ m and there does not exist a positive integer n satisfying （ t = t（n） ） t^n＋1≡t（modm）.For a number x≥3, let G（x）=∑m≤tg（m） In this paper, we obtain the asymptotic formula： .G（x）=αx^2＋O（xlogx）,ax x→∞ Our result improves the corresponding result with an error term O（xlog^2 x） of Yang Zhaohua obtained in 1986展开更多
The author recently published a paper which claimed that an ordinal interpretation of numbers had limited applicability for cryptography. A further examination of this subject, in particular to what extent an ordinal ...The author recently published a paper which claimed that an ordinal interpretation of numbers had limited applicability for cryptography. A further examination of this subject, in particular to what extent an ordinal interpretation is useful for recurrence sequences, is needed. Hilbert favored an interpretation of the natural numbers that placed their ordinal properties prior to their cardinal properties [1] [2]. The author examines ordinal uses of the integers in number theory in order to discuss the possibilities and limitations of this approach. The author hopes this paper will be useful in clarifying or even correcting some matters that were discussed in his paper of January of 2018. I was trained informally in philosophical realism, and while I think idealism too has a place, at this time in my life I believe that the weight of evidence and usefulness is more on the side of philosophical materialism. I hope this discussion will help supplement for my readers the material in Number in Mathematical Cryptography. I still maintain that a lack of clarity on these matters has hindered progress in cryptography;and it has taken time for me to better understand these things. I hope others who have interest and ability will assist in making these matters clearer. My intention was to work in pure mathematics, and the transition to an applied mindset was difficult for me. As a result, I feel most comfortable in a more middle-of-the road attitude, but have had to slowly move to a more precise analysis of the physical quantities involved. I hope my readers will be patient with my terminology, which is still evolving, and my discussion of things which are more indirectly related, and which are necessary for my expression. These are important things for the mathematical community to understand, and I hope smarter and more knowledgeable people will address my errors, and improve upon the things I might have correct. I am discussing sequences which are sometimes a use of both ordinal and cardinal numbers.展开更多
The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamilt...The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.展开更多
For multi-dimensional integer-valued sequence, a new proof of the upper-bound and lower-bound estimations on the product of all its components is given in this paper. Those estimations are very important to characteri...For multi-dimensional integer-valued sequence, a new proof of the upper-bound and lower-bound estimations on the product of all its components is given in this paper. Those estimations are very important to characterize the Kondratiev space of random test function, in which space it is convenient to study random distribution,random partial differential equation and other problems.展开更多
WHEN am I ever going to use this?GEOGRAPHY Badwater, in Death Valley, California, is the lowest point in North America, while Mt. Mc Knlei y in Alaska, is the highest point. The graph shows their elevations and extrem...WHEN am I ever going to use this?GEOGRAPHY Badwater, in Death Valley, California, is the lowest point in North America, while Mt. Mc Knlei y in Alaska, is the highest point. The graph shows their elevations and extreme temperatures.1. What does an elevation of-86 meters represent?2. What does a temperature of-35°represent?With sea level as the starting point of 0, you can express 86 meters below sea level as 0-86, or-86. A negative number is a number less than 0.Negative numbers like-86, positive numbers like+125, and zero are members of the set of integers. Integers can be represented as points on a number line.展开更多
Let L/F be a finite Galois extension of number fields of degree n and let p be a prime which does not divide n.We shall study the p^(j)-rank of K_(2i)(O_(L))via its Galois module structure following the approaches of ...Let L/F be a finite Galois extension of number fields of degree n and let p be a prime which does not divide n.We shall study the p^(j)-rank of K_(2i)(O_(L))via its Galois module structure following the approaches of Iwasawa and Komatsu–Nakano.Along the way,we generalize previous observations of Browkin,Wu and Zhou on K_(2)-groups to higher even K-groups.We also give examples to illustrate our results.Finally,we apply our discussion to refine a result of Kitajima pertaining to the p-rank of even K-groups in the cyclotomic Z_(l)-extension,where l≠p.展开更多
This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specific...This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.展开更多
In this paper,we determine the Sombor index and some other vertex-degree-based topological indices for the nil-clean graphs of Z_(n),the ring of integers modulo n.
Integer factorization (IFP), also called prime factorization, is an important problem in number theory, cryptography, and quantum computation. Factoring large integers to attack the RSA cryptosystem is intractable for...Integer factorization (IFP), also called prime factorization, is an important problem in number theory, cryptography, and quantum computation. Factoring large integers to attack the RSA cryptosystem is intractable for powerful supercomputers, let alone classical computers. In 1994, Shor [1]presented an algorithm that potentially enabled a quantum computer to find prime factors in polynomial time.展开更多
Let P（x） denote the greatest prime factor of ∏<sub>x【n≤x+x<sup>1/2</sup></sub>n. In this paper, we shall prove that P（x）】x<sup>0.728</sup>holds true for sufficiently large x.
Let △（x）be the error term in the asymptotic formula for the counting function of square-full integers.In the present paper it is proved that △（x）=O(x<sup>27/4+ε</sup>),which improves on the exponent...Let △（x）be the error term in the asymptotic formula for the counting function of square-full integers.In the present paper it is proved that △（x）=O(x<sup>27/4+ε</sup>),which improves on the exponent 33/5 obtained by X.D.CAO.展开更多
Let k≥2 is a fixed integer. A natural number n is called k-full if p^k|n whenever p is a prime factor of n. Let A_k(x) denote the number of k-full integers not exceeding x. A. Ivi(?) proved on the Lindelof hypothesis
This letter is a continuation of previous work(Scientia Sinica, 27(1984), 1: 16—23). By using number-theoretic and model-theoretic arguments, we establish the existence of Goldbach and nonGoldbach extension rings of ...This letter is a continuation of previous work(Scientia Sinica, 27(1984), 1: 16—23). By using number-theoretic and model-theoretic arguments, we establish the existence of Goldbach and nonGoldbach extension rings of certain cyclotomic rings of integers and of some related rings of integers. The results obtained are as follows (The following Lemma is cited from the author’s previous work to help the reader to comprehend Theorem 1).展开更多
Let L（x） denote the number of square-full integers not exceeding x. It is proved in [1] thatL（x）～（ζ（3/2）/ζ（3））x<sup>1/2</sup>+（ζ（2/3）/ζ（2））x<sup>1/3</sup> as x→∞,where ζ...Let L（x） denote the number of square-full integers not exceeding x. It is proved in [1] thatL（x）～（ζ（3/2）/ζ（3））x<sup>1/2</sup>+（ζ（2/3）/ζ（2））x<sup>1/3</sup> as x→∞,where ζ（s） denotes the Riemann zeta function. Let △（x） denote the error function in the asymptotic formula for L（x）. It was shown by D. Suryanaryana<sup>[2]</sup> on the Riemann hypothesis （RH） that1/x integral from n=1 to x |△（t）|dt=O(x<sup>1/10+s</sup>)for every ε】0. In this paper the author proves the following asymptotic formula for the mean-value of △（x） under the assumption of R. H.integral from n=1 to T (△<sup>2</sup>(t/t<sup>6/5</sup>)) dt～c log T,where c】0 is a constant.展开更多
文摘In block ciphers,the nonlinear components,also known as sub-stitution boxes(S-boxes),are used with the purpose of inducing confusion in cryptosystems.For the last decade,most of the work on designing S-boxes over the points of elliptic curves has been published.The main purpose of these studies is to hide data and improve the security levels of crypto algorithms.In this work,we design pair of nonlinear components of a block cipher over the residue class of Gaussian integers(GI).The fascinating features of this structure provide S-boxes pair at a time by fixing three parameters.But the prime field dependent on the Elliptic curve(EC)provides one S-box at a time by fixing three parameters a,b,and p.The newly designed pair of S-boxes are assessed by various tests like nonlinearity,bit independence criterion,strict avalanche criterion,linear approximation probability,and differential approximation probability.
基金extend their appreciation to the Deanship of Scientific Research at King Khalid University,for funding this work through the General Research Groups Program under Grant No.R.G.P.2/109/43.
文摘In block ciphers,the nonlinear components,also known as substitution boxes(S-boxes),are used with the purpose to induce confusion in cryptosystems.For the last decade,most of the work on designing S-boxes over the points of elliptic curves,chaotic maps,and Gaussian integers has been published.The main purpose of these studies is to hide data and improve the security levels of crypto algorithms.In this work,we design pair of nonlinear components of a block cipher over the residue class of Eisenstein integers(EI).The fascinating features of this structure provide S-boxes pair at a time by fixing three parameters.However,in the same way,by taking three fixed parameters only one S-box is obtained through a prime field-dependent Elliptic curve(EC),chaotic maps,and Gaussian integers.The newly designed pair of S-boxes are assessed by various tests like nonlinearity,bit independence criterion,strict avalanche criterion,linear approximation probability,and differential approximation probability.
基金Supported by the National Natural Science Foundation of China(Grant No.11771176)。
文摘In this paper, the automorphism group is completely determined, of the unitary Cayley graph, the unit graph and the total graph, over the ring of Gaussian integers modulo a prime power.
文摘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.
文摘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.
文摘Let m be a positive integer, g（m） be the number of integers t for which 1 ≤ t ≤ m and there does not exist a positive integer n satisfying （ t = t（n） ） t^n＋1≡t（modm）.For a number x≥3, let G（x）=∑m≤tg（m） In this paper, we obtain the asymptotic formula： .G（x）=αx^2＋O（xlogx）,ax x→∞ Our result improves the corresponding result with an error term O（xlog^2 x） of Yang Zhaohua obtained in 1986
文摘The author recently published a paper which claimed that an ordinal interpretation of numbers had limited applicability for cryptography. A further examination of this subject, in particular to what extent an ordinal interpretation is useful for recurrence sequences, is needed. Hilbert favored an interpretation of the natural numbers that placed their ordinal properties prior to their cardinal properties [1] [2]. The author examines ordinal uses of the integers in number theory in order to discuss the possibilities and limitations of this approach. The author hopes this paper will be useful in clarifying or even correcting some matters that were discussed in his paper of January of 2018. I was trained informally in philosophical realism, and while I think idealism too has a place, at this time in my life I believe that the weight of evidence and usefulness is more on the side of philosophical materialism. I hope this discussion will help supplement for my readers the material in Number in Mathematical Cryptography. I still maintain that a lack of clarity on these matters has hindered progress in cryptography;and it has taken time for me to better understand these things. I hope others who have interest and ability will assist in making these matters clearer. My intention was to work in pure mathematics, and the transition to an applied mindset was difficult for me. As a result, I feel most comfortable in a more middle-of-the road attitude, but have had to slowly move to a more precise analysis of the physical quantities involved. I hope my readers will be patient with my terminology, which is still evolving, and my discussion of things which are more indirectly related, and which are necessary for my expression. These are important things for the mathematical community to understand, and I hope smarter and more knowledgeable people will address my errors, and improve upon the things I might have correct. I am discussing sequences which are sometimes a use of both ordinal and cardinal numbers.
文摘The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.
基金Supported by Nature Science Foundation of Hainan Province (Grant No. 118MS040,2018CXTD338)NSF of Higher Education Institutions of Hainan Province (Grant No. Hnky2018ZD-6)National Natural Science Foundation of China (Grant No. 11861029)。
文摘For multi-dimensional integer-valued sequence, a new proof of the upper-bound and lower-bound estimations on the product of all its components is given in this paper. Those estimations are very important to characterize the Kondratiev space of random test function, in which space it is convenient to study random distribution,random partial differential equation and other problems.
文摘WHEN am I ever going to use this?GEOGRAPHY Badwater, in Death Valley, California, is the lowest point in North America, while Mt. Mc Knlei y in Alaska, is the highest point. The graph shows their elevations and extreme temperatures.1. What does an elevation of-86 meters represent?2. What does a temperature of-35°represent?With sea level as the starting point of 0, you can express 86 meters below sea level as 0-86, or-86. A negative number is a number less than 0.Negative numbers like-86, positive numbers like+125, and zero are members of the set of integers. Integers can be represented as points on a number line.
基金Supported by National Natural Science Foundation of China(Grant No.11771164)the Fundamental Research Funds for the Central Universities of CCNU(Grant No.CCNU20TD002)。
文摘Let L/F be a finite Galois extension of number fields of degree n and let p be a prime which does not divide n.We shall study the p^(j)-rank of K_(2i)(O_(L))via its Galois module structure following the approaches of Iwasawa and Komatsu–Nakano.Along the way,we generalize previous observations of Browkin,Wu and Zhou on K_(2)-groups to higher even K-groups.We also give examples to illustrate our results.Finally,we apply our discussion to refine a result of Kitajima pertaining to the p-rank of even K-groups in the cyclotomic Z_(l)-extension,where l≠p.
文摘This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.
基金supported by the National Natural Science Foundation of China(Nos.12261001,12461001)the Research Foundation Ability Enhancement Project for Young and Middle aged Teachers in Guangxi Universities(No.025KY0477)。
文摘In this paper,we determine the Sombor index and some other vertex-degree-based topological indices for the nil-clean graphs of Z_(n),the ring of integers modulo n.
文摘Integer factorization (IFP), also called prime factorization, is an important problem in number theory, cryptography, and quantum computation. Factoring large integers to attack the RSA cryptosystem is intractable for powerful supercomputers, let alone classical computers. In 1994, Shor [1]presented an algorithm that potentially enabled a quantum computer to find prime factors in polynomial time.
基金Project supported by the Tian Yuan Item in the National Natural Science Foundation of China.
文摘Let P（x） denote the greatest prime factor of ∏<sub>x【n≤x+x<sup>1/2</sup></sub>n. In this paper, we shall prove that P（x）】x<sup>0.728</sup>holds true for sufficiently large x.
基金Project supported by the National Natural Science Foundation of China
文摘Let △（x）be the error term in the asymptotic formula for the counting function of square-full integers.In the present paper it is proved that △（x）=O(x<sup>27/4+ε</sup>),which improves on the exponent 33/5 obtained by X.D.CAO.
文摘Let k≥2 is a fixed integer. A natural number n is called k-full if p^k|n whenever p is a prime factor of n. Let A_k(x) denote the number of k-full integers not exceeding x. A. Ivi(?) proved on the Lindelof hypothesis
文摘This letter is a continuation of previous work(Scientia Sinica, 27(1984), 1: 16—23). By using number-theoretic and model-theoretic arguments, we establish the existence of Goldbach and nonGoldbach extension rings of certain cyclotomic rings of integers and of some related rings of integers. The results obtained are as follows (The following Lemma is cited from the author’s previous work to help the reader to comprehend Theorem 1).
文摘Let L（x） denote the number of square-full integers not exceeding x. It is proved in [1] thatL（x）～（ζ（3/2）/ζ（3））x<sup>1/2</sup>+（ζ（2/3）/ζ（2））x<sup>1/3</sup> as x→∞,where ζ（s） denotes the Riemann zeta function. Let △（x） denote the error function in the asymptotic formula for L（x）. It was shown by D. Suryanaryana<sup>[2]</sup> on the Riemann hypothesis （RH） that1/x integral from n=1 to x |△（t）|dt=O(x<sup>1/10+s</sup>)for every ε】0. In this paper the author proves the following asymptotic formula for the mean-value of △（x） under the assumption of R. H.integral from n=1 to T (△<sup>2</sup>(t/t<sup>6/5</sup>)) dt～c log T,where c】0 is a constant.
