Browsing by Author "Shankar, B.R."

Now showing 1 - 14 of 14

(2,1)-Lagged fibonacci generators using elliptic curves over finite fields
(2009) Shankar, B.R.; Karuna, Kamath, K.
A novel pseudorandom sequence generator is presented in this paper. The genesis of this new generator is evolved from the concept of Lagged Fibonacci generator[1] applied to points on elliptic curves over a finite field. It is observed that the generator has a long period. Also a successful statistical testing of the randomness attributes of the given generator, in accordance with the National Institute of Standards and Technology test suite, admits to a key stream source that is in conformance with the Advanced Encryption Standard for data encryption. � 2009 IEEE.
(2,1)-Lagged fibonacci generators using elliptic curves over finite fields
(2009) Shankar, B.R.; Karuna Kamath, K.
A novel pseudorandom sequence generator is presented in this paper. The genesis of this new generator is evolved from the concept of Lagged Fibonacci generator[1] applied to points on elliptic curves over a finite field. It is observed that the generator has a long period. Also a successful statistical testing of the randomness attributes of the given generator, in accordance with the National Institute of Standards and Technology test suite, admits to a key stream source that is in conformance with the Advanced Encryption Standard for data encryption. Â© 2009 IEEE.
An Asymptotic Expansion for a Twisted Lambert Series Associated to a Cusp Form and the Möbius Function: Level Aspect
(Birkhauser, 2022) Maji, B.; Sathyanarayana, S.; Shankar, B.R.
Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert series associated with a cusp form over the full modular group and the Möbius function. In this paper, we investigate the Lambert series ∑n=1∞[af(n)ψ(n)∗μ(n)ψ′(n)]exp(-ny), where af(n) is the nth Fourier coefficient of a cusp form f over any congruence subgroup, and ψ and ψ′ are primitive Dirichlet characters. This extends the earlier work to the case of higher level subgroups and also gives a character analogue. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
Backward dynamics, Inverse limit Theory and p-adic integers
(2010) Palimar, S.; Shankar, B.R.
The concept of Backward dynamics and Inverse limit Theory is discussed. The natural arithmetic structure for these dynamics is given as ?p, the ring of p-adic integers. In particular we study the backward dynamics of the logistic map in p-adic metric.
Building Voting Systems for a Fairer Future: Exploring Blockchain based E-voting with Ethereum for National Elections
(Institute of Electrical and Electronics Engineers Inc., 2024) Prasad, S.V.; Shetty D, P.; Shankar, B.R.
The increasing adoption of Blockchain technology, spurred by the success of cryptocurrencies, has gained substantial traction across various sectors. A notable application of Blockchain technology is in electronic voting (e-voting), where decentralized nodes enhance the security and integrity of the voting process. Traditional voting methods suffer from shortcomings such as result delays, susceptibility to tampering, hijacking and destruction of voting machines. Given the scalability challenges of blockchains, a single blockchain network cannot feasibly cover all constituencies in a country. Therefore, a more effective approach is to implement multiple smaller independent blockchain networks, with each constituency having its own network and blockchain. This paper discusses the concept of one network and one blockchain for one constituency, which can be replicated for every other constituencies in the country to scale up. It explores a web3-based e-voting system utilizing private Ethereum blockchain technology, focusing on a network architecture and the design that features a DApp (Decentralized Application) application with a user-friendly interface for voting in polling booths and a governing Smart Contract. Voters can cast their votes using unique identifiers like Aadhaar or UID credentials. The outcomes of this proposed e-voting system demonstrate promising and viable performance for governmental elections. However, it is advisable to conduct trials in local elections or general body elections within institutions to validate its efficacy and reliability before wider adoption in larger democratic elections. Â© 2024 IEEE.
Mersenne primes in real quadratic fields
(2012) Palimar, S.; Shankar, B.R.
The concept of Mersenne primes is studied in real quadratic fields with class number one. Computational results are given. The field ?(?2) is studied in detail with a focus on representing Mersenne primes in the form x2 + 7y2. It is also proved that x is divisible by 8 and y ? ±3 (mod 8), generalizing a result of F. Lemmermeyer, first proved by H. W. Lenstra and P. Stevenhagen using Artin's reciprocity law.
ON THE MINIMUM CARDINALITY OF MPTQ SETS
(Colgate University, 2024) Shetty, A.; Shankar, B.R.
Given a finite set A ⊂ R, we define A + A = {a + a′ | a, a′ ∈ A} and A − A = {a − a′ | a, a′ ∈ A}. A set A is said to be an MSTD (More Sums than Differences) set if |A+A| > |A−A|. We define A.A = {aa′ | a, a′ ∈ A} and A/A = {a/a′ | a, a′ ∈ A, a′ ≠ 0}. Analogous to MSTD sets, H V Chu defines a set A ⊂ R ∖ {0} to be an MPTQ (More Products than Quotients) set if |A.A| > |A/A|. It is known by the exponentiation of MSTD sets that there exist MPTQ sets of cardinality 8. In an attempt to determine the smallest cardinality of an MPTQ set, Chu proved that an MPTQ set of real numbers must have at least 5 elements. In this work, we prove that a set of real numbers with cardinality 5 is not an MPTQ set. So we conclude an MPTQ set of real numbers must contain at least 6 elements. We have identified certain cases of sets with cardinality 6 that are not MPTQ sets. Further, we give an infinite family of MPTQ sets that are not the exponential of an MSTD set. © 2024, Colgate University. All rights reserved.
Pell surfaces and elliptic curves
(Ramanujan Mathematical Society, 2016) Manasa, K.J.; Shankar, B.R.
Let Em be the elliptic curve y2 = x3 - m, where m is a squarefree positive integer and - m = 2,3 (mod 4). Let Cl(K)[3] denote the 3-torsion subgroup of the ideal class group of the quadratic field K = Q ?-m). Let S3: y2 + mz2 = x3 be the Pell surface. We show that the collection of primitive integral points on S3 coming from the elliptic curve Em do not form a group with respect to the binary operation given by Hambleton and Lemmermeyer. We also show that there is a group homomorphism ? fromrational points of Em to Cl(K)[3] using 3-descent on Em, whose kernel contains 3Em(Q). We also explain how our homomorphism ?, the homomorphism ? of Hambleton and Lemmermeyer and the homomorphism ? of Soleng are related.
R-ORTHOGONALITY OF LATIN SQUARES USING BIVARIATE PERMUTATION POLYNOMIALS
(Jangjeon Research Institute for Mathematical Sciences and Physics, 2022) Bhatta, G.R.; Shankar, B.R.; Poojary, P.
Cryptographic applications of Latin squares require to study them in various aspects. In this paper, the formation and observation of Latin squares using bivariate permutation polynomials over some finite rings are established with respect to their properties like self orthogonalization, r-orthogonalization, and r-mirror orthogonalization. We also identified why some particular cases fail to form self orthogonal Latin squares, and we illustrate it by giving examples. © 2022 Jangjeon Research Institute for Mathematical Sciences and Physics. All rights reserved.
Sequences of numbers via permutation polynomials over some finite rings
(Universidad Catolica del Norte, 2020) Bhatta, G.R.V.; Shankar, B.R.; Mishra, V.N.; Poojary, P.
A polynomial can represent every function from a finite field to itself. The functions which are also permutations of the field give rise to permutation polynomials, which have potential applications in cryptology and coding theory. Permutation polynomials over finite rings are studied with respect to the sequences they generate. The sequences obtained through some permutation polynomials are tested for randomness by carrying out known statistical tests. Random number generation plays a major role in cryptography and hence permutation polynomials may be used as random number generators. © 2020. All rights reserved.
Shrinking generators based on ?-LFSRs
(Elsevier B.V., 2020) Bishoi, S.K.; Senapati, K.; Shankar, B.R.
The word-based LFSRs called ?-LFSRs are very attractive as they take advantage of the modern word-based processor and thus increase the throughput. Secondly, the bitstream produced by ?-LFSR has excellent statistical properties with a high period except for low linear complexity. In order to increase the linear complexity, the concept of both bit-oriented shrinking and self-shrinking generators is introduced in case of ?-LFSRs. In both the cases, the lower bound for the period as well as for the linear complexity of the bitstream are shown to be exponential. Further, we have experimented and investigated more results on the periodicity and statistical properties of the bitstream in self-shrinking ?-LFSRs. This helps to find and prove the exact period of the bitstream produced by self-shrinking generators. © 2020 Elsevier B.V.
SUM AND DIFFERENCE SETS IN GENERALIZED QUATERNION GROUPS
(Jangjeon Research Institute for Mathematical Sciences and Physics, 2024) Neetu; Shankar, B.R.
Given a group G, we say that a set [Formula presented] has more sums than differences (MSTD) if |A+A| > |A – A|, has more differences than sums (MDTS) if |A+A| < |A–A|, or is balanced if |A+A| = |A–A|. A problem of recent interest has been to understand the frequencies of these types of subsets. It is known that for arbitrary finite groups G, almost all subsets [Formula presented] are balanced sets as [Formula presented]. Recently for the generalized dihedral groups [Formula presented], it is conjectured that there are more MSTD sets than MDTS sets. In this paper, we investigate the behavior of the sum and difference sets of [Formula presented], where Q4n denotes generalized quaternion groups and show that the generalized quaternion group Q4n has at least 22n subsets which are MSTD. We also analyze the expectation for |A – A| where [Formula presented], proving an explicit formula for |A – A| when n is prime. © 2024 Jangjeon Research Institute for Mathematical Sciences and Physics. All rights reserved.
Variations of diagonal cyclicity of Latin squares formed by permutation polynomials
(Forum-Editrice Universitaria Udinese SRL, 2021) Bhatta, V.G.R.; Shankar, B.R.; Poojary, P.; Manasa, K.J.; Mishra, V.N.
Cryptographic applications of Latin squares require to study them in various aspects. The Latin squares, which are due to bivariate polynomials, show some interesting patterns of entries. In this paper, we discussed the diagonally cyclic nature of Latin squares over some small finite rings with the help of the bivariate permutation polynomials, which formed them. © 2021 Forum-Editrice Universitaria Udinese SRL. All rights reserved.
WHEN SETS CAN OR CANNOT BE PRODUCT-DOMINANT
(Jangjeon Research Institute for Mathematical Sciences and Physics, 2024) Shetty, A.; Shankar, B.R.
Given a finite set [Formula presented], we define [Formula presented] A set A is said to be sum-dominant or MSTD (More Sums than Differences) if |A + A| > |A – A| and a set [Formula presented] is said to be product-dominant or MPTQ (More Products than Quotients) if |A.A| > |A/A|. In this paper, we shall discuss several properties of MPTQ sets, investigate techniques of generating an infinite family of MPTQ sets, and identify some characterizations under which a finit e set of numbers can or cannot be product-dominant. We confirm the existence of MPTQ sets of perfect squares and justify that nth powers of prime numbers do not contain any MPTQ set. We extend the notion of MPTQ sets to the multiplicative group [Formula presented] and recognize their correspondence with the MSTD sets in [Formula presented]. © 2024 Jangjeon Research Institute for Mathematical Sciences and Physics. All rights reserved.