Search

Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of 2 modulo F n , for every positive integer n not divisible by 3, where F n denotes the nth Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of a modulo F n , for every fixed integer a ≥ 3 and for all positive integers n with gcd (a , F n) = 1 . Our proof makes use of the so-called base- φ expansion of real numbers. [ABSTRACT FROM AUTHOR]

The synthesis of quantum circuits for multiplicative inverse over GF (2 8) are discussed in this paper. We first convert the multiplicative inverse operation in GF (2 8) to arithmetic operations in the composite field GF ((2 4) 2) , and then discuss the expressions of the square calculation, the inversion calculation and the multiplication calculation separately in the finite field GF (2 4) , where the expressions of multiplication calculation in GF (2 4) are given directly in GF (2 4) and given through being transformed into the composite field GF ((2 2) 2) . Then the quantum circuits of these calculations are realized one by one. Finally, two quantum circuits for multiplicative inverse over GF (2 8) are synthesized. They both use 21 qubits, the first quantum circuit uses 55 Toffoli gates and 107 CNOT gates and the second one uses 37 Toffoli gates and 209 CNOT gates. As an example of the application of multiplication inverse, we apply these quantum circuits to the implementations of the S-box quantum circuit of the AES cryptographic algorithm. Two quantum circuits for implementing the S-box of the AES cryptographic algorithm are presented. The first quantum circuit uses 21 qubits, 55 Toffoli gates, 131 CNOT gates and 4 NOT gates and the second one uses 21 qubits, 37 Toffoli gates, 233 CNOT gates and 4 NOT gates. Through the evaluation of quantum cost, the two quantum circuits of the S-box of AES cryptographic algorithm use less quantum resources than the existing schemes. [ABSTRACT FROM AUTHOR]

Abstract Background Interpreting graphs of continuous safety variables can be complicated because differences in age, gender, and testing site methodologies data may give rise to multiple reference limits. Furthermore, data below the lower limit of normal are compressed relative to those points above the upper limit of normal. The objective of this study is to develop a graphing technique that addresses these issues and is visually intuitive. Methods A mock dataset with multiple reference ranges is initially used to develop the graphing technique. Formulas are developed for conditions where data are above the upper limit of normal, normal, below the lower limit of normal, and below the lower limit of normal when the data value equals zero. After the formulae are developed, an anonymized dataset from an actual set of trials for an approved drug is evaluated comparing the technique developed in this study to standard graphical methods. Results Formulas are derived for the novel graphing method based on multiples of the normal limits. The formula for values scaled between the upper and lower limits of normal is a novel application of a readily available scaling formula. The formula for the lower limit of normal is novel and addresses the issue of this value potentially being indeterminate when the result to be scaled as a multiple is zero. Conclusions The formulae and graphing method described in this study provides a visually intuitive method to graph continuous safety data including laboratory values, vital sign data.

This article describes and analyzes all existing algorithms for computing $x=a^{-1}\pmod {p^k}$ x = a - 1 (mod p k) for a prime $p$ p , and also introduces a new algorithm based on the exact solution of linear equations using $p$ p -adic expansions. The algorithm starts with the initial value $c=a^{-1}\pmod {p}$ c = a - 1 (mod p) and iteratively computes the digits of the inverse $x=a^{-1}\pmod {p^k}$ x = a - 1 (mod p k) in base $p$ p . The mod 2 version of the algorithm is more efficient than all existing algorithms for small values of $k$ k . Moreover, it stands out as being the only one that works for any $p$ p , any $k$ k , and digit-by-digit. While the new algorithm is asymptotically worse off, it requires the minimal number of arithmetic operations (just a single addition) per step, as compared to all existing algorithms. [ABSTRACT FROM AUTHOR]

Let m and n be positive integers and let Fm and Fn be the mth and nth Fibonacci numbers. In this article, we find the Zeckendorf representation of the multiplicative inverse of Fm modulo Fn when m is small or when m is closed to n and (m, n) = 2. [ABSTRACT FROM AUTHOR]

Let m and n be positive integers and let Fm and Fn be the mth and nth Fibonacci numbers. In this article, we find the Zeckendorf representation of the multiplicative inverse of Fm modulo Fn when m is small or when m is closed to n and (m, n) = 2. [ABSTRACT FROM AUTHOR]

Hubert, Mauduit and Sárközy introduced pseudorandom measures for finite pseudorandom binary lattices. Gyarmati, Mauduit, Sárközy and Stewart presented some natural and flexible constructions, which are the two-dimensional extensions and modifications of a few one-dimensional constructions. The upper estimates for the pseudorandom measures of their binary lattices are based on the principle that character sums or exponential sums in two variables can be estimated by fixing one of the variables. In this paper, we constructed two large families of n dimensional pseudorandom binary lattices by using the multiplicative inverse modulo p , and study the properties: pseudorandom measure, collision and avalanche effect. [ABSTRACT FROM AUTHOR]

This paper describes an attack on the Rivest, Shamir and Adleman (RSA) cryptosystem utilizing the modulus N=p2q where p and q are two large balanced primes. Let e1,e2

Multiplicative inversion in finite fields is an essential operation in many cryptographic applications such as elliptic curve and pairing-based cryptography. While the classical extended Euclidean algorithm involves expensive division operations, the binary extended Euclidean and Kaliski’s algorithms use simple shift, addition and subtraction operations. The Montgomery inverse operation is applied when the Montgomery multiplication operation is used for fast arithmetic. As the inversion operation is applied to sensitive data, a constant-time inversion algorithm is useful against a class of side-channel attacks. In this paper, we show different ways of computing the Montgomery inverse of a given integer and compare their complexity in terms of the number of additions/subtractions and shift operations. We also propose a simple parallel algorithm to compute Montgomery inverse, which can be useful in multi-core processors where data sharing among cores is relatively inexpensive. Finally, we propose two efficient constant-time Montgomery inversion algorithms, which are useful as countermeasures against side-channel attacks. [ABSTRACT FROM AUTHOR]

Conventional approaches for encryption are unsuitable for simultaneously achieving reliability and confidentiality. Threshold-based secret sharing scheme deals with the problem of sharing a secret information into a group of n users and secret can be recovered only when a sufficient users come together, we can reconstruct a secret information from t , where t < n . There is a requirement to evolve a secure secret sharing scheme so that the reconstruction is possible only when all shares are available. In this paper, we propose a secure ( n , n ) -Multi Secret Sharing (MSS) scheme using Chinese Remainder Theorem (CRT). The proposed scheme shares secret information among n users and for recovery all n users are needed, if one of the user is absent then we could not reconstruct the secret information. The proposed scheme does not reveals the partial secret information, as the randomness in shares is more. The proposed scheme outperforms the existing techniques in terms of randomness and security. [ABSTRACT FROM AUTHOR]

Multiplicative inverses under an irreducible polynomial over Galois Field GF(pm) play important role in cryptography. The substitution box of Advanced Encryption Standard is designed in 1999 with multiplicative inverses under the first irreducible polynomial over GF(28). In the present paper, a recently published pseudorandom number generator GF7 designed over GF(73) in 2016, is intuitively incorporated into the Data Encryption Standard algorithm to modify it by increasing its randomness as well as security. Following the method proposed by National Institute of Standards and Technology, the statistical randomness testing is done on a large number of output ciphers obtained by the algorithm. The statistical result shows tremendous improvement in randomness for the proposed modified algorithm. [ABSTRACT FROM AUTHOR]