On Binary Representation of Integers

PRIMUS ◽  
2019 ◽  
Vol 29 (5) ◽  
pp. 474-486
Author(s):  
Vardges Melkonian
Keyword(s):  
Binary Representation ◽  
Representation Of Integers

Every recursive linear ordering has a copy in DTIME-SPACE(n,log(n))

1990 ◽  
Vol 55 (1) ◽  
pp. 260-276 ◽  
Author(s):  
Serge Grigorieff
Keyword(s):  
Complexity Theory ◽  
Order Type ◽  
Linear Ordering ◽  
Binary Representation ◽  
Relational Structures ◽  
Time Space ◽  
Representation Of Integers ◽  
Recursive Structures ◽  
Primitive Recursive ◽  
Recursive Relations

This paper is a contribution to the following natural problem in complexity theory:(*) Is there a complexity theory for isomorphism types of recursive countable relational structures? I.e. given a recursive relational structure ℛ over the set N of nonnegative integers, is there a nontrivial lower bound for the time-space complexity of recursive structures isomorphic (resp. recursively isomorphic) to ℛ?For unary recursive relations R, the answer is trivially negative: either R is finite or coinfinite or 〈N, R〉 is recursively isomorphic to 〈N, {x ϵ N: x is even}〉.The general problem for relations with arity 2 (or greater) is open.Related to this problem, a classical result (going back to S. C. Kleene [4], 1955) states that every recursive ordinal is in fact primitive recursive.In [3] Patrick Dehornoy, using methods relevant to computer science, improves this result, showing that every recursive ordinal can be represented by a recursive total ordering over N which has linear deterministic time complexity relative to the binary representation of integers. As he notices, his proof applies to every recursive total order type α such that the isomorphism type of α is not changed if points are replaced by arbitrary finite nonempty subsets of consecutive points.In this paper we extend Dehornoy's result to all recursive total orderings over N and get minimal complexity for both time and space simultaneously.

scholarly journals On Minimum Weight Binary Representation of Integers and Continued Fractions with Application to Computer Arithmetic

1981 ◽  
Vol 10 (130) ◽  
Author(s):  
David W. Matula ◽  
Peter Kornerup
Keyword(s):  
Rational Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Minimum Weight ◽  
Computer Arithmetic ◽  
Binary Representation ◽  
Representation Of Integers

We develop the concept of minimum weight binary continued fraction representation of a rational number as an extension of minimum weight binary radix representation of an integer. The relation of these representations to the attainment of optimum efficiency in the shift and add or subtract model of binary computer arithmetic is discussed.

scholarly journals Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shahram Jahani ◽  
Azman Samsudin ◽  
Kumbakonam Govindarajan Subramanian
Keyword(s):  
Number Theory ◽  
Public Key ◽  
Binary Representation ◽  
Public Key Cryptosystem ◽  
Digital Information ◽  
Public Key Cryptosystems ◽  
Multiplication Algorithm ◽  
Representation Of Integers ◽  
Integer Multiplication

Public-key cryptosystems are broadly employed to provide security for digital information. Improving the efficiency of public-key cryptosystem through speeding up calculation and using fewer resources are among the main goals of cryptography research. In this paper, we introduce new symbols extracted from binary representation of integers called Big-ones. We present a modified version of the classical multiplication and squaring algorithms based on the Big-ones to improve the efficiency of big integer multiplication and squaring in number theory based cryptosystems. Compared to the adopted classical and Karatsuba multiplication algorithms for squaring, the proposed squaring algorithm is 2 to 3.7 and 7.9 to 2.5 times faster for squaring 32-bit and 8-Kbit numbers, respectively. The proposed multiplication algorithm is also 2.3 to 3.9 and 7 to 2.4 times faster for multiplying 32-bit and 8-Kbit numbers, respectively. The number theory based cryptosystems, which are operating in the range of 1-Kbit to 4-Kbit integers, are directly benefited from the proposed method since multiplication and squaring are the main operations in most of these systems.

scholarly journals Binary Representation of Natural Numbers

2018 ◽  
Vol 26 (3) ◽  
pp. 223-229
Author(s):  
Hiroyuki Okazaki
Keyword(s):  
Binary Representation ◽  
Natural Numbers ◽  
Arithmetic Operations ◽  
Representation Of Integers ◽  
System 2 ◽  
Mizar Mathematical Library

Summary Binary representation of integers [5], [3] and arithmetic operations on them have already been introduced in Mizar Mathematical Library [8, 7, 6, 4]. However, these articles formalize the notion of integers as mapped into a certain length tuple of boolean values. In this article we formalize, by means of Mizar system [2], [1], the binary representation of natural numbers which maps ℕ into bitstreams.

scholarly journals Parity as a Property of Integers

2018 ◽  
Vol 26 (2) ◽  
pp. 91-100
Author(s):  
Rafał Ziobro
Keyword(s):  
History Of Mathematics ◽  
The Other ◽  
Binary Representation ◽  
Irrational Numbers ◽  
Decimal System ◽  
Human Eye ◽  
Common Opinion ◽  
Representation Of Integers ◽  
History Of ◽  
Definition Of

Summary Even and odd numbers appear early in history of mathematics [9], as they serve to describe the property of objects easily noticeable by human eye [7]. Although the use of parity allowed to discover irrational numbers [6], there is a common opinion that this property is “not rich enough to become the main content focus of any particular research” [9]. On the other hand, due to the use of decimal system, divisibility by 2 is often regarded as the property of the last digit of a number (similarly to divisibility by 5, but not to divisibility by any other primes), which probably restricts its use for any advanced purposes. The article aims to extend the definition of parity towards its notion in binary representation of integers, thus making an alternative to the articles grouped in [5], [4], and [3] branches, formalized in Mizar [1], [2].

scholarly journals MBR-SIFT: A mirror reflected invariant feature descriptor using a binary representation for image matching

PLoS ONE ◽  
2017 ◽  
Vol 12 (5) ◽  
pp. e0178090 ◽  
Author(s):  
Mingzhe Su ◽  
Yan Ma ◽  
Xiangfen Zhang ◽  
Yan Wang ◽  
Yuping Zhang
Keyword(s):  
Image Matching ◽  
Feature Descriptor ◽  
Binary Representation ◽  
Invariant Feature

Efficient Conversion Technique from Redundant Binary to NonRedundant Binary Representation

2017 ◽  
Vol 26 (09) ◽  
pp. 1750135 ◽  
Author(s):  
Ranjan Kumar Barik ◽  
Manoranjan Pradhan ◽  
Rutuparna Panda
Keyword(s):  
Final Stage ◽  
Hardware Description Language ◽  
Binary Representation ◽  
Description Language ◽  
Power Product ◽  
Efficient Conversion ◽  
Look Ahead ◽  
Hardware Description ◽  
Sign Extension ◽  
Cadence Tool

Redundant Binary (RB) to Two’s Complement (TC) converter offers nonredundant representation. However, the sign bit of TC representation has to be handled using nonstandard hardware blocks. The concept of Inverted encoding of negative weighted bits (IEN) eliminates the need of sign extension and offers design only using predefined hardware blocks. NonRedundant Binary (NRB) representation refers to both conventional and IEN representations. The NRB representation is also useful considering problem related to shifting in Carry Save (CS) representation of a RB number. In this paper, we have proposed two new conversion circuits for RB to NRB representation. The proposed circuits of the RB to NRB converter are coded in Verilog Hardware Description language (HDL) and synthesized using the Encounter(R) RTL Compiler RC13.10 v13.10-s006_1 of Cadence tool considering ASIC platform. Considering 64 bits’ operand, the delay power product performances of proposed one-bit and two-bit computations offer improvement of almost 29.9% and 47%, respectively as compared to Carry-Look-Ahead (CLA). The proposed one-bit converter is also applied in the final stage of the Modified Redundant Binary Adder (MRBA). The 32-bit MRBA offers a delay improvement of 7.87% replacing conventional converter with proposed one-bit converter in same FPGA 4vfx12sf363-12 device.

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