Reliable multiprecision arithmetic for number theory

Reliable multiprecision arithmetic for number theory

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Personal Sign In. For IEEE to continue sending you helpful information on our products and services, please consent to our updated Privacy Policy. Email Address. Sign In. Access provided by: anon Sign Out. The Rise of Multiprecision Arithmetic Abstract: There is a growing demand for and availability of multiprecision arithmetic: floating point arithmetic supporting multiple, possibly arbitrary, precisions.

For an increasing body of applications, including in supernova simulations, electromagnetic scattering theory, and computational number theory, double precision arithmetic is insufficient to provide results of the required accuracy. On the other hand, for climate modelling and deep learning half precision about four significant decimal digits has been shown to be sufficient in some studies.

reliable multiprecision arithmetic for number theory

We discuss a number of topics involving multiprecision arithmetic, including: The need for, availability of, and ways to exploit, higher precision arithmetic e. How to derive linear algebra algorithms that will run in any precision, as opposed to be being optimized as some key algorithms are for double precision. For solving linear systems with the use of iterative refinement, the benefits of suitably combining three different precisions of arithmetic say, half, single, and double.

How a new form of preconditioned iterative refinement can be used to solve very ill conditioned sparse linear systems to high accuracy. Article :. DOI: Need Help?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.

reliable multiprecision arithmetic for number theory

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.

The proposed multiplication algorithm is also 2. 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. The growth of digital technologies has an exponential trend and as a consequence the need of information security also increases even more than before [ 12 ]. Cryptography is an essential tool in providing a reasonable solution for this necessity.

The modern field of cryptography consists of two main areas, the symmetric-key cryptography and the public-key cryptography. The same key is used in symmetric-key cryptosystems to encrypt and decrypt a message, while the public-key cryptosystems use two keys in their protocols. Most of the public-key cryptosystems [ 3 ] use modular exponentiation in their calculation. For example, Diffie and Hellman introduced the first key exchange scheme in that is based on the modular exponentiation [ 4 ].

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Few years later inone of the most used public-key cryptosystems, RSA [ 3 ], is also based on the modular exponentiation. ElGamal key exchange [ 5 ] is another example of public key that has been developed based on the modular exponentiation. Modular exponentiation,is a one-way function because the inverse of a modular exponentiation is a known hard problem [ 6 — 8 ]. Calculating modular exponentiation for a large exponent and large modulo is a costly operation and therefore improving its efficiency has become an important research issue for researchers in cryptography and mathematics.

There are two main approaches currently being employed in order to improve the efficiency of modular exponentiation: improving the involved operations, exponentiation, and division, separately, and improving both of the operations simultaneously. Residue number system RNS [ 11 ] and Montgomery modular multiplication [ 12 ] are examples of the first approach, while binary and m -ary exponentiation or Barrett reduction [ 13 ] are instances from the second approach.

This paper focuses on the second approach, by proposing a new number representation, which will improve the squaring and multiplication operations, two of the three main operations in calculating modular exponentiation [ 14 ]. The naive approach of calculating the exponentiation is by doing repetitive multiplication, which is not an efficient way for calculating large exponent.

A better alternative for calculating exponentiation is by employing binary exponentiation; that is, ifwherethen. The term can be obtained by squaring the th term. On the other hand, the binary method requires only squarings and multiplications on averagewhere see Algorithm 1. Consequently, improving the multiplication and squaring operations as found in algorithm such as the right-to-left algorithm and its variants [ 6 — 8 ] will inherently improve the efficiency of the exponentiation calculation [ 7 ].

Input : positive integers and. In spite of all the differences in these methods, which sometimes make them apparently unrelated to each other, these methods have been founded based on the same idea, that is, how to represent a polynomial to behave efficiently in calculations. The classical method uses coefficient representation, while the other three methods use point-value representation.

This representation conversion enables us to reduce the cost of convolution from of classical method to a lower cost for point-to-point multiplication. However, because of the preprocessing overheads such as the divide and conquer, evaluation, and interpolation, the operating cost of these algorithms is actually much higher, making them useful only when the integers are extremely large. Consequently, only classical and Karatsuba multiplication algorithms and their combination are being used in current cryptosystem.

This is especially true after considering circumstances such as memory constraints and the practical finite field size. In positional numeral system [ 15 ], the natural way of multiplying numbers, known as classical multiplication algorithm, is by multiplying each digit of the multiplicand by each digit of the multiplier and then adding up all the properly shifted results.

This method requires a multiplication table for single digits available to the algorithm.My research is centered on computer algebra and rigorous numerical computing.

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I'm especially interested in fast and reliable algorithms for arbitrary-precision arithmetic, complex analysis, and numerical evaluation of special functions including hypergeometric functions, L-functions, and modular forms. Interested in doing a master's thesis or PhD in this area? Feel free to contact me. No guarantees that funding will be available. A big part of my work is to develop free scientific software. This list is available in BibTeX format txt file.

I also have a Google Scholar profile. I've taken part in Google Summer of Code once as a student and three times as a mentor:. In summer andI worked on Sage and mpmath as a contractor for the American Institute of Mathematicsthanks to funding provided by William Stein.

Fredrik Johansson Contact: fredrik. Mathematical software A big part of my work is to develop free scientific software.

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The Mathematical Functions Grimoire Fungrim is a website and software library of symbolic formulas and tables for mathematical functions. Arb is a C library for arbitrary-precision ball arithmetic, supporting a wide range of numerical operations with automatic, rigorous error bounds.

It is also my main research project since Arb is designed with computer algebra and computational number theory in mind, extending the principles behind FLINT to the domain of real and complex numbers. It is used in a growing number of projects and has been cited in about 50 papers. My record computation of the partition function with Arb requiring billion-digit arithmetic! It is easy to use and provides a range of advanced numerical functionality.

I started mpmath in and remain the main author and maintainer. A small but useful part of the scientific Python stack, mpmath is a dependency of several other packages including SymPy and Sage and has been mentioned in over papers. See one-liners for pi for a demo.Besides the features listed explicitly, each module implementing a specific type also provides methods for arithmetic operations add, sub, mul and so onconversions between types, random generation, and printing.

Modular Arithmetic

Some utility modules are omitted from the list. The Antic library adds support for algebraic number theory. Antic includes the following modules. The Arb library adds support for fast arbitrary-precision real and complex numbers with rigorous error control using ball interval arithmetic. It supports polynomials, power series, matrices, and evaluation of special functions.

Arb includes the following modules. Contact: William Hartflint-devel mailing list. Mathematical constants pi, e, gamma, Catalan, Elementary functions sqrt, pow, exp, log, sin, Rapid computation of special trigonometric values Special functions gamma, digamma, log gamma, rising factorials, Riemann zeta function fmpcb - complex numbers Complex floating-point arithmetic based on the fmprb type All operations support arbitrary precision with automatic, rigorous error bounds Interval predicates Elementary functions sqrt, pow, exp, log, sin, In particular, it allows working with recursive dense multivariate polynomials and matrices over generic rings.Introducing new learning courses and educational videos from Apress.

Start watching. It also gives both a technical overview and an implementation of the Rijndael algorithm that was selected as the Advanced Encryption Standard by the U. Author Michael Welschenbach avoids complexities by explaining cryptography and its mathematical basis in terms a programmer can easily understand.

This book offers a comprehensive yet relentlessly practical overview of the fundamentals of modern cryptography. Readers will learn, step by step, how to implement a platform-independent library for the all-important multiprecision arithmetic used in modern cryptography.

This is followed by an implementation of the cryptographic algorithms themselves.

Cryptography in C and C++

Skip to main content Skip to table of contents. Advertisement Hide. Front Matter Pages i-xix. Front Matter Pages Pages Interface Semantics. The Fundamental Operations. Modular Arithmetic: Calculating with Residue Classes.

Bitwise and Logical Functions. Input, Output, Assignment, Conversion. Dynamic Registers. Basic Number-Theoretic Functions. Large Random Numbers.

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Error Handling. Approaches for Further Extensions.In computer sciencearbitrary-precision arithmeticalso called bignum arithmeticmultiple-precision arithmetic, or sometimes infinite-precision arithmeticindicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit ALU hardware, which typically offers between 8 and 64 bits of precision.

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Several modern programming languages have built-in support for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than store values as a fixed number of binary bits related to the size of the processor registerthese implementations typically use variable-length arrays of digits.

Arbitrary precision is used in applications where the speed of arithmetic is not a limiting factor, or where precise results with very large numbers are required. A common application is public-key cryptographywhose algorithms commonly employ arithmetic with integers having hundreds of digits. Another example is in rendering fractal images with an extremely high magnification, such as those found in the Mandelbrot set.

Arbitrary-precision arithmetic can also be used to avoid overflowwhich is an inherent limitation of fixed-precision arithmetic. Similar to a 5-digit odometer 's display which changes from toa fixed-precision integer may exhibit wraparound if numbers grow too large to represent at the fixed level of precision. Some processors can instead deal with overflow by saturationwhich means that if a result would be unrepresentable, it is replaced with the nearest representable value.

With bit unsigned saturation, adding any positive amount to would yield Some processors can generate an exception if an arithmetic result exceeds the available precision. Where necessary, the exception can be caught and recovered from—for instance, the operation could be restarted in software using arbitrary-precision arithmetic. In many cases, the task or the programmer can guarantee that the integer values in a specific application will not grow large enough to cause an overflow.

Such guarantees may be based on pragmatic limits: a school attendance program may have a task limit of 4, students. A programmer may design the computation so that intermediate results stay within specified precision boundaries. Some programming languages such as LispPythonPerlHaskell and Ruby use, or have an option to use, arbitrary-precision numbers for all integer arithmetic. Although this reduces performance, it eliminates the possibility of incorrect results or exceptions due to simple overflow.

It also makes it possible to guarantee that arithmetic results will be the same on all machines, regardless of any particular machine's word size. The exclusive use of arbitrary-precision numbers in a programming language also simplifies the language, because a number is a number and there is no need for multiple types to represent different levels of precision.Help with Search.

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Science Math Number Theory Software Pari-GP 8. Elliptic Curves and Modular Forms 5. Primality Proving 6.

reliable multiprecision arithmetic for number theory

Computers Algorithms Computational Algebra. Science Math Algebra Software. Science Math Software. It has several builtin functions for algorithmic number theory. A calculator program for doing arbitrary precision integer arithmetic, written in ANSI C and Yacc, with a number of built-in functions for number theory, by Keith Matthews.

Elliptic curve factoring software. Encyclopedia of Integer Sequences Seeker. EIS-Seeker is a libguile-based scheme interpreter devoted to the study of the Encyclopedia of Integer Sequences database.

It is intended to help find new relations between sequences.

MPIR: Multiple Precision Integers and Rationals

Supports arbitrary precision integer arithmetic calculations with interactive execution: it is a superior version of the basic bc program which comes with the UNIX operating system. Keith Matthews has written several number-theoretic programs. Implementations of Extended LLL.

By Wilberd van der Kallen. An application for numerically and graphically analyzing dynamical systems. KANT is a software package for sophisticated computations in number fields and in global function fields. Kaprekar Series Generator. A java application for determining the Kaprekar Constants and Series for numbers in various radixes. Library for large-integer arithmetic, plus some number theory modules. A large, well-supported software package for computationally hard problems in algebra, number theory, geometry and combinatorics.

Maple Worksheets by John Cosgrave. A selection of Maple V. Math Archives Software. Multiprecision Software Directory.