Designs, Codes and Cryptography, ISSN 0925-1022, 1/2003, Volume 28, Issue 1, pp. 33 - 44

...}}$$ mod p gives a square root of a when p ≡ 3 mod 4. Let us write p − 1 = 2 n s with s odd. A fast algorithm due to Shanks, with n steps, allows us to compute a square root of a modulo p...

Information and Communication, Circuits | finite fields | Data Structures, Cryptology and Information Theory | Convex and Discrete Geometry | Shanks algorithm | Mathematics | square root mod p | Combinatorics | Electronic and Computer Engineering | MATHEMATICS, APPLIED | COMPUTER SCIENCE, THEORY & METHODS | Algorithms

Information and Communication, Circuits | finite fields | Data Structures, Cryptology and Information Theory | Convex and Discrete Geometry | Shanks algorithm | Mathematics | square root mod p | Combinatorics | Electronic and Computer Engineering | MATHEMATICS, APPLIED | COMPUTER SCIENCE, THEORY & METHODS | Algorithms

Journal Article

International Journal of Computational and Mathematical Sciences, ISSN 2010-3905, 06/2009, Volume 3, Issue 3, pp. 101 - 109

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 03/1999, Volume 45, Issue 2, pp. 807 - 809

In this article it is shown how Gauss' (1981) famous cyclotomic sum formula can be used for extracting square-roots modulo N.

Jacobian matrices | Polynomials | Cryptography | Equations | Gaussian processes | Square-roots modulo n | Gauss sums | Factoring | factoring | square-roots modulo N | FINITE-FIELDS | COMPUTER SCIENCE, INFORMATION SYSTEMS | cryptography | ENGINEERING, ELECTRICAL & ELECTRONIC | Factor tables | Research | Square root | Information theory

Jacobian matrices | Polynomials | Cryptography | Equations | Gaussian processes | Square-roots modulo n | Gauss sums | Factoring | factoring | square-roots modulo N | FINITE-FIELDS | COMPUTER SCIENCE, INFORMATION SYSTEMS | cryptography | ENGINEERING, ELECTRICAL & ELECTRONIC | Factor tables | Research | Square root | Information theory

Journal Article

Studia Scientiarum Mathematicarum Hungarica, ISSN 0081-6906, 12/2013, Volume 50, Issue 4, p. 470

Journal Article

Journal of Discrete Mathematical Sciences and Cryptography, ISSN 0972-0529, 01/2016, Volume 19, Issue 1, pp. 93 - 101

..., to the ring of Gaussian integers ℤ[i]. Its security relies on the integer factorization problem and extraction of square roots of a Gaussian integer over ℤ n...

Zero-knowledge | Gaussian integers | Identification protocol | RSA modulo | Gaussian primes | Square root extraction

Zero-knowledge | Gaussian integers | Identification protocol | RSA modulo | Gaussian primes | Square root extraction

Journal Article

Designs, Codes and Cryptography, ISSN 0925-1022, 10/2011, Volume 61, Issue 1, pp. 41 - 69

...>}$$ are obtained, where p and q are distinct odd primes; n ≥ 1 is an integer and q has order $${\frac{\phi(2p^{n})}{2}}$$ modulo 2p n . The generator polynomials, the dimension...

Information and Communication, Circuits | Generator polynomials | 16S34 | Data Encryption | Group rings | QR codes | 12E20 | Mathematics | Data Structures, Cryptology and Information Theory | Primitive idempotents | Discrete Mathematics in Computer Science | Square root bound | 94B05 | Coding and Information Theory | Combinatorics | Cyclic codes | 20C05 | 94B65 | Dual codes

Information and Communication, Circuits | Generator polynomials | 16S34 | Data Encryption | Group rings | QR codes | 12E20 | Mathematics | Data Structures, Cryptology and Information Theory | Primitive idempotents | Discrete Mathematics in Computer Science | Square root bound | 94B05 | Coding and Information Theory | Combinatorics | Cyclic codes | 20C05 | 94B65 | Dual codes

Journal Article

DESIGNS CODES AND CRYPTOGRAPHY, ISSN 0925-1022, 10/2011, Volume 61, Issue 1, pp. 41 - 69

...(2p(n))/2 modulo 2p(n). The generator polynomials, the dimension, the minimum distance of the minimal cyclic codes of length 2p(n...

Generator polynomials | MATHEMATICS, APPLIED | Primitive idempotents | Square root bound | Group rings | QR codes | COMPUTER SCIENCE, THEORY & METHODS | Cyclic codes | Dual codes

Generator polynomials | MATHEMATICS, APPLIED | Primitive idempotents | Square root bound | Group rings | QR codes | COMPUTER SCIENCE, THEORY & METHODS | Cyclic codes | Dual codes

Journal Article

Journal of Systems Architecture, ISSN 1383-7621, 2008, Volume 54, Issue 10, pp. 957 - 966

Square root is an operation performed by the hardware in recent generations of processors. The hardware implementation of the square root operation is achieved...

CMOS | Square root | VLSI | Non-restoring algorithm | Expandability | Standard cell | CMOs | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | IMPLEMENTATIONS | ALGORITHM | Very-large-scale integration | Algorithms | Design and construction | Complementary metal oxide semiconductors

CMOS | Square root | VLSI | Non-restoring algorithm | Expandability | Standard cell | CMOs | COMPUTER SCIENCE, HARDWARE & ARCHITECTURE | IMPLEMENTATIONS | ALGORITHM | Very-large-scale integration | Algorithms | Design and construction | Complementary metal oxide semiconductors

Journal Article

Japan Academy Proceedings Series A: Mathematical Sciences, ISSN 0386-2194, 01/2012, Volume 88A, Issue 1, p. 16

... [Z.sub.p]-extension. Denote by [h.sup.-.sub.n] the relative class number of the n-th layer [k.sub.n...

Fields, Algebraic | Square root | Numbers, Prime | Research

Fields, Algebraic | Square root | Numbers, Prime | Research

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 01/2014, Volume 83, Issue 285, pp. 435 - 446

.... In the particular case m=2 O(\mathsf {M}(n)\log (p) + \mathsf {C}(n)\log (n)) \mathbb{F}_p \mathsf {M}(n) \mathsf {C}(n) \mathsf {M}(n) = O(n\log (n) \log \log (n)) \mathsf {C...

Integers | Computer science | Algorithms | Algebra | Cubes | Polynomials | Exponentiation | Logarithms | Cryptography | Factorization | Finite field arithmetic | Square roots | Root extraction | MODULAR COMPOSITION | square roots | MATHEMATICS, APPLIED | MATRIX MULTIPLICATION | finite field arithmetic | FAST POLYNOMIAL FACTORIZATION | ALGORITHM | COMPUTATION

Integers | Computer science | Algorithms | Algebra | Cubes | Polynomials | Exponentiation | Logarithms | Cryptography | Factorization | Finite field arithmetic | Square roots | Root extraction | MODULAR COMPOSITION | square roots | MATHEMATICS, APPLIED | MATRIX MULTIPLICATION | finite field arithmetic | FAST POLYNOMIAL FACTORIZATION | ALGORITHM | COMPUTATION

Journal Article

Applicable Algebra in Engineering, Communication and Computing, ISSN 0938-1279, 3/2019, Volume 30, Issue 2, pp. 135 - 145

In this paper, we present a refinement of the Cipolla–Lehmer type algorithm given by H. C. Williams in 1972, and later improved by K. S. Williams and K. Hardy...

Finite field | 68W40 | Artificial Intelligence | Primitive root | Theory of Computation | Adleman–Manders–Miller algorithm | 11T06 | 11Y16 | Computer Hardware | r -th root | Computer Science | Symbolic and Algebraic Manipulation | Cipolla–Lehmer algorithm | r-th root | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Adleman-Manders-Miller algorithm | Cipolla-Lehmer algorithm | COMPUTING SQUARE ROOTS | COMPUTER SCIENCE, THEORY & METHODS | COMPUTATION | Algebra | Algorithms

Finite field | 68W40 | Artificial Intelligence | Primitive root | Theory of Computation | Adleman–Manders–Miller algorithm | 11T06 | 11Y16 | Computer Hardware | r -th root | Computer Science | Symbolic and Algebraic Manipulation | Cipolla–Lehmer algorithm | r-th root | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | Adleman-Manders-Miller algorithm | Cipolla-Lehmer algorithm | COMPUTING SQUARE ROOTS | COMPUTER SCIENCE, THEORY & METHODS | COMPUTATION | Algebra | Algorithms

Journal Article

Results in Mathematics, ISSN 1422-6383, 3/2019, Volume 74, Issue 1, pp. 1 - 33

We investigate the relationship between various choice principles and $$n\hbox {th}$$ nth -root functions in rings...

root functions in integral domains | finite choice | 13A99 | cycle choice | Mathematics, general | 03E25 | Mathematics | bounded multiple choice | consistency results | Square root functions in rings | axiom of choice | MATHEMATICS | MATHEMATICS, APPLIED

root functions in integral domains | finite choice | 13A99 | cycle choice | Mathematics, general | 03E25 | Mathematics | bounded multiple choice | consistency results | Square root functions in rings | axiom of choice | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Designs, Codes and Cryptography, ISSN 0925-1022, 6/2015, Volume 75, Issue 3, pp. 483 - 495

In this paper, we present a new cube root algorithm in the finite field $$\mathbb {F}_{q}$$ F q with $$q$$ q a power of prime, which extends the Cipolla–Lehmer...

Information and Communication, Circuits | Finite field | 68W40 | Data Encryption | Mathematics | Cube root | Adleman–Manders–Miller algorithm | 11T06 | Data Structures, Cryptology and Information Theory | 11Y16 | Discrete Mathematics in Computer Science | Coding and Information Theory | Linear recurrence relation | Combinatorics | Cipolla–Lehmer algorithm | Tonelli–Shanks algorithm | MATHEMATICS, APPLIED | Tonelli-Shanks algorithm | Adleman-Manders-Miller algorithm | Cipolla-Lehmer algorithm | COMPUTING SQUARE ROOTS | COMPUTER SCIENCE, THEORY & METHODS | COMPUTATION | Algorithms

Information and Communication, Circuits | Finite field | 68W40 | Data Encryption | Mathematics | Cube root | Adleman–Manders–Miller algorithm | 11T06 | Data Structures, Cryptology and Information Theory | 11Y16 | Discrete Mathematics in Computer Science | Coding and Information Theory | Linear recurrence relation | Combinatorics | Cipolla–Lehmer algorithm | Tonelli–Shanks algorithm | MATHEMATICS, APPLIED | Tonelli-Shanks algorithm | Adleman-Manders-Miller algorithm | Cipolla-Lehmer algorithm | COMPUTING SQUARE ROOTS | COMPUTER SCIENCE, THEORY & METHODS | COMPUTATION | Algorithms

Journal Article

SIAM Journal on Computing, ISSN 0097-5397, 2008, Volume 38, Issue 5, pp. 1987 - 2006

...: Given a division-free straight-line program producing an integer N, decide whether N > 0. In the Blum-Shub-Smale model, polynomial-time computation over the reals...

Straight-line programs | Blum-Shub-Smale model | BPP | Counting hierarchy | Sum of square roots problem | Arithmetic circuits | MATHEMATICS, APPLIED | NUMBER | arithmetic circuits | INFORMATION | MACHINES | counting hierarchy | straight-line programs | THRESHOLD CIRCUITS | COMPUTER SCIENCE, THEORY & METHODS | sum of square roots problem | PROBABILISTIC ALGORITHMS | COMPUTATION

Straight-line programs | Blum-Shub-Smale model | BPP | Counting hierarchy | Sum of square roots problem | Arithmetic circuits | MATHEMATICS, APPLIED | NUMBER | arithmetic circuits | INFORMATION | MACHINES | counting hierarchy | straight-line programs | THRESHOLD CIRCUITS | COMPUTER SCIENCE, THEORY & METHODS | sum of square roots problem | PROBABILISTIC ALGORITHMS | COMPUTATION

Journal Article

Compositio mathematica, ISSN 0010-437X, 07/2013, Volume 149, Issue 7, pp. 1175 - 1202

Let $N/ F$ be an odd-degree Galois extension of number fields with Galois group $G$ and rings of integers...

Galois module | Galois Gauss sum | Dwork's power series | Self-dual integral normal basis | Square root of the inverse different | Weakly ramified extensions | weakly ramified extensions | MATHEMATICS | self-dual integral normal basis | EXTENSIONS | square root of the inverse different | Mathematics | Integers | Construction | Integrals | Modules | Inverse | Decomposition | Power series | Rings (mathematics) | Mathematics - Number Theory

Galois module | Galois Gauss sum | Dwork's power series | Self-dual integral normal basis | Square root of the inverse different | Weakly ramified extensions | weakly ramified extensions | MATHEMATICS | self-dual integral normal basis | EXTENSIONS | square root of the inverse different | Mathematics | Integers | Construction | Integrals | Modules | Inverse | Decomposition | Power series | Rings (mathematics) | Mathematics - Number Theory

Journal Article

ACM Transactions on Computation Theory (TOCT), ISSN 1942-3454, 11/2012, Volume 4, Issue 4, pp. 1 - 15

The sum of square roots over integers problem is the task of deciding the sign of a nonzero sum, S = ∑ i=1 n Δ i · √ a i , where Δ i ∈ {+1, −1...

polynomials | Sum of square roots | arithmetic circuits | Polynomials | Arithmetic circuits | Computational geometry

polynomials | Sum of square roots | arithmetic circuits | Polynomials | Arithmetic circuits | Computational geometry

Journal Article

SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, 01/2001, Volume 22, Issue 4, pp. 1112 - 1125

The standard inverse scaling and squaring algorithm for computing the matrix logarithm begins by transforming the matrix to Schur triangular form in order to...

Denman-Beavers iteration | Inverse scaling and squaring method | Matrix logarithm | Matrix square root | Padé approximation | matrix logarithm | matrix square root | MATHEMATICS, APPLIED | inverse scaling and squaring method | Pade approximation | SQUARE-ROOT | SIGN FUNCTION

Denman-Beavers iteration | Inverse scaling and squaring method | Matrix logarithm | Matrix square root | Padé approximation | matrix logarithm | matrix square root | MATHEMATICS, APPLIED | inverse scaling and squaring method | Pade approximation | SQUARE-ROOT | SIGN FUNCTION

Journal Article

The European Physical Journal C, ISSN 1434-6044, 08/2005, Volume 42, Issue 3, pp. 303 - 308

... elastic scattering with applications to the T evatron, RHIC and LHC G. Pancheri 1,a , Y. Srivastava 2,3,b , N. Staﬀolani 2,c 1 Laboratori Nazionali di INFN, Frascati...

Elementary Particles and Nuclei | Particle Acceleration and Detection, Beam Physics | Nuclear Physics, Heavy Ions, Hadrons | Physics, general | Nuclear Fusion | Physics | Elementary Particles, Quantum Field Theory | QCD | ANGLE ANTIPROTON-PROTON | BEHAVIOR | TOTAL CROSS-SECTION | PBARP | SQUARE-ROOT-S | PHYSICS, PARTICLES & FIELDS | Scattering amplitude | Absorption | Sum rules | Cross sections | Elastic scattering

Elementary Particles and Nuclei | Particle Acceleration and Detection, Beam Physics | Nuclear Physics, Heavy Ions, Hadrons | Physics, general | Nuclear Fusion | Physics | Elementary Particles, Quantum Field Theory | QCD | ANGLE ANTIPROTON-PROTON | BEHAVIOR | TOTAL CROSS-SECTION | PBARP | SQUARE-ROOT-S | PHYSICS, PARTICLES & FIELDS | Scattering amplitude | Absorption | Sum rules | Cross sections | Elastic scattering

Journal Article

The European Physical Journal H, ISSN 2102-6459, 5/2018, Volume 43, Issue 2, pp. 119 - 183

A revolution in elementary particle physics occurred during the period from the ICHEP1968 to the ICHEP1982 with the advent of the parton model from discoveries...

History and Philosophical Foundations of Physics | History of Science | Quantum Physics | Physics, general | Measurement Science and Instrumentation | Physics | PROTON-PROTON COLLISIONS | DIRECT-PHOTON PRODUCTION | PB-PB COLLISIONS | PHYSICS, MULTIDISCIPLINARY | PT DIRECT PHOTON | QUARK-GLUON PLASMA | PI-O PRODUCTION | HISTORY & PHILOSOPHY OF SCIENCE | NUCLEUS-NUCLEUS COLLISIONS | LARGE-TRANSVERSE-MOMENTUM | INELASTIC ELECTRON-PROTON | SQUARE-ROOT-S | Quarks | Particle accelerators | Collisions (Nuclear physics) | Specific gravity | Analysis | Nuclear physics | NUCLEAR PHYSICS AND RADIATION PHYSICS

History and Philosophical Foundations of Physics | History of Science | Quantum Physics | Physics, general | Measurement Science and Instrumentation | Physics | PROTON-PROTON COLLISIONS | DIRECT-PHOTON PRODUCTION | PB-PB COLLISIONS | PHYSICS, MULTIDISCIPLINARY | PT DIRECT PHOTON | QUARK-GLUON PLASMA | PI-O PRODUCTION | HISTORY & PHILOSOPHY OF SCIENCE | NUCLEUS-NUCLEUS COLLISIONS | LARGE-TRANSVERSE-MOMENTUM | INELASTIC ELECTRON-PROTON | SQUARE-ROOT-S | Quarks | Particle accelerators | Collisions (Nuclear physics) | Specific gravity | Analysis | Nuclear physics | NUCLEAR PHYSICS AND RADIATION PHYSICS

Journal Article

Designs, Codes and Cryptography, ISSN 0925-1022, 3/2015, Volume 74, Issue 3, pp. 559 - 569

We show how to perform basic operations (arithmetic, square roots, computing isomorphisms) over finite fields of the form $$\mathbb F _{q^{2^k}}$$ F q 2 k in...

Information and Communication, Circuits | Finite field | 68W30 | Square root | Data Encryption | Mathematics | Complexity | Data Structures, Cryptology and Information Theory | 11Y16 | Discrete Mathematics in Computer Science | 12Y05 | Coding and Information Theory | Combinatorics | Algebraic closure

Information and Communication, Circuits | Finite field | 68W30 | Square root | Data Encryption | Mathematics | Complexity | Data Structures, Cryptology and Information Theory | 11Y16 | Discrete Mathematics in Computer Science | 12Y05 | Coding and Information Theory | Combinatorics | Algebraic closure

Journal Article

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