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In computational number theory, Cipolla's algorithm is a technique for solving a congruence of the form x 2 ≡ n ( mod p ) , {\displaystyle x^{2}\equiv
Cipolla's_algorithm
Topics referred to by the same term
SQ2), a main-belt asteroid Cipolla (surname) Cipolla di Giarratana, a variety of onion Cipolla's algorithm Search for "cipolla" on Wikipedia. All pages
Cipolla
finding algorithm Cipolla's algorithm Tonelli–Shanks algorithm Multiplication algorithms: fast multiplication of two numbers Karatsuba algorithm Schönhage–Strassen
List_of_algorithms
Algorithm for computing greatest common divisors
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers
Euclidean_algorithm
Italian mathematician (1880–1947)
developed (among other things) a theory for sequences of sets and Cipolla's algorithm for finding square roots modulo a prime number. He also solved the
Michele_Cipolla
Algorithm for integer multiplication
The Karatsuba algorithm is a fast multiplication algorithm for integers. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. It is a
Karatsuba_algorithm
Algorithm used in modular arithmetic
The Tonelli–Shanks algorithm (referred to by Shanks as the RESSOL algorithm) is used in modular arithmetic to solve for r in a congruence of the form r2
Tonelli–Shanks_algorithm
Quantum algorithm for integer factorization
Shor's algorithm is a quantum algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor
Shor's_algorithm
Method for division with remainder
A division algorithm is an algorithm which, given two integers N and D (respectively the numerator and the denominator), computes their quotient and/or
Division_algorithm
Multiplication algorithm
The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers, published by Arnold Schönhage and Volker Strassen
Schönhage–Strassen_algorithm
Algorithm in computational number theory
Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Decomposition of a number into a product
efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty
Integer_factorization
Algorithm to multiply two numbers
multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient
Multiplication_algorithm
Algorithm for computing the greatest common divisor
The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor
Binary_GCD_algorithm
Algorithm for computing logarithms
theory, the Pohlig–Hellman algorithm, sometimes credited as the Silver–Pohlig–Hellman algorithm, is a special-purpose algorithm for computing discrete logarithms
Pohlig–Hellman_algorithm
Method for computing the relation of two integers with their greatest common divisor
and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common
Extended_Euclidean_algorithm
Algorithm in computational number theory
kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced
Pollard's_kangaroo_algorithm
Integer factorization algorithm
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and
Pollard's_rho_algorithm
Special-purpose algorithm for factoring integers
Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning
Pollard's_p_−_1_algorithm
Mathematical algorithm
Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, analogous to Pollard's
Pollard's rho algorithm for logarithms
Pollard's_rho_algorithm_for_logarithms
Probabilistic algorithm for computing discrete logarithms
In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms. Dedicated to the discrete
Index_calculus_algorithm
Mathematical procedure
a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}=0.\,} An integer relation algorithm is an algorithm for finding integer relations. Specifically, given a set of real
Integer_relation_algorithm
Method in number theory
In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials
Berlekamp–Rabin_algorithm
Integer factorization algorithm
theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms. It was invented by
Williams's_p_+_1_algorithm
Fast greatest common divisor algorithm
GCD algorithm, named after D. H. Lehmer, is a fast GCD algorithm for multiple-precision arithmetic, which improves on the simpler Euclidean algorithm by
Lehmer's_GCD_algorithm
Efficient algorithm to count points on elliptic curves
Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields. The algorithm has applications in elliptic curve cryptography
Schoof's_algorithm
British computer vision researcher
Exposure used his algorithms to convert a small scale solid form into a geometrical system suitable for large scale fabrication. Cipolla was elected a Fellow
Roberto_Cipolla
Pocklington's algorithm is a technique for solving a congruence of the form x 2 ≡ a ( mod p ) , {\displaystyle x^{2}\equiv a{\pmod {p}},} where x and
Pocklington's_algorithm
Standard division algorithm for multi-digit numbers
In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit numbers that is simple enough to perform by hand. It breaks
Long_division
Number-theoretic algorithm
In computational number theory, Cornacchia's algorithm is an algorithm for solving the Diophantine equation x 2 + d y 2 = m {\displaystyle x^{2}+dy^{2}=m}
Cornacchia's_algorithm
Probabilistic primality test
composite return probably prime Using fast algorithms for modular exponentiation, the running time of this algorithm is O(k·log3 n), where k is the number
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Ancient algorithm for generating prime numbers
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking
Sieve_of_Eratosthenes
Algorithm in number theory
(also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method
Dixon's_factorization_method
Multiplication algorithm
ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as long multiplication after the multiplier and multiplicand
Ancient Egyptian multiplication
Ancient_Egyptian_multiplication
Probabilistic primality test
or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar
Miller–Rabin_primality_test
Algorithm for multiplying large numbers
introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers
Toom–Cook_multiplication
Algorithm for determining whether a number is prime
Adleman–Pomerance–Rumely primality test is an algorithm for determining whether a number is prime. Unlike other, more efficient algorithms for this purpose, it avoids the
Adleman–Pomerance–Rumely primality test
Adleman–Pomerance–Rumely_primality_test
Algorithm checking for prime numbers
test and the cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena
AKS_primality_test
Factorization algorithm
the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity
General_number_field_sieve
Integer factorization algorithm
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field
Quadratic_sieve
Algorithm for integer factorization
elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose
Lenstra elliptic-curve factorization
Lenstra_elliptic-curve_factorization
Largest integer that divides given integers
|a|. This case is important as the terminating step of the Euclidean algorithm. The above definition is unsuitable for defining gcd(0, 0), since there
Greatest_common_divisor
Special-purpose integer factorization algorithm
number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special
Special_number_field_sieve
Greatest integer less than or equal to square root
y {\displaystyle y} and k {\displaystyle k} be non-negative integers. Algorithms that compute (the decimal representation of) y {\displaystyle {\sqrt {y}}}
Integer_square_root
Algorithm for generating prime numbers
In mathematics, the sieve of Pritchard is an algorithm for finding all prime numbers up to a specified bound. Like the ancient sieve of Eratosthenes,
Sieve_of_Pritchard
Algorithm for solving the discrete logarithm problem
branch of mathematics, the baby-step giant-step is a meet-in-the-middle algorithm for computing the discrete logarithm or order of an element in a finite
Baby-step_giant-step
Integer factorization algorithm
x-y} will give a non-trivial factor of N {\displaystyle N} . A practical algorithm for finding pairs ( x , y ) {\displaystyle (x,y)} which satisfy x 2 ≡
Shanks's square forms factorization
Shanks's_square_forms_factorization
Algorithms to generate prime numbers
In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently. These are used in various applications
Generation_of_primes
Korkine–Zolotarev (KZ) lattice basis reduction algorithm or Hermite–Korkine–Zolotarev (HKZ) algorithm is a lattice reduction algorithm. For lattices in R n {\displaystyle
Korkine–Zolotarev lattice basis reduction algorithm
Korkine–Zolotarev_lattice_basis_reduction_algorithm
Exponentation in modular arithmetic
multiplicative inverse d of b modulo m (for instance by using extended Euclidean algorithm). More precisely: c = be mod m = d−e mod m, where e < 0 and b ⋅ d ≡ 1
Modular_exponentiation
Algorithm for generating prime numbers
In mathematics, the sieve of Atkin is a modern algorithm for finding all prime numbers up to a specified integer. Compared with the ancient sieve of Eratosthenes
Sieve_of_Atkin
Algorithm for generating prime numbers
Sundaram is a variant of the sieve of Eratosthenes, a simple deterministic algorithm for finding all the prime numbers up to a specified integer. It was discovered
Sieve_of_Sundaram
Integer factorization algorithm
most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer n
Trial_division
Problem of inverting exponentiation in groups
Index calculus algorithm Number field sieve Pohlig–Hellman algorithm Pollard's rho algorithm for logarithms Pollard's kangaroo algorithm (aka Pollard's
Discrete_logarithm
System of rapid mental calculation
This article presents some methods devised by Trachtenberg. Some of the algorithms Trachtenberg developed are for general multiplication, division and addition
Trachtenberg_system
factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning that it is suitable for factoring any integer
Continued fraction factorization
Continued_fraction_factorization
Cyclic algorithm to solve indeterminate quadratic equations
The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly
Chakravala_method
Integer factorization algorithm
In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field
Rational_sieve
converse is not necessarily true. Grantham's stated goal when developing the algorithm was to provide a test that primes would always pass and composites would
Quadratic_Frobenius_test
Approach to dimensionality reduction
Laboratories. The MPCA algorithm written in Matlab (MPCA+LDA included). The UMPCA algorithm written in Matlab (data included). The UMLDA algorithm written in Matlab
Multilinear_subspace_learning
Probabilistic primality test
no value. Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log2n log log
Fermat_primality_test
Methods to test or prove primality
Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin in the same year. The algorithm was altered and improved by several collaborators
Elliptic_curve_primality
Creation of a 3D model from a set of images
extrinsic parameters, without which, at some level, no arrangement of algorithms will work. The dotted line between Calibration and Depth determination
3D reconstruction from multiple images
3D_reconstruction_from_multiple_images
Algorithm for checking if a number is prime
exponentiation algorithm like binary or addition-chain exponentiation). The algorithm can be written in pseudocode as follows: algorithm lucas_primality_test
Lucas_primality_test
Primality test for certain numbers
based on the Lucas–Lehmer primality test. It is the fastest deterministic algorithm known for numbers of that form.[citation needed] For numbers of the form
Lucas–Lehmer–Riesel_test
Study of algorithms for performing number theoretic computations
mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating
Computational_number_theory
Test if a Mersenne number is prime
odd prime. The primality of p can be efficiently checked with a simple algorithm like trial division since p is exponentially smaller than Mp. Define a
Lucas–Lehmer_primality_test
Probabilistic primality testing algorithm
primality test is a probabilistic or possibly deterministic primality testing algorithm that determines whether a number is composite or is a probable prime.
Baillie–PSW_primality_test
Mathematical lemma
Extended Euclidean Lehmer's Modular square root Cipolla Pocklington's Tonelli–Shanks Berlekamp Other algorithms Chakravala Cornacchia Exponentiation by squaring
Bhaskara's_lemma
Topic in computer science and language technology
subdiscipline of computer vision,[citation needed] it employs mathematical algorithms to interpret gestures. Gesture recognition offers a path for computers
Gesture_recognition
Algorithm to solve the discrete logarithm problem
In mathematics, the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has
Function_field_sieve
Process of steering a ship from a starting point to a destination
Wallenstein. Weintrit, Adam; Neumann, Tomasz (7 June 2011). Methods and Algorithms in Navigation: Marine Navigation and Safety of Sea Transportation. CRC
Marine_navigation
Integer that is a perfect square modulo some integer
known formula. Tonelli (in 1891) and Cipolla found efficient algorithms that work for all prime moduli. Both algorithms require finding a quadratic nonresidue
Quadratic_residue
Factorization method based on the difference of two squares
of the quadratic sieve and general number field sieve, the best-known algorithms for factoring large semiprimes, which are the "worst-case". The primary
Fermat's_factorization_method
Algorithm for generating numbers coprime with first few primes
list of initial prime numbers constitute complete parameters for the algorithm to generate the remainder of the list. These generators are referred to
Wheel_factorization
Primality test for numbers of a certain form
in contrast to the probably prime results typical of other Monte Carlo algorithms such as the Miller-Rabin test. An approximate upper bound error probability
Proth's_theorem
(1998). "The FERET database and evaluation procedure for face-recognition algorithms". Image and Vision Computing. 16 (5): 295–306. doi:10.1016/s0262-8856(97)00070-x
List of datasets in computer vision and image processing
List_of_datasets_in_computer_vision_and_image_processing
Mathematical for factoring integers
made Euler's factorization method disfavoured for computer factoring algorithms, since any user attempting to factor a random integer is unlikely to know
Euler's_factorization_method
Composite number that passes Fermat's probable primality test
example, public-key cryptography algorithms such as RSA require the ability to quickly find large primes. The usual algorithm to generate prime numbers is
Fermat_pseudoprime
Primality test for Fermat numbers
F_{n}} by repeated squaring. This makes the test a fast polynomial-time algorithm. However, Fermat numbers grow so rapidly that only a handful of Fermat
Pépin's_test
Archaeological theory
Evolution: Calculation and Contingency, Bruce G. Trigger, 1998, p. 101 Carlo M. Cipolla, Before the Industrial revolution: European Society and Economy 1000–1700
Cultural_diffusion
Positive integer of the form (2^(2^n))+1
F_{n}} by repeated squaring. This makes the test a fast polynomial-time algorithm. But Fermat numbers grow so rapidly that only a handful of them can be
Fermat_number
Number-theoretic algorithm
Extended Euclidean Lehmer's Modular square root Cipolla Pocklington's Tonelli–Shanks Berlekamp Other algorithms Chakravala Cornacchia Exponentiation by squaring
Pocklington_primality_test
Field of study in computer vision
In computer vision, articulated body pose estimation is the task of algorithmically determining the pose of a body composed of connected parts (joints
Articulated body pose estimation
Articulated_body_pose_estimation
2024 film score by Zilgi
Zilgi is appropriately dread-inducing." Regarding the sound design, Matt Cipolla of The Film Stage stated "Eugenio Battaglia’s sound design lends as much
Longlegs_(soundtrack)
Use of oxygen as a medical treatment
Island (FL): StatPearls Publishing. PMID 28613494. Retrieved 2021-11-12. Cipolla MJ (2009). Control of Cerebral Blood Flow. Morgan & Claypool Life Sciences
Oxygen_therapy
University of Technology (Finland) who develops mathematical methods and algorithms for asteroid shape and spin modeling. JPL · 11815 11816 Vasile 1981 EX32
Meanings of minor-planet names: 11001–12000
Meanings_of_minor-planet_names:_11001–12000
Approach to optimizing robustness to failure
Optimization and Its Applications, Kluwer. B. Rustem and M. Howe, 2002, Algorithms for Worst-case Design and Applications to Risk Management, Princeton University
Info-gap_decision_theory
Central concept in Marxian critique of political economy
to 25% with AI", Crombie website [68]; Cem Dilmegani, "Dynamic Pricing Algorithms: Top 3 Models." AIMultiple website, 12 Aug 2025 [69]; "Personalized Pricing:
Value-form
Interdisciplinary field
such as a seed region or rough outline of the region to segment. An algorithm can then iteratively refine such a segmentation, with or without guidance
Medical_image_computing
Appeal), presented "citizen" lists, with candidates selected through an algorithm aimed at ensuring gender parity and regional representation, following
Constituent Assembly of Valais
Constituent_Assembly_of_Valais
Field of design concerned with the influence of design on behavior
Sustainable Use: Changing consumer behaviour through product design. In: Cipolla, C., Peruccio, P. (eds.) Changing the Change. Torino, Italy: Allemandi
Behavioural_design
Multidisciplinary art studio in Modena, Italy
ongoing project exploring botanical illustrations through machine learning algorithms: premiered at Cosmo Caixa in Barcelona (ES), it has later been adapted
Fuse*
to a new generation of ultra-fast Wi-Fi. 11 August – A deep learning algorithm is reported to be capable of visually identifying thousands of plant species
2017_in_science
CIPOLLAS ALGORITHM
CIPOLLAS ALGORITHM
CIPOLLAS ALGORITHM
CIPOLLAS ALGORITHM
Female
Celtic
, soft.
Girl/Female
Tamil
Usha Lakshi | உஷாலாகà¯à®·à¯€
Morning, Dawn
Boy/Male
Indian
Divine of Power
Boy/Male
Tamil
Lord of sweetness
Girl/Female
Indian, Kannada
Glorious; Goddess Parvathi
Boy/Male
American, British, English, Norse
From the Warrior's Settlement
Boy/Male
Dutch, German, Hebrew
Gift from God; God is Gracious
Surname or Lastname
English and German (also Gümbel)
English and German (also Gümbel) : from the Germanic personal name Gumbald, composed of the elements gund ‘battle’ + bald ‘bold’, ‘brave’; it was taken to Britain from France by the Normans.
Boy/Male
Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Ocean
Boy/Male
British, English
Ash-tree Meadow
CIPOLLAS ALGORITHM
CIPOLLAS ALGORITHM
CIPOLLAS ALGORITHM
CIPOLLAS ALGORITHM
CIPOLLAS ALGORITHM
n.
A genus of solanaceous herbs with funnelform or salver-shaped corollas. Two species are common in cultivation, Petunia violacera, with reddish purple flowers, and P. nyctaginiflora, with white flowers. There are also many hybrid forms with variegated corollas.
n.
A genus of herbaceous plants with pretty salver-shaped corollas, usually blue or violet; leadwort.
a.
Having a corolla or corollas; like a corolla.
n.
The art of calculating with any species of notation; as, the algorithms of fractions, proportions, surds, etc.
a.
Having corollas of five petals with long claws inclosed in a tubular, calyx, as the pink
pl.
of Cupola
n.
Any plant of the genus Veronica, mostly low herbs with pale blue corollas, which quickly fall off.
v. i.
To develop bells or corollas; to take the form of a bell; to blossom; as, hops bell.
a.
Having the appearance of being labiate; -- said of certain polypetalous corollas.
n.
The art of calculating by nine figures and zero.
n.
Alt. of Algorithm