ⓘ Free online encyclopedia. Did you know? page 280




                                               

Trigonometric function

In mathematics, the trigonometric functions are a set of functions which relate angles to the sides of a right triangle. There are many trigonometric functions, the 3 most common being sine, cosine, tangent, followed by cotangent, secant and cose ...

                                               

Hyperbolic functions

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined for the unit hyperbola rather than on the unit circle: just as the points form a circle with a unit radius, the points form the right half of ...

                                               

3 (number)

The number three is one more than two and one less than four. It is the first Mersenne prime. Three is an important number for many cultures. It is also a prime number. It is the first odd prime. In Roman numerals, it is III.

                                               

40 (number)

Forty is the number that is after thirty-nine and before forty-one. The prime factors of forty are 2 and 5. 2 * 2 * 2 * 5 = 40 Forty is also the only number in the English language which is spelt in alphabetical order.

                                               

42 (number)

Forty-two is a number. It comes between forty-one and forty-three, and is an even number. It is divisible by 1, 2, 3, 6, 7, 14, 21, and 42. 42 is an extremely interesting number to mathematicians, because of its many features. For example: It is ...

                                               

7 (number)

To test if an integer is divisible by 7 or not, Repeat the second step again and again until it is clearly that the number is divisible by 7 or not. Take off the last digit and multiply it by 2. Subtract the root number from the product. Example ...

                                               

Acre

For the state in Brazil, see Acre state. An acre is an English unit of area, which is also used in the United States and some Commonwealth countries. It is most often used to describe areas of land. By the official definition it is 4.046.8564224 ...

                                               

Attack model

Attack models or attack types specify how much information a cryptanalyst has access to when breaking or cracking an encrypted message, commonly known as codebreaking or cracking the code. Some common attack models are: Chosen-plaintext attack Ch ...

                                               

Backpropagation

Backpropagation is a method of training neural networks to perform tasks more accurately. The algorithm was first used for this purpose in 1974 in papers published by Werbos, Rumelhart, Hinton, and Williams. The term backpropagation is short for ...

                                               

Basis (linear algebra)

In linear algebra, a basis is a set of vectors in a given vector space with certain properties: One can get any vector in the vector space by multiplying each of the basis vectors by different numbers, and then adding them up. If any vector is re ...

                                               

Bernoulli distribution

The Bernoulli distribution is a probability distribution. It takes a value of 1 with probability p and a value of 0 with probability 1- p. It is sometimes written as Ber ⁡ {\displaystyle \operatorname {Ber} }. It is used in probability theory and ...

                                               

Bertrands postulate

Bertrands postulate states that if n > 3 is an integer, then there always exists at least one prime number p with n < p < 2 n − 2. This statement was first made in 1845 by Joseph Bertrand. Bertrand verified his statement for all numbers ...

                                               

Binomial distribution

In probability and statistics, the binomial distribution is a probability distribution which models the probabilities of having a certain number of successes among n identical trials. It is also written as Bin ⁡ {\displaystyle \operatorname {Bin} ...

                                               

Block cipher

In cryptography, a block cipher is a symmetric key cipher which operates on a groups of bits of fixed length, called blocks, using an exact transformation. During encryption, a block cipher algorithm might take a 128-bit block of plaintext as inp ...

                                               

Block size (cryptography)

In modern cryptography, symmetric key ciphers are generally divided into stream ciphers and block ciphers. Block ciphers operate on a fixed length string of bits. The length of this bit string is the block size. Both the input and output are the ...

                                               

Brute force attack

A brute force attack is a term in cryptanalysis. It means trying to break a coded cyphertext by trying a lot of possibilities with fast computers. For example, a large number of the possible keys are tried in the key space. If successful, this de ...

                                               

Cardinal number

Cardinal numbers are numbers that say how many of something there are, for example: one, two, three, four, five, six. They are sometimes called counting numbers. The cardinality of a set is the cardinal number that tells us, roughly speaking, the ...

                                               

Carmichael number

In number theory a Carmichael number is a composite positive integer n {\displaystyle n}, which satisfies the congruence b n − 1 ≡ 1 {\displaystyle b^{n-1}\equiv 1{\pmod {n}}} for all integers b {\displaystyle b} which are relatively prime to n { ...

                                               

Chain rule

In differential calculus, the chain rule is a way of finding the derivative of a function. It is used where the function is within another function. This is called a composite function. More specifically, if F x {\displaystyle Fx} equals the comp ...

                                               

Chi-square distribution

In probability theory and statistics, the chi-square distribution is one of the most widely used theoretical probability distributions. Chi-square distribution with ν {\displaystyle \nu } degrees of freedom is written as χ 2 {\displaystyle \chi ^ ...

                                               

Chinese remainder theorem

The chinese remainder theorem is a theorem from number theory. It is about congruence. The original form was: How many soldiers are there in Han Xins army? – If you let them parade in rows of 3 soldiers, two soldiers will be left. If you let them ...

                                               

Chosen-plaintext attack

A chosen-plaintext attack is a model for cryptanalysis which assumes that the attacker can choose random plaintexts to be encrypted and obtain the corresponding ciphertexts. The goal of the attack is to gain some further information which reduces ...

                                               

Ciphertext-only attack

In cryptography, a ciphertext-only attack or known ciphertext attack is an attack model for cryptanalysis where the attacker is assumed to have access only to a set of ciphertexts. The attack is completely successful if the corresponding plaintex ...

                                               

Communication Theory of Secrecy Systems

Communication Theory of Secrecy Systems is a paper published by Claude Shannon discussing cryptography from the viewpoint of information theory. It is one of the foundational treatments of modern cryptography. It is also a proof that all theoreti ...

                                               

Composite number

A composite number is a natural number, that can be found by multiplying prime numbers. For example, the number 9 can be found by multiplying 3 by 3. Another example is the number 12, which is obtained by multiplying 3, 2 and 2. All natural numbe ...

                                               

Cone

In common speaking and geometry, a cone is a solid object that one gets when one rotates a right triangle around one of its two short sides, the cones axis. The disk made by the other short side is called the base, and the point of the axis which ...

                                               

Confusion and diffusion

In cryptography, confusion and diffusion are two properties of the operation of a secure cipher. Confusion and diffusion were identified by Claude Elwood Shannon in his paper, "Communication Theory of Secrecy Systems" published in 1949. In Shanno ...

                                               

Consecutive integer

Consecutive numbers are numbers that follow each other in order. They have a difference of 1 between every two numbers. In a set of consecutive numbers, the mean and the median are equal. If n is a number, then the next numbers will be n+1 and n+ ...

                                               

Converse (logic)

In mathematics and logic, a converse is a variant of an implication. More specifically, given an implication of the form P → Q {\displaystyle P\to Q}, the converse is the statement Q → P {\displaystyle Q\to P}. While a converse is similar to its ...

                                               

Cryptosystem

There are two different meanings of the word cryptosystem. The other is a general meaning which is how the word cryptosystem is understood by the public. One is used by the cryptographic community.

                                               

De Moivres formula

In mathematics, de Moivres formula or de Moivres theorem is an equation named after Abraham de Moivre. It states that for any real number x and integer n, cos ⁡ x + i sin ⁡ x n = cos ⁡ n x + i sin ⁡ n x {\displaystyle \cos x+i\sin x^{n}=\cos nx+i ...

                                               

Decagon

The amount of space a regular decagon takes up is Area = 5 2 a 2 5 + 2 5. {\displaystyle {\text{Area}}={\frac {5}{2}}a^{2}{\sqrt {5+2{\sqrt {5}}}}.} a is the length of one of its sides. An alternative formula is A = 2.5 d t {\displaystyle A=2.5dt ...

                                               

Differential cryptanalysis

Differential cryptanalysis is a general form of cryptanalysis applicable to block ciphers, but also can be applied to stream ciphers and cryptographic hash functions. It is the study of how differences in the input can affect the resultant differ ...

                                               

Digon

In geometry, a digon is a polygon with two sides and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in ell ...

                                               

Diophantus

Diophantus of Alexandria was an ancient Greek mathematician. He is well know for his series of books called Arithmetica. He mainly focused on solving algebraic equations. His work was built upon by Pierre de Fermat, a French mathematician, who cl ...

                                               

Divergence

In mathematics, divergence is a differential operator that associates a vector field with a scalar field. In a vector field, each point of the field is associated with a vector; in a scalar field, each point of the field is associated with a scal ...

                                               

Dodecahedron

A dodecahedron is a solid shape that has 12 surfaces. Each face is a pentagon. In total there are 20 corners and 30 edges altogether. A dodecahedron which looks exactly the same from all faces is also a platonic solid.

                                               

Double factorial

Double factorial is a method of calculating how many times a number is repeated in a geometric equation. This way, we can calculate the number of times a product is used in its life-time. The double factorial of n is written as n!! {\displaystyle ...

                                               

EFF DES cracker

In cryptography, the EFF DES cracker is a machine built by the Electronic Frontier Foundation to perform a brute force search of DES ciphers key space - that is, to decrypt an encrypted message by trying every possible key. The aim in doing this ...

                                               

Equilateral triangle

In geometry, an equilateral triangle is a triangle where all three sides are the same length and all three angles are also the same and are each 60°. If an altitude is drawn, it bisects the side to which it is drawn, in turn leaving two separate ...

                                               

ESTREAM

eSTREAM is a project to identify "new stream ciphers that might become suitable for widespread adoption", organized by the EU ECRYPT network. It was set up because of the failure of all six stream ciphers submitted to the NESSIE project. The call ...

                                               

Euclidean algorithm

The Euclidean algorithm is an algorithm. It can be used to find the biggest number that divides two other numbers.

                                               

Eulers totient function

In number theory, the totient of a positive integer n is the number of positive integers smaller than n which are coprime to n. It is often written as ϕ {\displaystyle \phi }. For example, ϕ 8 = 4 {\displaystyle \phi 8=4}, because there are four ...

                                               

Euler–Mascheroni constant

In mathematics, Euler-Mascheroni constant is a number that appears in analysis and number theory. It first appeared in the work of Swiss mathematician Leonhard Euler in the early 18th century. It is usually represented with the Greek letter γ {\d ...

                                               

Exponential distribution

The exponential distribution with rate λ {\displaystyle \lambda }, written Exp ⁡ {\displaystyle \operatorname {Exp} }, is a probability distribution defined only on the positive real numbers. It is the continuous analogue of geometric distributio ...

                                               

Exponentiation by squaring

Exponentiating by squaring is an algorithm. It is used for quickly working out large integer powers of a number. It is also known as the square-and-multiply algorithm or binary exponentiation. It uses the binary expansion of the exponent. It is o ...

                                               

Googolplex

A googolplex is the number 10 googol, equivalent to 10 100 or 10 Ten duotrigintillion or 10 10.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000. Like the number go ...

                                               

Great circle

A great circle the largest possible circle that can be drawn on a sphere, one that divides the surface into equal halves, called hemispheres. It is a circle that has the same diameter as the sphere it was drawn on. These curves are geodesics in t ...

                                               

Hexacontagon

Since 60 = 2 × 3 × 5, a regular hexacontagon is constructible using a compass and straightedge. As a truncated triacontagon, it can be constructed by an edge-bisection of a regular triacontagon.

                                               

Hexadecagon

The area of a regular hexadecagon with edge length t is A = 4 t 2 cot ⁡ π 16 = 4 t 2 1 + 2 + 4 + 2 = 4 t 2 + 1 4 − 2 + 1. {\displaystyle {\begin{aligned}A=4t^{2}\cot {\frac {\pi }{16}}=&4t^{2}\left1+{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}\right\ ...