algorithm - A series of steps used to complete a task.
Alice - The name traditionally used for the first user of cryptography in a system; Bob's friend.
attack - Either a successful or unsuccessful attempt at breaking part or all of a cryptosystem.
basis - The maximal linearly independent set of vectors, that is the set, which ceases to be linearly dependent after adding to it of any other vector froom this space, is called the basis of the vector space. If some basis in the vector space contains n elements, then every other basis contains n elements and the space is called n - dimensional. In such space any (n+1) vectors form the linearly dependent set.
bit - A binary digit, either 0 or 1.
Bob - The name tradionally used for the second user of cryptography in a system; Alice's friend.
cipher - An encrypyion - decryption algorithm.
composite number - Number, which has divisors, different from one and itself is called composite. (Number 1 refers neither to prime, nor to composite numbers).
co-prime - Two integers are co-prime if they have no common factors, i. e. (14, 25) = 1.
cryptanalysis - The art and science of breaking encryption ir anty form of cryptography. See attack.
cryptography - The art and science of using mathematics to secure information and create a high degree of trust in the electronic realm. see also public key.
cryptology - The branch of mathematics concerned with cryptography and cryptanalysis.
cryptosystem - An encryption decryption algorithm (cipher), together with all possible plaintexts, ciphertexts and keys.
decryption - The inverse (reverse) of encryption.
encryption - The transformation of plaintext into an apparently less readable form (called ciphertext) through a mathematical process. The ciphertext may be read by anyone who has the key that decrypts (undoes the encryption) the ciphertext.
factor - Given an integer N, any number that divides it is called a factor.
factoring - The breaking down of an integer into its prime factors. This is a hard problem.
field - The set M is called field, if there are defined two operations - addition and multiplication, which are satisfying the following conditions:
1. a + b = b + a, where a and b are elements of M (commutativity of addition);
2. a + (b + g) = (a + b) + g, where a, b and g are elements of M (associativity of addition);
3. There exists zero - an element denoted by 0, which has the following property: for any element a from M the equality a + 0 = a is fulfilled;
4. For any element a from M there exists an oposite one, that is, such an element -a, for which a + (-a) = 0;
5. ab = ba (commutativity of multiplication);
6. a(bg) = (ab)g (associativity of multiplication);
7. There exists unity element, denoted by 1, which has the following property: for every element a from M the equality a×1 = a is fulfilled;
8. For every element a from M, different from zero , there exists an opposite one, that is, such an element a-1, for which a×a-1 = 1;
9. (a + b)g = ag + bg (distributivity of multiplication relatively to addition).
finite field - There exist fields, which consist of finite quentity of elements. The example of such set is a set of remainders gotten when dividing by some prime number p, with correspondingly introduced rules of addition and multiplication. It's denoted by Zp.
function - A mathematical relationship between two values called the input and the output, such that for each input there is precisely one output.
GCD - Greatest Common Divisor.
Internet - The connection of computer networks from all over the world forming a worldwide network.
key - A string of bits used widely in cryptography, allowing people to encrypt and decrypt data; a key can be used to perform other mathematical operations as well. Given a cipher, a key determines the mapping of the plaintext to the ciphertext. See also private key, public key.
linear dependence - In the vector space the set X = {x1, x2, ... , xn}, which consists of finite number of vectors, is called linearly dependent, if there exists such scalars a1, a2, ... , an not simultaneously equal to zero, that the equality a1x1 + a2x2 + ... + an xn = 0 holds.
linear independence - In the vector space the set X = {x1, x2, ... , xn}, which consists of finite number of vectors, is called linearly independent, if for some scalers a1, a2, ... , an the equality a1x1 + a2x2 + ... + an xn= 0 holds only when a1 = a2 = ... = an = 0.
modular arithmetic - A form of arithmetic where integers are considered equal if they leave the same remainder when divided by the modulus.
modulus - The integer used to divide out by in modular arithmetic.
number field sieve - A method of factoring, currently the fastest general purpose factoring algorithm published. It was used to factor RSA-130.
number theory - A branch of mathematics that investigates the relationships and properties of numbers.
one-way function - A function that is easy to compute in one direction but quite difficult to reverse compute (compute in the opposite direction.)
plaintext - The data to be encrypted.
prime factor - A prime number that is a factor of another number is called a prime factor of that number.
prime number - Any integer greater than 1 that is divisible only by 1 and itself (Number 1 refers neither to prime, nor to composite numbers).
private key - In public-key cryptography, this key is the secret key. It is primarily used for decryption but is also used for encryption with digital signatures.
protocol - A series of steps that two or more parties agree upon to complete a task.
public key - In public-key cryptography this key is made public to all, it is primarily used for encryption but can be used for verifying signatures.
public-key cryptography - Cryptography based on methods involving a public key and a private key.
quadratic sieve - A method of factoring an integer, developed by Carl Pomerance.
RSA algorithm - A public-key cryptosystem based on the factoring problem. RSA stands for Rivest, Shamir and Adleman, the developers of the RSA public-key cryptosystem and the founders of RSA Data Security, Inc.
scalars - The elements of the field are often called scalars.
trap door one-way function - A one-way function that has an easy-to-compute inverse if you know certain secret information. This secret information is called the trap door.
vector - The elements of the vector space are called vectors.
vector space - The vector space F is defined on some field of scalars and represents a set, elements of which are called vectors and in which there exists ruler of addition of elements and their multiplication by elements of field of scalars, and the following conditions are fulfilled:
1. Addition of vectors is commutative:
x + y = y + x for any x, y from F ;
2. Addition of vectors is associative:
x + (y + z) = (x + y) + z for any x, y and z from F ;
3. There exists a zero-vector, denoted by 0 such,that for any vector x from F the equality x + 0 = x holds;
4. For unrestricted vector x there exists an opposite one, denoted by -x, which satisfies the condition x + (-x) = 0;
5. For unrestricted scalars a, b and vector x the condition a(bx) = (ab)x fulfills;
6. For unrestricted vector x the condition 1×x = x fulfills;
7. a(x + y) = ax + ay - distributivity of multiplication relatively to addition of vectors;
8. (a + b)x = ax + bx - distributivity of multiplication relatively to addition of scalars.