Algebraic Structure
Groups:
A group (G) is a set of elements with binary operations “•” denoted as G = < {}, • > that satisfies four following properties:
Closure If a and b are elements of G then c = a • b is also an element of G. this means that the result of applying the operation on any two elements is the set is another element in the set.
Associativity If a, b and c are the elements of G, then (a•b)•c = a•(b•c). In other words, it does not matter in which order we apply the operation on more than two elements.
Existence of Identity For all a in G, there exists an element e, called the identity element such that e•a = a•e = a.
Existence of Inverse For each element a on G, there exists an element a’, called the inverse of a, such that a•a’ = a’•a = e.
Abelian Group (Commutative Group): A group is called Abelian Group or Commutative Group if the operator satisfies the four properties for groups plus an extra property, commutative.
Commutativity For all a and b in G, we have a•b=b•a. note that this property needs to be satisfied only for Abelian group.
As an example The set of residue integers with the addition operator, G =
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Closure: The result of adding two integers in Zn is another integer in Zn.
Associativity: The result of (3+2)+4 = 3+(2+4).
Commutativity: The result of 3+5 = 5+3.
Identity: The identity element is 0, 4+0 = 0+4 = 4.
Inverse: Every elements of the set has an additive inverse. This is the complement of the element; inverse of the 3 is -3 and vice versa. The inverse is used to perform the subtraction using addition.
Finite Group A group is called a finite group if the set has a finite number of elements.
Infinite Group A group is called a infinite group if the set has infinite number of elements.
Order of a Group The order of a group denoted by |G| is the number of elements present in the set. If the group is infinite then the order of the group is also infinite.
Subgroups A group H is a subgroup of G if H has the same operation as group G and the set of the group H is the subset of the set in G. In other word if G =
Cyclic Subgroups If a subgroup of a group can be generated using the power of the elements from the set of group G. The subgroup is called the cyclic subgroup. The term power here means repeatedly applying the group operation to the element.
As an example
Four cyclic subgroup can be made from the group G =
G = < {0, 1, 2, 3, 4, 5}, + >
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H1 = < {0}, + >
00 mod 6 = 0 01 mod 6 = 0 . .
H2 = < {0, 1, 2, 3, 4, 5}, + >
10 mod 6 = 0 11 mod 6 = 1 12 mod 6 = (1+1) mod 6 =2 13 mod 6 = (1+1+1) mod 6 =3 14 mod 6 = (1+1+1+1) mod 6 =4 15 mod 6 = (1+1+1+1+1) mod 6 =5 16 mod 6 = (1+1+1+1+1+1) mod 6 =0 . . OR 50 mod 6 = 0 51 mod 6 = 5 52 mod 6 = (5+5) mod 6 =4 53 mod 6 = (5+5+5) mod 6 =3 54 mod 6 = (5+5+5+5) mod 6 =2 55 mod 6 = (5+5+5+5+5) mod 6 =1 56 mod 6 = (5+5+5+5+5+5) mod 6 =0 . .
H3 = < {0, 2, 4}, + >
20 mod 6 = 0 21 mod 6 = 2 22 mod 6 = (2+2) mod 6 = 4 23 mod 6 = (2+2+2) mod 6 = 0 . . OR 40 mod 6 = 0
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41 mod 6 = 4 42 mod 6 = (4+4) mod 6 = 2 43 mod 6 = (4+4+4) mod 6 = 0 . .
H4 = < {0, 3}, + >
30 mod 6 = 0 31 mod 6 = 3 32 mod 6 = (3+3) mod 6 = 0 . .
Generator element An element a of group G is called generator element if the repeated operation on that element produces all the elements of the group G. In the above example the element 1 and 5 is called generator element. Also notice that these two elements are inverse to each other, in other word inverse of generator elements for a group is also a generator element.
Cyclic Groups A group G is called cyclic group if there exists at-least one generator element. As in the group G = < Z6, + > there exists two generator element 1 and 5 it is a Cyclic group.
Lagrange’s Theorem Assume G is a group and H is a subgroup of G. If the order of G is |G| and order of H is |H|then based on Lagrange’s theorem, |H| divides |G|.
Ring:
A ring is a algebraic structure denoted as R = < {}, •, □ > consist of two operations. The first operation must satisfy all five properties required for an abelian group and the second operation must be distributed over the first and satisfy only the first two properties of the group these are closure and Associativity. Here distributive properties
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state that for all a, b, c elements of R, we have a □ (b • c) = (a □ b) • c and (a • b) □ c = (a □ b) • (b □ c).
A ring is called commutative if the commutative property is also satisfied for the second operation. Actually the 1st operation is a pair of operation such as addition and subtraction and the 2nd operation is a single operation such as multiplication but not division.
For example the set Z with the operations, Addition and Multiplication is a commutative ring. It can be showed by R = < Z, +, * >. Addition satisfies all the five properties. Multiplication satisfies only three properties. Multiplication is distributed over addition as example, 5 * (3 + 2) = (5 * 3) + (5 * 2) = 25 here we can also perform subtraction in place of addition. But the division operation can’t be possible in place of multiplication. Because division operation can produces an element out of the set (12 / 5 = 2.4).
Field:
A field is donated by F = < {…}, •, □ > is a commutative ring in which the second operation satisfies all five properties define for the first operation except the identity of the first operation has no inverse.
Finite field:
In cryptography only finite fields are used extensively. A finite field is a field with finite number of elements. Galois showed that in a finite field the number of element should be pn where p is a prime number and n is the positive integer. By the name of Galois finite fields are known as Galois fields and denoted as GF(pn).
GF(p) Fields:
n In Galois fields GF(p ) when n = 1, we have GF(p). This field can be the set Zn, {0, 1, …, p-1}, with two arithmetic operations (+ and ×). In this set each element has an additive inverse and that nonzero elements have a multiplicative inverse.
For example, we can define GF(5) by the set Z5 with the addition and multiplication operation as < {0, 1, 2, 3, 4, 5}, +, × >
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GF(pn) Fields:
In computer the positive integers are stored in the computer as n-bit word in which n is usually 8, 16, 32, 64 and so on. This means that the range of integers is 0 to 2n-1. The modulus is 2n. To use a field we have two choices:
1. We can use GF(p) with the set Zp, p is the largest prime number less than 2n. In this method we have some problems; we can’t use integers from p to 2n-1. If n = 4, the largest prime less than 24 is 13, this means that we cannot use integers 13, 14, and 15. 2. We can work with GP(2n) and uses a set of 2n elements. The elements in this set are n bit words for example if n=3 the set is {000, 001, 010, 011, 100, 101, 110, 111}.
Summary:
Algebraic Structurte Operation Supported Supported Set * Group (+ -) or (* /) Zn or Zn Ring (+ -) and (*) Z
Field (+ -) and (* /) Zp
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