What is Modular Arithmetic in Information Security?

Modular arithmetic is a structure of arithmetic for integers, where numbers "wrap around" upon reaching a specific value. Modular arithmetic enables us to simply make groups, rings and fields which are the basic constructing piece of most modern public-key cryptosystems.

For instance, Diffie-Hellman need the multiplicative group of integers modulo a prime pp. There are different groups which can work. Modular or clock arithmetic is arithmetic on a circle rather than a number line modulo N, it can use only the twelve entire numbers from 0 through N-1.

Modular arithmetic is very well understood in method of algorithms for several basic operations. That is one of the reason why it can use finite fields (AES) in symmetric key cryptography. Cryptography needed complex problems. Some problems develop into a hard with modular arithmetic.

For example, logarithms are simply to calculate over all integers but can become hard to compute when it can introduce a modular reduction. Similarly with discovering roots. Mod-arithmetic is the central mathematical terms in cryptography.

Much of modern number theory and some practical problems are concerned with modular arithmetic. In arithmetic modulo N, it is concerned with arithmetic on the integers, where it can recognize all numbers which vary by an impose multiple of N. 

That is,

x=y mod N if x = y +mN for some integer m.

This recognition divides all the integers into N same classes. It can generally indicated these by their simplest members that is the number 0, 1, ….N-1.

If a is an integer and n is a positive integer, represent a mod n to be the remainder when a is divided by n. Then $\mathrm{a\, =\, \left \lfloor a/n\right \rfloor\, x\, n\, +\, \left ( a\, mod\, n \right );}$

Example − 11 mod 7 = 4

Theorem − n is an equivalence relation on the integers. An equivalence class includes those integers which have the equal remainder on division by n. The equivalence classes are also called a congruence classes modulo n. Instead of say the integers a and b are equivalent and it can said that they are congruent modulo n.

The set of all integers congruent to a modulo n is called the residue class [a].

The modulo operator has the following properties −

  • a ≡ b mod n if n|(a − b).

  • (a mod n) = (b mod n) implies a ≡ b mod n.

  • a ≡ b mod n implies b ≡ a mod n.

  • a ≡ b mod n and b ≡ c mod n imply a ≡ c mod n.

Properties of modular arithmetic operations

  • [(a mod n) + (b mod n)] mod n = (a + b) mod n

  • [(a mod n) - (b mod n)] mod n = (a - b) mod n

  • [(a mod n) x (b mod n)] mod n = (a x b) mod n

Let Zn = {0, 1, 2,… (n-1)}, be the set of residues modulus n.

Commutative laws
(w + x) mod n = (x + w) mod n
Associative laws
(w x x) mod n = (x x w) mod n

[(w + x)+y] mod n = [w+(x+y)] mod n
Distributive laws
[(w x x) x y] mod n = [w x (x x y)] mod n
[(w x (x + y)] mod n =[(w x x) + (w x y)] mod n

(0 + w) mod n = w mod n
Additive inverse (-w)
(1 x w) mod n = w mod n

For each w ∈ Zn, there exists a z such that w + z ≡ 0 mod n