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What is Euler’s Theorem in Information Security?
Euler's theorem is a generalization of Fermat's little theorem handling with powers of integers modulo positive integers. It increase in applications of elementary number theory, such as the theoretical supporting structure for the RSA cryptosystem.
This theorem states that for every a and n that are relatively prime −
$$\mathrm{a^{\phi \left ( n \right )}\, \equiv\, 1\left ( mod \, n \right ) }$$
where $\mathrm{\phi}$(n) is Euler's totient function, which counts the number of positive integers less than n that are relatively prime to n.
Consider the set of such integers −
R = {x1, x2, … x$\mathrm{\phi}$(n)}, i.e., each element xi of R is unique positive integer less than n with ged(xi, n) = 1. Then multiply each element by a and modulo n −
S = {(ax1mod n), (ax2mod n), … (ax$\mathrm{\phi}$(n)mod n)}
Because a is relatively prime to n and xi is relatively prime to n, axi must also be relatively prime to n. Therefore, all the members of S are integers that are less than n and that are relatively prime to n.
There are no duplicates in S.
If axi mod n and n = axj mod n then xi = xj
Therefore,
$$\mathrm{\Pi _{i=1}^{\phi \left ( n \right )}\left ( ax_{i}\, mod\, n \right )=\Pi _{i=1}^{\phi \left ( n \right )}\, x_{i}}$$
$$\mathrm{\Pi _{i=1}^{\phi \left ( n \right )}\, ax_{i}\equiv \Pi _{i=1}^{\phi \left ( n \right )}\, x_{i}\left ( mod\, n \right )}$$
$$\mathrm{a^{\phi \left ( n \right )}\, x\left [ \Pi _{i=1}^{\phi \left ( n \right )}\, x_{i} \right ]=\Pi _{i=1}^{\phi \left ( n \right )}\, x_{i}\left ( mod\, n \right )}$$
$$\mathrm{a^{\phi \left ( n \right )}\equiv 1\left ( mod\, n \right )}$$
Euler Totient Function
Euler’s Totient function is the mathematical multiplicative functions which count the positive integers up to the given integer generally known as as ‘n’ that are a prime number to ‘n’ and the function can be used to understand the number of prime numbers that exist up to the given integer ‘n’.
Euler’s Totient function is also called as Euler’s phi function. It plays an essential role in cryptography. It can discover the number of integers that are both smaller than n and relatively prime to n. These set of numbers defined by $\mathrm{Z_{n}^{*}}$ (number that are smaller than n and relatively prime to n).
Euler’s totient function is beneficial in several ways. It can be used in the RSA encryption system, which can be used for security goals. The function deals with the prime number theory, and it is beneficial in the computation of large calculations also. The function can be utilized in algebraic computation and simple numbers.
The symbol used to indicate the function is ϕ, and it is also known as phi function. The function includes more theoretical use instead of practical use. The sensible requirement of the function is limited.
The function can be better understood through the several practical examples instead of only theoretical explanations. There are several rules for computing the Euler’s totient function, and for different numbers, different rules are to be used.
The Euler totient function $\mathrm{\phi}$(n) calculates the number of elements in $\mathrm{Z_{n}^{*}}$ with the help of the following rules −
$\mathrm{\phi}$(1) = 0.
$\mathrm{\phi}$(P) = P − 1 if P is a Prime.
$\mathrm{\phi}$(m x n) = $\mathrm{\phi}$(m) x $\mathrm{\phi}$(n) if m and n are relatively prime.
$\mathrm{\phi}$(Pe) = Pe − Pe−1 (if P is a prime. )
The following four rules can be combined to obtain the value of $\mathrm{\phi}$(n), factorize n as
$$\mathrm{n=P_{1}^{e1}\, x\,P_{2}^{e2}x\cdot \cdot \cdot P_{k}^{ek}}$$
$$\mathrm{\phi \left ( n \right )=\left ( P_{1}^{e1}-P_{1}^{e1-1} \right )\left ( P_{2}^{e2}-P_{2}^{e2-1} \right )x\cdot \cdot \cdot x\left (P_{k}^{ek}-P_{k}^{ek-1} \right )}$$
The difficulty of finding $\mathrm{\phi}$(n) depends on the difficulty of finding the factorization of n.
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