How to solve problems on fermats theorem in cryptography?
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Fermat's Little Theorem is highly useful in number theory for simplifying the computation of exponents in modular arithmetic (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to Pierre de Fermat.
Statement
If ${a}$ is an integer, ${p}$ is a prime number and ${a}$ is not divisible by ${p}$, then $a^{p-1}\equiv 1 \pmod {p}$.
A frequently used corollary of Fermat's Little Theorem is $a^p \equiv a \pmod {p}$. As you can see, it is derived by multipling both sides of the theorem by $a$. The restated form is nice because we no longer need to restrict ourselves to integers ${a}$ not divisible by ${p}$.
This theorem is a special case of Euler's Totient Theorem, which states that if $a$ and $n$ are integers, then $a^{\phi(n)} \equiv 1 \pmod{n}$, where $\phi(n)$ denotes Euler's totient function. In particular, $\phi(p) = p-1$ for prime numbers $p$. In turn, this is a special case of Lagrange's Theorem.
In contest problems, Fermat's Little Theorem is often used in conjunction with the Chinese Remainder Theorem to simplify tedious calculations.
Proof
We offer several proofs using different techniques to prove the statement $a^p \equiv a \pmod{p}$. If $\text{gcd}\,(a,p) = 1$, then we can cancel a factor of $a$ from both sides and retrieve the first version of the theorem.
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Statement
If ${a}$ is an integer, ${p}$ is a prime number and ${a}$ is not divisible by ${p}$, then $a^{p-1}\equiv 1 \pmod {p}$.
A frequently used corollary of Fermat's Little Theorem is $a^p \equiv a \pmod {p}$. As you can see, it is derived by multipling both sides of the theorem by $a$. The restated form is nice because we no longer need to restrict ourselves to integers ${a}$ not divisible by ${p}$.
This theorem is a special case of Euler's Totient Theorem, which states that if $a$ and $n$ are integers, then $a^{\phi(n)} \equiv 1 \pmod{n}$, where $\phi(n)$ denotes Euler's totient function. In particular, $\phi(p) = p-1$ for prime numbers $p$. In turn, this is a special case of Lagrange's Theorem.
In contest problems, Fermat's Little Theorem is often used in conjunction with the Chinese Remainder Theorem to simplify tedious calculations.
Proof
We offer several proofs using different techniques to prove the statement $a^p \equiv a \pmod{p}$. If $\text{gcd}\,(a,p) = 1$, then we can cancel a factor of $a$ from both sides and retrieve the first version of the theorem.
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