Cryptography Concepts and Theorems

Public key cryptography systems like RSA rely on groups, while symmetric cryptography uses fields and rings.

The Euler's ϕ-function counts integers less than n that are divisible by n.

Fermat's Little Theorem states that if p is prime, then for any integer a coprime to p, a^(p-1) ≡ 1 (mod p).

Euler's Theorem applies only to prime numbers.

The Miller-Rabin primality test is a deterministic algorithm for checking the primality of numbers.

In the Miller-Rabin test, if p - 1 can be expressed as 2^s * d, with d being odd, the test will always confirm p as prime.

AES encryption primarily depends on modular arithmetic for its S-box operations.

The group of integers modulo n corresponds to the multiplication operation in public key cryptography.

Public Key Cryptography vs. Symmetric Cryptography

Public key cryptography, like RSA, often uses simpler structures like groups for its mathematical foundation.
Symmetric cryptography, like AES, uses more complex structures such as fields and rings.
A group is a set with one operation that satisfies closure, associativity, an identity element, and inverses.
A field is a structure with two operations (addition and multiplication) forming groups and where multiplication distributes over addition.
Symmetric cryptography often utilizes finite fields, crucial for the substitution layers in algorithms like AES.
Public key cryptography relies on groups, like the group of integers modulo n under multiplication, a simpler mathematical structure.

Euler’s ϕ-Function, Fermat’s Little Theorem, and Euler’s Theorem

Euler's ϕ-function calculates the number of integers less than n that are coprime to n.
For n = p * q (p and q are prime): ϕ(n) = (p-1)(q-1)
Fermat's Little Theorem: If p is prime and a is coprime to p, then a^p-1 ≡ 1 (mod p).
Euler's Theorem: For any integer a coprime to n, a^ϕ(n) ≡ 1 (mod n).
Fermat's theorem is a special case of Euler's theorem when n is prime.
Euler's theorem generalizes Fermat's theorem to composite numbers.
The ϕ-function provides the exponent in Euler's theorem, which matches p-1 in Fermat's theorem.

Miller-Rabin Primality Test

The Miller-Rabin test is a probabilistic algorithm that checks if a number is likely prime.
It works by first expressing p-1 = 2^s * d, where d is odd.
Then it selects a base a and computes powers a^d, (a^d)², ..., (a^d)^{2^s-1} (mod p).
A number is likely prime if a^d ≡ 1 (mod p) or one of the powers a^{2^rd} ≡ -1 (mod p) for 0 ≤ r < s.

Explore the foundational concepts of public key and symmetric cryptography, including the mathematical structures that support each. Delve into Euler's ϕ-function, Fermat's Little Theorem, and their applications in cryptography. This quiz tests your knowledge of essential cryptographic principles and theorems.