Thread by @SrinivasR1729

The significance of #mathematics in #cryptography is immense. 𝐂𝐫𝐲𝐩𝐭𝐨𝐠𝐫𝐚𝐩𝐡𝐲 involves the process of transforming messages to ensure secure communication. In this thread, we'll explore the integral role of math within cryptography & present some relevant equations (1/9)

One of the simplest examples of cryptography is the Caesar cipher, which shifts the letters of the alphabet by a fixed number. The encryption function can be written as: E(x) = (x + k) mod 26, where x is the letter's position, k is the shift, and mod is the modulo operation.(2/9)

Public-key cryptography is a more advanced technique that relies on mathematical concepts. RSA, a widely-used public-key cryptosystem, involves the generation of two large prime numbers, p and q. Their product, n = p * q, is used in encryption and decryption.(3/9)

In RSA, the public key is (n, e), where e is a number coprime with (p-1)(q-1). The private key is (n, d), where d is the inverse of e modulo (p-1)(q-1). To encrypt a message m, the sender calculates: c ≡ m^e (mod n). To decrypt, the recipient computes: m ≡ c^d (mod n).
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Elliptic curve cryptography (ECC) is another approach that relies on elliptic curve math. An elliptic curve can be represented by the equation: y^2 = x^3 + ax + b. ECC provides similar security as RSA but with shorter key lengths, making it more efficient. (5/9)

In ECC, a public key is generated by multiplying a private key (a scalar) by a base point G on the elliptic curve. The resulting point on the curve, Q, is the public key. Q = dG, where d is the private key and G is the base point. (6/9)

Diffie-Hellman key exchange protocol, which allows secure communication between parties, is based on modular arithmetic. Given a prime number p & a primitive root g, each party selects a private key a or b and shares g^a mod p or g^b mod p & shared secret is g^(ab) mod p. (7/9)

Cryptography also involves hashing functions, which map data to fixed-size output. One example is the SHA-256 algorithm, used in Bitcoin. These functions rely on bitwise operations and modular arithmetic, providing a way to verify data integrity without revealing content. (8/9)

Mathematics is the foundation of cryptography. From the Caesar cipher to modern cryptographic techniques, math plays a vital role in securing our digital world. As technology advances, so too will the mathematical methods that keep our information safe. (9/9)