Yesterday's meeting explored the mathematical foundations and cryptographic applications of Elliptic Curve Cryptography (ECC), from the core concepts of modular arithmetic and group theory to the specific algorithms that power modern security, such as ECDH and ECDSA. Afterwards, members participated in a hands-on workshop to implement these concepts.

Math Foundations

Our presentation began with the building blocks required for ECC:

• •Modular Arithmetic: We reviewed modular inverses and how they only exist if the number and the modulus are relatively prime (gcd(a, n) = 1). We practiced finding these inverses using Fermat's Little Theorem (a^(p-2) = a^(-1) mod p).
• •Group Theory: A set is considered a group under a binary operation if it satisfies closure, associativity, identity, and inverse properties. We specifically looked at Cyclic Groups, where every element can be generated by repeatedly applying the operator to a single "generator" element.
• •Finite Fields: We explored arithmetics within fields of prime order, noting that "division" in a finite field does not suffer from precision loss and follows consistent rules for additive and multiplicative inverses.

Properties of Elliptic Curves

Next, we discussed elliptic curves defined by the formula y² = x³ + ax + b. Key takeaways included:

• •Requirements: To be useful for cryptography, a curve must have no singular points and include a "point at infinity" to act as the identity element.
• •Point Addition: We explored the geometric interpretation of adding points on a curve - drawing a line through P and Q, finding the third intersection point, and reflecting it across the x-axis.

ECC Algorithms

The meeting then transitioned to ECC algorithms and their real-world use cases:

• •ECDH (Elliptic Curve Diffie-Hellman): A key exchange protocol allowing two parties to agree on a shared secret over an insecure channel by combining their private keys with public generator points.
• •ECDSA (Elliptic Curve Digital Signature Algorithm): A method for creating unforgeable digital signatures that can be verified using a signer's public key.
• •Dual_EC_DRBG: We analyzed this infamous NSA-designed random bit generator, discussing how a hidden mathematical relationship between its public parameters could serve as a backdoor.