Elliptic Curve & Post-Quantum Cryptography

The math behind modern keys

1:00

How elliptic-curve cryptography achieves RSA-level security with far smaller keys — and why quantum computers will break both, making post-quantum cryptography the urgent replacement.

By the end, you can

Explain what a one-way trapdoor function is and give the ECC and RSA examples.
Describe how elliptic-curve point addition works geometrically.
Explain scalar multiplication and why double-and-add is efficient.
State the ECDLP and explain why it is the security foundation of ECC.
Compare ECC and RSA key sizes at equivalent security levels (256-bit vs. 3072-bit).
Explain how Shor's algorithm breaks both ECC and RSA, and why ECC is no safer.
Distinguish Grover's weaker impact on symmetric crypto from Shor's full break of public-key crypto — Grover only gives a square-root speedup (doubling the key length restores security), while Shor factors and solves discrete logs efficiently, breaking RSA and ECC outright.
Describe the harvest-now-decrypt-later threat and identify which data is most at risk.
Name the three main PQC families and the hard problem each exploits.
Identify the three NIST PQC standards finalized in August 2024 and their roles.
Explain the purpose and security guarantee of hybrid classical + PQC key exchange.

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