Homework 1: Public-Key Cryptography

1. Experimental Observations (Varying k)

By experimenting with different scalar values ($k$), the resulting coordinates $x$ and $y$ on the discrete plane modulo 17 demonstrate apparent uniformity. The points do not cluster linearly but are scattered pseudo-randomly across the finite field space. There is zero linear correlation between input and output: advancing the private key from $k$ to $k+1$ causes the resulting point to jump to a completely unrelated geometric position, exhibiting extreme unpredictability.

2. One-Way Functions vs Cryptographic Hash Functions

Relevant for ECC (Cryptographic One-Way Functions):

• Hardness of Inversion (ECDLP): The chaotic scattering of points ensures that deriving $k$ from $Q$ requires exhaustive search.
• Algebraic Structure: The algorithm preserves a strict underlying additive homomorphism. We mathematically know that $k_1G + k_2G = (k_1 + k_2)G$. This structure is mandatory for protocols like ECDSA.

Relevant specifically for Cryptographic Hash Functions:

• Destruction of Structure: A hash function must not possess exploitable algebraic relationships (homomorphisms are a flaw in hashing).
• Strict Avalanche Criterion: Hash functions are explicitly designed so that a 1-bit change in an input of arbitrary length diffuses uniformly across a fixed-length bit array. In ECC, the domain is strictly mapped to valid points in the curve's subgroup.