Homomorphic Time-Lock Puzzles & Verifiable Delay Studio

Architect non-interactive cryptographic time capsules with Rivest-Shamir-Wagner (RSW96) sequential squaring. Simulate trapdoor key generation via Euler's totient φ(N), test in-browser sequential squarings, and evaluate parallel-resistant unsealing horizons.

The Mathematics of RSW96 Time Capsules

Time-lock puzzles bridge number theory with computational physics:

• Hidden Order Groups: When factoring $N = p cdot q$ is intractable, the group order $phi(N)$ is unknown to all parties except the puzzle creator.
• The Trapdoor Shortcut: Knowing $phi(N)$ allows reducing the exponent $2^T$ modulo $phi(N)$. Since $2^T pmod{phi(N)}$ takes $O(log T)$ bit operations via double-and-add, a creator can prepare a 10-year puzzle in under 5 milliseconds.
• Irreducible Sequentiality: Without $phi(N)$, finding $g^{2^T} pmod N$ requires computing each intermediate square $x_{i+1} = x_i^2 pmod N$. Every squaring requires the full 1024-bit or 2048-bit result of the previous operation, making parallel distribution mathematically impossible.

VDF Integration: Wesolowski & Pietrzak Proofs

In decentralized networks (e.g. Ethereum beacon chain randomness, sealed-bid auctions), observers must verify the puzzle solution without repeating the delay:

• Wesolowski Proofs: The solver computes a single group element $pi = g^{lfloor 2^T / ell floor} pmod N$, where $ell$ is a Fiat-Shamir prime challenge $ell = H(g, y, T)$.
• Logarithmic Verification: Any node verifies the equation $pi^ell cdot g^{(2^T mod ell)} equiv y pmod N$ with only a few modular exponentiations ($< 2$ milliseconds).
• Homomorphic Time-Locks (HTLP): Puzzles can be combined homomorphically ($C_1 cdot C_2 pmod N$) such that solving the composite puzzle unlocks the sum of the secret plaintexts without revealing intermediate inputs.