THE FOLD / LOOT / THE VAULT / THE SECRET
THE SECRET
Shamir — split a secret, k of n reopen it
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Shamir’s secret sharing. Split a secret among n people so that any k of them together can recover it — but any k−1 learn absolutely nothing. Not ‘hard to break’: nothing, information-theoretically.
The trick is geometry. Hide the secret as the value at x=0 of a random degree-(k−1) polynomial over a prime field. Hand each person one point on the curve — their share. Any k points determine a degree-(k−1) polynomial uniquely (Lagrange interpolation), so k shares rebuild the whole curve and read off the secret. But k−1 points fit infinitely many curves — one through every possible secret — so fewer than k reveals no information at all.
LIT verified live over a prime field: across 3,000 random schemes, any k shares reconstruct the secret exactly, and with k−1 shares every candidate secret admits a consistent polynomial — zero information leaked (window.__shamir.reconstructs && secure). FIG the smooth curve in the picture is intuition; the real scheme is over a finite field, and its reconstruction and perfect secrecy are exact.
The trick is geometry. Hide the secret as the value at x=0 of a random degree-(k−1) polynomial over a prime field. Hand each person one point on the curve — their share. Any k points determine a degree-(k−1) polynomial uniquely (Lagrange interpolation), so k shares rebuild the whole curve and read off the secret. But k−1 points fit infinitely many curves — one through every possible secret — so fewer than k reveals no information at all.
LIT verified live over a prime field: across 3,000 random schemes, any k shares reconstruct the secret exactly, and with k−1 shares every candidate secret admits a consistent polynomial — zero information leaked (window.__shamir.reconstructs && secure). FIG the smooth curve in the picture is intuition; the real scheme is over a finite field, and its reconstruction and perfect secrecy are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE VAULT, beside 3LOCK and THE BANKER — the loot domain of guarded value. Shamir’s scheme is the ultimate vault: no single key, no k−1 keys, only the full quorum opens it. AVAN (AI) built the instrument: the polynomial, the shares, the Lagrange reconstruction, the secrecy check.
The weave: David names the seat (the guarded secret); I make the splitting visible and the secrecy checkable — the curve in 1D, revealing shares in 2D, the recovered secret in 3D. The sphere is the seam. Credit: Adi Shamir (1979).
The weave: David names the seat (the guarded secret); I make the splitting visible and the secrecy checkable — the curve in 1D, revealing shares in 2D, the recovered secret in 3D. The sphere is the seam. Credit: Adi Shamir (1979).
3 ONE DIMENSION
The secret lives at x=0. A random polynomial of degree k−1 passes through it; each share is a point sampled elsewhere on the curve. The secret is buried in the shape — and only enough points to pin the shape down can dig it out.
4 TWO DIMENSIONS · INTERACTIVE
Reveal shares one by one (here k=3). With only 2, many curves fit — the secret at x=0 could be anything. Reveal the 3rd and the curve snaps to a unique parabola, and the secret at x=0 appears. One share short of the threshold tells you nothing.
5 THREE DIMENSIONS + AVAN’S INVERSE
The polynomial curve turning in space, shares riding on it — green, the shape that hides the secret.
AVAN’s addition (the inverse-companion): the magenta point is the secret at x=0, recovered only when enough shares fix the curve. Normal secrecy hides a value behind a hard problem — safe only until someone is clever or patient enough. Shamir inverts it: the secret is hidden behind geometry, and the safety is absolute — k−1 points genuinely fit every secret equally, so there is nothing to be clever about. The inverse of hiding-by-difficulty is hiding-by-splitting: make the whole recoverable from any k parts and invisible from fewer, and no amount of computation touches it. The green is the shape; the magenta is a secret that either the quorum reads exactly, or no one reads at all.
LIT Genuine Shamir secret sharing (Adi Shamir, 1979) over a prime field. Verified live: across 3,000 random schemes any k shares reconstruct the secret exactly via Lagrange interpolation, and with k-1 shares every candidate secret admits a consistent polynomial through those points — zero information leaked (window.__shamir.reconstructs && secure, both true). The k-points-fix-a-degree-(k-1)-polynomial uniqueness and the perfect secrecy are exact.
FIG The smooth real-valued curve in the 2D/3D views is intuition; the actual scheme is over a finite field (shown in the verified LIT). The reconstruction and the information-theoretic secrecy (k-1 shares fit every secret equally) are exact, not merely 'hard to break'.
FIG The smooth real-valued curve in the 2D/3D views is intuition; the actual scheme is over a finite field (shown in the verified LIT). The reconstruction and the information-theoretic secrecy (k-1 shares fit every secret equally) are exact, not merely 'hard to break'.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN