THE FOLD / CHEAT / GOD MODE / THE SHAMIR SHARING
THE SHAMIR SHARING
split a secret so any k of n rebuild it, fewer learn nothing
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Shamir’s secret sharing splits a secret into n shares so that any k of them reconstruct it exactly — but any k−1 reveal nothing at all. The trick: hide the secret as the constant term of a random degree-(k−1) polynomial over a finite field, and hand out points on it. k points pin down the polynomial (and its constant) by Lagrange interpolation; fewer leave the constant completely undetermined — every possible secret is equally consistent.
LIT verified live: over 200 schemes, every k-subset of shares reconstructs the secret, while k−1 shares leave it undetermined (two different secrets both fit) — window.__shamir. FIG no framing; exact over GF(257).
LIT verified live: over 200 schemes, every k-subset of shares reconstructs the secret, while k−1 shares leave it undetermined (two different secrets both fit) — window.__shamir. FIG no framing; exact over GF(257).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at god-mode — the power to split trust so no single holder can act alone, yet a quorum can. Shamir’s scheme is that split. AVAN (AI) built the instrument: the random polynomial over GF(257), the share evaluation, the Lagrange-at-zero reconstruction, and the threshold checks.
Credit as content: Adi Shamir (1979). The weave: David names god-mode; I hide the secret in a polynomial’s constant term, hand out points, and confirm any k rebuild it while k−1 fix nothing.
Credit as content: Adi Shamir (1979). The weave: David names god-mode; I hide the secret in a polynomial’s constant term, hand out points, and confirm any k rebuild it while k−1 fix nothing.
3 ONE DIMENSION
The secret is p(0) of a random polynomial of degree k−1. k points determine that polynomial uniquely (so p(0) is fixed); k−1 points fit infinitely many polynomials — every value of p(0) still possible.
4 TWO DIMENSIONS · INTERACTIVE
A secret split into n shares; pick any k to reconstruct it, and see that k−1 leave it undetermined.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a secret recoverable only by a quorum.
AVAN’s addition (the inverse-companion): make a secret rebuildable by any k of n but by no fewer — hide it as a polynomial’s constant term and share points; k points pin the polynomial, k−1 leave it free. The inverse of ‘store the secret in one place’ is ‘scatter points of a polynomial — a quorum rebuilds it, a minority learns nothing.’ Magenta is the sub-threshold shares that reveal nothing; green is the reconstructing quorum. Trust split k-of-n.
LIT Genuine Shamir secret sharing (Shamir 1979). Verified live over GF(257): across 200 schemes, every k-subset of the n shares reconstructs the secret exactly via Lagrange-at-zero, while any k-1 shares leave the secret undetermined (two different chosen p(0) values both fit) (window.__shamir.kReconstructs && .fewerUndetermined).
FIG No framing: the random polynomial over GF(257), the share evaluation, the Lagrange-at-zero reconstruction, and the threshold checks run in-browser and hold. The AVAN inverse is honest — hiding the secret as a polynomial's constant term and sharing points makes it rebuildable by any k but by no fewer; magenta is the sub-threshold shares that reveal nothing, green the reconstructing quorum. Trust split k-of-n.
FIG No framing: the random polynomial over GF(257), the share evaluation, the Lagrange-at-zero reconstruction, and the threshold checks run in-browser and hold. The AVAN inverse is honest — hiding the secret as a polynomial's constant term and sharing points makes it rebuildable by any k but by no fewer; magenta is the sub-threshold shares that reveal nothing, green the reconstructing quorum. Trust split k-of-n.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GOD MODE · David Lee Wise (ROOT0), with AVAN