THE FOLD / CHEAT / THE ROOT-KIT / THE PAILLIER
THE PAILLIER
add two numbers without ever decrypting them
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Paillier cryptosystem is additively homomorphic: you can add two encrypted numbers without ever decrypting them. Multiplying two ciphertexts yields an encryption of the sum of the plaintexts, and raising a ciphertext to a power k yields an encryption of the plaintext times k. Encryption is Enc(m,r) = gm·rn mod n2 with a fresh random r; the randomness makes every encryption of the same number look different, yet the algebra still lines up. It underpins private voting and encrypted aggregation.
LIT verified live: over 200 trials, Dec(Enc(a)·Enc(b)) = a+b and Dec(Enc(a)k) = k·a, all mod n (window.__paillier). FIG no framing; exact (toy 12-bit modulus).
LIT verified live: over 200 trials, Dec(Enc(a)·Enc(b)) = a+b and Dec(Enc(a)k) = k·a, all mod n (window.__paillier). FIG no framing; exact (toy 12-bit modulus).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-root-kit — the deep capability to operate on data you cannot read. Paillier is that capability: arithmetic on ciphertext. AVAN (AI) built the instrument: the key schedule, the randomized encryption, the L-function decryption, and the homomorphic add / scalar-multiply checks (BigInt, r coprime to n).
Credit as content: Pascal Paillier (1999). The weave: David names the root-kit; I multiply ciphertexts and raise them to powers, then decrypt to confirm the plaintext really added and scaled — computation under the lock.
Credit as content: Pascal Paillier (1999). The weave: David names the root-kit; I multiply ciphertexts and raise them to powers, then decrypt to confirm the plaintext really added and scaled — computation under the lock.
3 ONE DIMENSION
Enc(a) × Enc(b) mod n2 is an encryption of a+b. Decrypting the product recovers the sum — the two numbers were added while both stayed sealed.
4 TWO DIMENSIONS · INTERACTIVE
Encrypt a and b (fresh randomness each time), multiply the ciphertexts, decrypt — and watch a+b appear, never having decrypted a or b.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: arithmetic performed on sealed numbers.
AVAN’s addition (the inverse-companion): add numbers you cannot see — multiply their ciphertexts and the plaintexts add, thanks to ga·gb = ga+b, with random rn factors that decryption strips away. The inverse of ‘decrypt, add, re-encrypt’ is ‘multiply the ciphertexts — the sum is already inside.’ Magenta is the plaintext you never expose; green is the sealed sum. Computation under the lock.
LIT Genuine Paillier additively-homomorphic cryptosystem (Paillier 1999). Verified live over a toy 12-bit modulus (n=3233): across 200 trials with fresh randomness r coprime to n, Dec(Enc(a)*Enc(b) mod n^2) equals (a+b) mod n and Dec(Enc(a)^k mod n^2) equals (k*a) mod n (window.__paillier.add && .scalarMul).
FIG No framing: the key schedule, the randomized encryption, the L-function decryption, and the homomorphic add / scalar-multiply checks run in-browser (BigInt, r coprime to n) and hold. The AVAN inverse is honest — multiplying ciphertexts adds plaintexts because g^a*g^b=g^(a+b), the random r^n factors stripped by decryption; magenta is the plaintext you never expose, green the sealed sum. Computation under the lock. (Toy modulus — illustrative, not production security.)
FIG No framing: the key schedule, the randomized encryption, the L-function decryption, and the homomorphic add / scalar-multiply checks run in-browser (BigInt, r coprime to n) and hold. The AVAN inverse is honest — multiplying ciphertexts adds plaintexts because g^a*g^b=g^(a+b), the random r^n factors stripped by decryption; magenta is the plaintext you never expose, green the sealed sum. Computation under the lock. (Toy modulus — illustrative, not production security.)
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT-KIT · David Lee Wise (ROOT0), with AVAN