hemalkavaiya / fhe

Implementation of Encryption and Decryption in Homomorphic Encryption Algorithm

FHE

Implementation of Encryption and Decryption in Homomorphic Encryption Algorithm.

Used Pyfhel library in python

which includes

1. BFV Scheme which is more suited for arithmetic on integers and decrypt the exact result.
2. CKKS Scheme which is more suited for arithmetic on real numbers and give approximate close result. and it also provides bootstrapping for CKKS.

Pyfhel Uses Microsoft SEAL as a Backend.

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Object of Pyfhel is Created using

HE = Pyfhel()

Keys are Generated using

HE.keyGen()

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For Example let x = 3 and y = 5 in BFV Scheme :-

Cipher text:

cx, cy = obj.encrypt(x=[3], y=[5])

Addition, Multiplication and Subtraction:

csum = cx + cy
cmul = cx * cy
csub = cx - cy

Ciphertext-ciphertext multiplications increase the size of the polynoms representing the resulting ciphertext. To prevent this growth, the relinearization technique is used (typically right after each c-c mult) to reduce the size of a ciphertext back to the minimal size. For this, a special type of public key called Relinearization Key is used.

Initializing Relinearization Key

HE.relinKeyGen()

And to relinearize a multiplication ~ sign is used

Before using ~cmul :- <Pyfhel Ciphertext at 0x1d8043f0840, scheme=bfv, size=3/3, noiseBudget=114>

~cmul

After using ~cmul :- <Pyfhel Ciphertext at 0x1d8043f0840, scheme=bfv, size=2/3, noiseBudget=114>

Decryption :-

BFV Scheme gives exact result

obj.decrypt("csum (cx + cy)", csum)
Decryption of csum (cx + cy) > [8 0 0 ... 0 0 0]

obj.decrypt("csub (cx - cy)", csub)
Decryption of csub (cx - cy) > [-2  0  0 ...  0  0  0]

obj.decrypt("cmul (cx * cy)", cmul)
Decryption of cmul (cx * cy) > [15  0  0 ...  0  0  0]

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Example with CKKS Scheme :-

Equation

Encrypt(3xy + x) = Encrypt(3xy) + Encrypt(x) = [3 * Encrypt(x) * Encrypt(y)] + Encrypt(x)

Encryption

cx, cy = obj.encrypt(x=[1.1, 2.2], y=[3.3, 4.4])

3xy = [3 * Encrypt(x) * Encrypt(y)]

_3xy = ~(3 * cx * cy)

Encrypt(3xy + x) = Encrypt(3xy) + Encrypt(x)

eq = _3xy + cx

Decryption :-

obj.decrypt("Equation Result", eq)
Decryption of Equation Result > [ 1.19936477e+01  3.12497888e+01  1.12081209e-06 ...  1.08117709e-05 -1.27877279e-06 -3.55203472e-06]

Decryption of above equation with BFV Scheme (x = 3, y = 5) :-

obj.decrypt("Equation Result", eq)
Decryption of Equation Result > [48  0  0 ...  0  0  0]