Partially Homomorphic Encryption – RSA
Prof Bill Buchanan, The Cyber Academyhttp://asecuritysite.com
Homomorphic Encryption
Fully homomorphic: DGHV, BGV, NTRU, LWE Partially homomorphic: RSA, Pallier, ElGamal
RSA
Link
Pick two prime number (P and Q)
P = 11, Q = 3
N = P x Q = 33
PHI = (P-1) x (Q-1) =20
Pick e so that it is relative prime to PHI
eg 3, 7, 9, etc.
Let e=3
(d x e) mod PHI = 1
(d x 3) mod PHI = 1
d = 7
RSAE = Me mod NM = Ed mod N
So with M=4, e=3, d=7 and N=33
E = 43 mod 33 = D = 337 mod 33 =
Homomorphic Encryption (Multiply)
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In RSA we can perform a multiplication
Cipher1 = V1e mod NCipher 2 = V2e mod N
Cipher1 x Cipher 2 = V1e x V2e mod N = (V1 x V2)e mod N
Homomorphic Encryption (Division)
Link
In RSA we can perform a division
Cipher1 = V1e mod NCipher 2 = V2e mod N
Cipher1 Cipher 2 = V1e ÷ V2e mod N = (V1 ÷ V2)e mod N =(V1 x V2-1)e mod N Calculate inverse V2 (mod N) here
Partially Homomorphic Encryption – RSA
Prof Bill Buchanan, The Cyber Academyhttp://asecuritysite.com