the reference answer GPT ᴼ ᴾ ᴱ ᴺ ᴬ ᴵ < br / > multiplicative homomorphic yption scheme can be through the use of certain mathematical operation technology, converting plaintext multiplication operation in the cipher text addition operation. This scheme can perform multiplication without exposing any encrypted information.
The following is a simple multiplicative homomorphic encryption scheme:
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Choose two large prime numbers p and q, and calculate n = p * q, as the public key.
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Select a number g, satisfying that g is the original root of n, as the public key.
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Select an integer a at random and compute g to the a-modulo n as the private key.
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Encrypted plaintext x: Select a random number r and calculate y = g^x * r^n mod n, where y is ciphertext.
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Encrypted plaintext y: Select a random number s and calculate z = g^y * s^n mod n, where z is ciphertext.
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Calculate the product: Calculate w = y * z mod n, where w is the ciphertext of a*b.
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Decrypt: Decrypt w as plain text w', calculate w' = w^(1/a) mod n, where a is the private key.
can prove that this scheme can implement multiplicative homomorphic encryption, that is, for ciphertext y and z, the decrypted plaintext is y' and z', Y '* z =(g ^ * r ^ n) x ^ ab * ^ ^ ^ y * s n(g) ab mod n =(g ^(abx + aby) * r ^(abn) * s ^(abn) mod n = g ^(abx + aby) *(r ^ n) ^ ab *(s^n)^ab mod n = g^ab(x+y) * y * z mod n = a*b
The main advantage of this scheme is that it is easy to use and can implement multiplicative homomorphic encryption. The disadvantage is that the security is not high enough, and the private key can be solved by attack. In practice, more complex schemes are needed to improve security.
