Polly Cracker scheme is one of the first SWHE that applies two operations, i.e., multiplication and addition over the ciphertexts. Background By the term Polly Cracker-type cryptosystem, we mean a family of cryptosystems starting from the early 1990s that propose to base their security on the difficulty of computing Grobner bases. In its public key version and the most simple form, the public key is an ideal I in a polynomial ring (given by sufficiently many polynomials of degree b from I) and the secret key is a Grobner basis for I consisting of polynomials of degree d≤ b. let I be some ideal in P := F[x0, . . . , xn−1]. Denote an injective function mapping bit strings to elements in the quotient ring P/I by Encode (·), and its inverse by Decode (·). If Decode (Encode (m0) ◦ Encode (m1)) = m0 ◦ m1 for ◦ ∈ {+, ·}
we can encrypt a message m as c = f + Encode (m) for f randomly chosen in I.
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