'Introduction to Fully Homomorphic Encryption' vitalikbuterin encryption
introduced by Brakerski and Vaikuntanathan in 2011, and show how to bootstrap it. Here, we will move away from keys and ciphertexts being. Given a key k=k1,k2....kn, to encrypt a message m, construct a vector c=c1,c2...cn such that the inner product =c1k1+c2k1+...+cnkn, modulo some fixed number p, equals m+2e where m is the message , and e is a small "error" term.
So this outer-product approach works. But there is, as you may have already noticed, a catch: the size of the ciphertext, and the key, grows quadratically.procedure. The holder of the private key k provides, as part of the public key, a "relinearization key", which you can think of as "noisy" encryptions of k⊗k under k.
And what do we have in our relinearization key? A bunch of elements of the form 2d∗kp∗kq, noisy-encrypted under k, for every possible combination of p and q! Having all the powers of two in our relinearization key allows us to generate any ∗kp∗kq by just adding up ≤log powers of two together for each pair.enc+enc+enc+Note that each noisy-encryption in the relinearization key has some even error 2e, and the equation itself has some error: if =2e1+m1 and =2e2+m2, then = ∗= 2+m1m2.
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