Outsourcing computation, privately Homomorphic evaluation function: Eval: f, Enc(x) -> Enc(f(x)) Fully homomorphic encryption Fully homomorphic = correctness for any efficient f = correctness for universal set Approximate eigenvector method 基于GSW13的特征向量的构造，我们可以在ciphertext上计算加法和乘法，然后可以通过secret key恢复出message的计算结果。但是这个方法不安全，因为特征向量很容易被找到。 **idea：**使用approximate eigenvectors，在secret key右乘cipher text Learning with errors (LWE) R05 Rearranging notation **basic idea: ** we have a matrix A, we can generate a matrix s, such that sA=$\eta$ Encryption scheme from LWE Encryption Decryption it can be generalized to matrices.

Microsoft SEAL Ref https://www.youtube.com/watch?v=XaYEHnaAg8M

What’s is CKKS Plain Computation Encrypted Computation bool, int (uint 64), modulo p BGV, BFV, TFHE double (float) CKKS Approximate arithmetic For floating-point arithmetic, we keep the bits of significand to be the same. For fixed-point arithmetic, we keep the scaling factor to be the same. Algorithms in CKKS Encoding and decoding 在编码过程中需要使用比较大的scaling factor，这样可以保证更大的精度。 Encrypt and decrypt Ref https://www.youtube.com/watch?v=iQlgeL64vfo