Perform Addition on a Product using the S.E.A.L. Library

I am trying to perform an operation that is of the form: (A * B) + C. The multiplication works fine, as all the numbers have the same scale at that point, but the product of A * B has a different scale than C. It makes sense that multiplication would change the scale, but I was wondering if there was a way to perform an operation like this using the SEAL library.

Environment information:

1. Language: C++
2. Encryption Scheme: CKKS
3. Small encoded doubles (eg. 0.4531)
4. Scale used for encoding: pow(2.0, 60) like the example

Thank you in advance and let me know if further information is needed.

There are multiple ways of getting this to work. For example, suppose ciphertexts A, B, C all have the same scale Z. Then A * B will have scale Z^2. At this point you should also relinearize A * B unless you have a strong reason not to.

To compute A * B + C, you could for instance:

• re-encode C (if you have the plaintext) with scale Z^2 and use that instead;
• use multiply_plain to multiply C with a scalar 1.0 plaintext with scale Z to increase the scale to Z^2 but keep the value the same (there is an overload for CKKSEncoder::encode for this);
• rescale A * B first so it has scale Z^2/q_k where q_k is the last prime in the coeff_modulus. Now, you could re-encode C to have scale exactly Z^2/q_k (if you have the plaintext), or multiply C with a scalar 1.0 plaintext as explained above to change the scale to exactly Z^2/q_k;
• if Z is close to q_k so that Z^2/q_k ~ Z, then after rescaling you might just be able to use double &Ciphertext::scale() to set the scale of A * B to exactly C.scale() at the cost of a small multiplicative error ~ Z/q_k. For example, instead of scale 2^60 for A, B, C you could use static_cast<double>(parms.coeff_modulus().back()). Then Z^2/q_k = Z (exactly) and the addition works immediately without any scale switching. Of course, this doesn't work so well anymore after a second multiply+rescale as the second to last prime can no longer be equal to Z (all primes in coeff_modulus must be distinct).