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I'm trying to use the Microsoft SEAL library in order to do Matrix multiplication. That's why I'm trying to find a way to compute the Dot Product of 2 vectors.

My issue is that the CKKS encoder in SEAL encodes entire vectors. So if I had a 2D vector of floats I get a 1D vector of Plaintext (and then a 1D vector of Ciphertext after encryption).

The operations that I am able to do are: addition, component-wise multiplication, exponential, and rotation. In order to do a dot product of vectors I need to multiply the components of the first vector by the components of the second vector and sum them up. If I want to multiply 2 matrices, I can transpose the second matrix and multiply the rows together but I am unable to compute the sum of the elements inside the Ciphertext. Is it possible to get the sum of those elements in the Ciphertext? Should I change my approach?

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    $\begingroup$ I don't understand.. Can you already do a homomorphic dot product? Because if you can do so, then you can of course multiply matrices. If you cannot, then what about using the rotation n-1 times to add all the components into the first entry? $\endgroup$
    – Hilder Vitor Lima Pereira
    Feb 17, 2020 at 7:59
  • $\begingroup$ @HilderVitorLimaPereira No I cannot do a homomorphic dot product in SEAL. However you are correct, there was an issue on Github that was just answered that solved the issue with the method you proposed. $\endgroup$
    – Marwan N
    Feb 17, 2020 at 13:32

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As suggested by Hilder Vitor Lima Pereira in the comments and KyoohyungHan on Github in the issue https://github.com/microsoft/SEAL/issues/138, it is possible to sum the elements with rotations:

For example, {1, 2, 3, 4} is encrypted in a ciphertext. You can get encrypted {3, 4, 1, 2} using homomorphic rotation by index 2. With homomorphic addition, you have encrypted {4, 6, 4, 6}. After that, you can get encrypted {6, 4, 6, 4} using homomorphic rotation by index 1. Finally, you have encrypted {10, 10, 10, 10} with homomorphic addition which is the sum of the element in a ciphertext.

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