Additional Times T in BGV multiplication evaluation

[bluechen8]

In side the tensorStandard function (which evaluates ciphertext-ciphertext multiplication), I noticed that one of the inputs times T * 2^{64} * 2^{64}. I understand that multiplying T is used to maintain the T^{-1} in the ciphertext (otherwise it will be T^{-2} after the multiplication. But why there are double Montgomery transformation? (2^{64} * 2^{64}) I thought one is enough (applied to T).

// Multiply by T * 2^{64} * 2^{64} -> result multipled by T and switched in the Montgomery domain
ringQ.MulRNSScalarMontgomery(tmp0.Value[0], eval.tMontgomery, c00)
ringQ.MulRNSScalarMontgomery(tmp0.Value[1], eval.tMontgomery, c01)

[Pro7ech]

This is a trick to switch the ciphertext in the Montgomery domain and multiply by T with a single Montgomery multiplication.

[bluechen8]

Thanks for the reply :)
Could you please elaborate this? For example, for two 2^{64}, one is to switch ciphertext op0 to Montgomery, and the other switches the T to Montgomery. But, ciphertext op0 should already be in Montgomery domain, right? Even if it is not, why not switching ciphertext op1 to the Montgomery domain?

[Pro7ech]

The Montgomery representation is (a * 2^{64} mod q) and the Montgomery multiplication is (a * b * 2^{-64} mod q), so by scaling T by 2^{128} mod q, we obtain (a * T * 2^{128} * 2^{-64} = a * T * 2^{64}) which is the Montgomery representation of a * T