Does the multiplicativeDepth determine the number of EvalMult(ciphertext, plaintext)?

FHE Questions
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1

I want to know how the multiplicationDepth affects the number of EvalMult(ciphertext, plaintext).
I set the multiplicationDepth = n. After performing n+1 plaintext-ciphertext multiplications, the result decrypted is 0.
So I want to know if I can perform more plaintext-ciphertext multiplications when I set the depth to n. Or how to set the scale factor?(The question seems to be similiar to About CKKS cipher-plain multiplication - FHE Questions - OpenFHE).
Here is a example.

using Ctx = lbcrypto::Ciphertext<lbcrypto::DCRTPoly>;
using Ptx = lbcrypto::Plaintext;
    uint32_t multDepth = 2;
    uint32_t scaleModSize = 40;
    uint32_t batchSize = 128;
    ScalingTechnique rescaleTech = FIXEDMANUAL;

    CCParams<CryptoContextCKKSRNS> parameters;
    parameters.SetMultiplicativeDepth(multDepth);
    parameters.SetScalingModSize(scaleModSize);
    parameters.SetBatchSize(batchSize);
    parameters.SetScalingTechnique(rescaleTech);

    CryptoContext<DCRTPoly> cc = GenCryptoContext(parameters);

    cc->Enable(PKE);
    cc->Enable(KEYSWITCH);
    cc->Enable(LEVELEDSHE);

    auto keys = cc->KeyGen();

    cc->EvalMultKeyGen(keys.secretKey);
    std::vector<double> x1 = {0.25, 0.5, 1.0, 2.0, 4.0};
    std::vector<double> x2 = {4, 2, 1, 0.5, 0.25};

    Plaintext ptxt1 = cc->MakeCKKSPackedPlaintext(x1);
    Plaintext ptxt2 = cc->MakeCKKSPackedPlaintext(x2);

    auto c1 = cc->Encrypt(keys.publicKey, ptxt1);
    auto c2 = cc->Encrypt(keys.publicKey, ptxt2);

    Ctx c3 = c1;
    Ptx ptx3;

    for (size_t i = 1; i < 5; ++i)
    {
        c3 = cc->EvalMult(c3, ptxt2);

        cc->Decrypt(keys.secretKey, c3, &ptx3);
        ptx3->SetLength(8);
        std::cout << ptx3 << std::endl;
    }

The output is
(1, 1, 1, 1, 1, 1.28776e-09, -5.05258e-10, -1.68733e-09, … ); Estimated precision: 30 bits

(4, 2, 1, 0.5, 0.25, -3.02947e-10, 8.86368e-10, -1.30289e-09, … ); Estimated precision: 29 bits

(3.51656e-06, 6.46768e-06, 8.11758e-07, 3.00899e-06, 2.803e-06, 9.14237e-07, 9.56325e-07, 4.59316e-06, … ); Estimated precision: 28 bits

(-3.45635e-10, 2.75771e-10, 1.30282e-10, -3.34742e-11, -1.57894e-10, -1.17082e-10, -2.21638e-10, 9.24746e-11, … ); Estimated precision: 31 bits

2

In OpenFHE, the same scaling factor is used for ciphertext-ciphertext and plaintext-ciphertext multiplications (for simplicity). This scaling factor is set using the scaleModSize variable in your example. Theoretically speaking, the scaling factor for plaintext-ciphertext multiplications can be set to a smaller number (by 10 bits or so) [if no ciphertext-ciphertext multiplications are performed at that level], but we use the same scaling factor (assuming the worst-case scenario). So the behavior you are describing is expected.

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