# BEHZ vs. HPS/default multiplication
**URL:** https://openfhe.discourse.group/t/behz-vs-hps-default-multiplication/497
**Category:** Library Questions
**Created:** [April 28, 2023, 4:29am UTC](https://openfhe.discourse.group/t/behz-vs-hps-default-multiplication/497 "2023-04-28T04:29:18Z")
**Posts on this page:** 6
**Page:** 1
### Author:  [@enehta](https://openfhe.discourse.group/u/enehta)
#### Post date: [April 28, 2023, 4:29am UTC](https://openfhe.discourse.group/t/behz-vs-hps-default-multiplication/497/1 "2023-04-28T04:29:18Z")
Hello everyone,
For the the parameters log p = 32, log q = 240, and n = 8192, we observed that with multiplication technique BEHZ we can get a multiplicative depth of 4 while with the default multiplication/HPS we only have 3. What’s the reason for this? According to [this paper](https://eprint.iacr.org/2019/1266) (also cited in the eprint document for OpenFHE), it says that BEHZ and have about the same noise level.
Many thanks for your help!!
---
### Author:  [@ypolyakov](https://openfhe.discourse.group/u/ypolyakov)
#### Post date: [April 28, 2023, 9:45pm UTC](https://openfhe.discourse.group/t/behz-vs-hps-default-multiplication/497/2 "2023-04-28T21:45:16Z")
Hi @enehta
Theoretically speaking, there should not be any noise growth difference between HPS and BEHZ. Could you post the example you are using for evaluating the noise growth so I could try it to recreate the issue?
Thanks,
Yuriy
---
### Author:  [@enehta](https://openfhe.discourse.group/u/enehta)
#### Post date: [April 30, 2023, 8:11pm UTC](https://openfhe.discourse.group/t/behz-vs-hps-default-multiplication/497/4 "2023-04-30T20:11:42Z")
Hi Yuriy,
Thanks for the response! Sure, I can:
```auto
CCParams parameters;
KeyPair keyPair;
CryptoContext cryptoContext;
parameters.SetPlaintextModulus(4293918721);
parameters.SetMultiplicativeDepth(3);
parameters.SetMaxRelinSkDeg(2);
parameters.SetSecurityLevel(HEStd_128_classic);
parameters.SetMultiplicationTechnique(BEHZ);
// parameters.SetMultiplicationTechnique(HPSPOVERQLEVELED);
parameters.SetKeySwitchTechnique(BV);
parameters.SetRingDim(8192);
cryptoContext = GenCryptoContext(parameters);
cryptoContext->Enable(PKE);
cryptoContext->Enable(LEVELEDSHE);
cryptoContext->Enable(ADVANCEDSHE);
keyPair = cryptoContext->KeyGen();
cryptoContext->EvalMultKeyGen(keyPair.secretKey);
std::vector toEncrypt(8192);
for (size_t i=0; i < 5; i++)
{
toEncrypt[i] = i;
}
for (size_t i=5; i < 8192; i++)
{
toEncrypt[i] = 0;
}
Ciphertext enc_test_vector = cryptoContext->Encrypt(keyPair.publicKey, cryptoContext->MakePackedPlaintext(toEncrypt));
for (int i = 0; i < 4; i++)
{
enc_test_vector = cryptoContext->EvalMult(enc_test_vector, enc_test_vector);
}
Plaintext plaintextMultResult;
cryptoContext->Decrypt(keyPair.secretKey, enc_test_vector, &plaintextMultResult);
std::cout << "Dec Result: " << plaintextMultResult << endl;
```
Thank you!
---
### Author:  [@ypolyakov](https://openfhe.discourse.group/u/ypolyakov)
#### Post date: [May 1, 2023, 11:45pm UTC](https://openfhe.discourse.group/t/behz-vs-hps-default-multiplication/497/5 "2023-05-01T23:45:06Z")
Thank you for posting the code. I ran some tests. The example works correctly for `BEHZ`, `HPS`, and `HPSPOVERQ`. It only fails for `HPSPOVERQLEVELED`. This behavior is expected because `HPSPOVERQLEVELED` uses the same noise estimation logic as parameter generation, when deciding to what internal modulus to switch for better efficiency (see [Revisiting Homomorphic Encryption Schemes for Finite Fields](https://eprint.iacr.org/2021/204) for more details). We never recommend using smaller multiplicative depths than the depth of the circuit to be computed. It may lead to decryption errors, and even potentially leak information about the secret key.
In summary, there is no contradiction to [Note on the noise growth of the RNS variants of the BFV scheme](https://eprint.iacr.org/2019/1266) Both `BEHZ` and `HPS` work the same way for these parameters.
---
### Author:  [@enehta](https://openfhe.discourse.group/u/enehta)
#### Post date: [May 2, 2023, 6:57am UTC](https://openfhe.discourse.group/t/behz-vs-hps-default-multiplication/497/6 "2023-05-02T06:57:35Z")
Dear Yuriy,
Thanks for quick response. I am not sure if I fully understand why HPSPOVERQLEVELED’s/leveled BFV’s failure is expected. I read that modulus switching is applied inside HE there for better efficiency. Does this implicitly mean, that the library is expecting more noise (i.e., decryption problems) already after less multiplications? I don’t see why it should increase the depth of the circuit (but I might miss something here.)
Thanks!
---
### Author:  [@ypolyakov](https://openfhe.discourse.group/u/ypolyakov)
#### Post date: [May 2, 2023, 3:39pm UTC](https://openfhe.discourse.group/t/behz-vs-hps-default-multiplication/497/7 "2023-05-02T15:39:23Z")
Hi @enehta
The circuit you are trying to execute has depth 4. The multiplicated depth you are setting is 3. In other words, you are trying an unsupported scenario. The leveled approach “distributes” the noise budget over the correct depth of 4. The other three modes do not use the leveled optimization, i.e., additional noise estimation inside homomorphic multiplication. In principle, I would not recommend using the depth of 3 in this case. If you still want to use it at your own risk, you could use any of the other three options (but the probability of decryption failure still may be high for these options as well).