# CKKS Bootstrapping during multiplication gives weird output

**URL:** <https://openfhe.discourse.group/t/ckks-bootstrapping-during-multiplication-gives-weird-output/1696>\
**Category:** FHE Questions\
**Created:** [November 6, 2024, 2:53pm UTC](https://openfhe.discourse.group/t/ckks-bootstrapping-during-multiplication-gives-weird-output/1696 "2024-11-06T14:53:18Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![bhavinmoriya](https://avatars.discourse-cdn.com/v4/letter/b/e68b1a/32.png) [@bhavinmoriya](https://openfhe.discourse.group/u/bhavinmoriya)\
**Post date:** [November 6, 2024, 2:53pm UTC](https://openfhe.discourse.group/t/ckks-bootstrapping-during-multiplication-gives-weird-output/1696/1 "2024-11-06T14:53:18Z")

</div>

Hi,

I have been playing with bootstrapping to multiply a given vector with 3. When I multiply by 3^20 the results get weird. May I know what is the problem. The code is a minor modification from the original code on OpenFHE.

```auto
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/*

Example for CKKS bootstrapping with sparse packing

*/

#define PROFILE

#include "openfhe.h"

using namespace lbcrypto;

void BootstrapExample(uint32_t numSlots);

int main(int argc, char *argv[])
{
    // We run the example with 8 slots and ring dimension 4096 to illustrate how to run bootstrapping with a sparse plaintext.
    // Using a sparse plaintext and specifying the smaller number of slots gives a performance improvement (typically up to 3x).
    BootstrapExample(8);
}

void BootstrapExample(uint32_t numSlots)
{
    // Step 1: Set CryptoContext
    CCParams<CryptoContextCKKSRNS> parameters;

    // A. Specify main parameters
    /* A1) Secret key distribution
     * The secret key distribution for CKKS should either be SPARSE_TERNARY or UNIFORM_TERNARY.
     * The SPARSE_TERNARY distribution was used in the original CKKS paper,
     * but in this example, we use UNIFORM_TERNARY because this is included in the homomorphic
     * encryption standard.
     */
    SecretKeyDist secretKeyDist = UNIFORM_TERNARY;
    parameters.SetSecretKeyDist(secretKeyDist);

    /* A2) Desired security level based on FHE standards.
     * In this example, we use the "NotSet" option, so the example can run more quickly with
     * a smaller ring dimension. Note that this should be used only in
     * non-production environments, or by experts who understand the security
     * implications of their choices. In production-like environments, we recommend using
     * HEStd_128_classic, HEStd_192_classic, or HEStd_256_classic for 128-bit, 192-bit,
     * or 256-bit security, respectively. If you choose one of these as your security level,
     * you do not need to set the ring dimension.
     */
    parameters.SetSecurityLevel(HEStd_NotSet);
    parameters.SetRingDim(1 << 12);

    /* A3) Key switching parameters.
     * By default, we use HYBRID key switching with a digit size of 3.
     * Choosing a larger digit size can reduce complexity, but the size of keys will increase.
     * Note that you can leave these lines of code out completely, since these are the default values.
     */
    parameters.SetNumLargeDigits(3);
    parameters.SetKeySwitchTechnique(HYBRID);

    /* A4) Scaling parameters.
     * By default, we set the modulus sizes and rescaling technique to the following values
     * to obtain a good precision and performance tradeoff. We recommend keeping the parameters
     * below unless you are an FHE expert.
     */
#if NATIVEINT == 128 && !defined( __EMSCRIPTEN__ )
    // Currently, only FIXEDMANUAL and FIXEDAUTO modes are supported for 128-bit CKKS bootstrapping.
    ScalingTechnique rescaleTech = FIXEDAUTO;
    usint dcrtBits = 78;
    usint firstMod = 89;
#else
    // All modes are supported for 64-bit CKKS bootstrapping.
    ScalingTechnique rescaleTech = FLEXIBLEAUTO;
    usint dcrtBits = 59;
    usint firstMod = 60;
#endif

    parameters.SetScalingModSize(dcrtBits);
    parameters.SetScalingTechnique(rescaleTech);
    parameters.SetFirstModSize(firstMod);

    /* A4) Bootstrapping parameters.
     * We set a budget for the number of levels we can consume in bootstrapping for encoding and decoding, respectively.
     * Using larger numbers of levels reduces the complexity and number of rotation keys,
     * but increases the depth required for bootstrapping.
     * We must choose values smaller than ceil(log2(slots)). A level budget of {4, 4} is good for higher ring
     * dimensions (65536 and higher).
     */
    std::vector<uint32_t> levelBudget = {3, 3};

    /* We give the user the option of configuring values for an optimization algorithm in bootstrapping.
     * Here, we specify the giant step for the baby-step-giant-step algorithm in linear transforms
     * for encoding and decoding, respectively. Either choose this to be a power of 2
     * or an exact divisor of the number of slots. Setting it to have the default value of {0, 0} allows OpenFHE to choose
     * the values automatically.
     */
    std::vector<uint32_t> bsgsDim = {0, 0};

    /* A5) Multiplicative depth.
     * The goal of bootstrapping is to increase the number of available levels we have, or in other words,
     * to dynamically increase the multiplicative depth. However, the bootstrapping procedure itself
     * needs to consume a few levels to run. We compute the number of bootstrapping levels required
     * using GetBootstrapDepth, and add it to levelsAvailableAfterBootstrap to set our initial multiplicative
     * depth.
     */
    uint32_t levelsAvailableAfterBootstrap = 10;
    usint depth = levelsAvailableAfterBootstrap + FHECKKSRNS::GetBootstrapDepth(levelBudget, secretKeyDist);
    // std::cout<<"FHECKKSRNS::GetBootstrapDepth(levelBudget, secretKeyDist);"<<FHECKKSRNS::GetBootstrapDepth(levelBudget, secretKeyDist);
    parameters.SetMultiplicativeDepth(depth);

    // Generate crypto context.
    CryptoContext<DCRTPoly> cryptoContext = GenCryptoContext(parameters);

    // Enable features that you wish to use. Note, we must enable FHE to use bootstrapping.
    cryptoContext->Enable(PKE);
    cryptoContext->Enable(KEYSWITCH);
    cryptoContext->Enable(LEVELEDSHE);
    cryptoContext->Enable(ADVANCEDSHE);
    cryptoContext->Enable(FHE);

    usint ringDim = cryptoContext->GetRingDimension();
    std::cout << "CKKS scheme is using ring dimension " << ringDim << std::endl
              << std::endl;

    // Step 2: Precomputations for bootstrapping
    cryptoContext->EvalBootstrapSetup(levelBudget, bsgsDim, numSlots);

    // Step 3: Key Generation
    auto keyPair = cryptoContext->KeyGen();
    cryptoContext->EvalMultKeyGen(keyPair.secretKey);
    // Generate bootstrapping keys.
    cryptoContext->EvalBootstrapKeyGen(keyPair.secretKey, numSlots);

    // Step 4: Encoding and encryption of inputs
    // Generate random input
    std::vector<double> x;
    std::random_device rd;
    std::mt19937 gen(rd());
    std::uniform_real_distribution<> dis(0.0, 1.0);
    for (size_t i = 0; i < numSlots; i++)
    {
        x.push_back(dis(gen));
    }
    x = {0.0222817, 0.18545, 0.209035, 0.0086332, 0.0220301, 0.875544, 0.219529, 0.376861};
    // Encoding as plaintexts
    // We specify the number of slots as numSlots to achieve a performance improvement.
    // We use the other default values of depth 1, levels 0, and no params.
    // Alternatively, you can also set batch size as a parameter in the CryptoContext as follows:
    // parameters.SetBatchSize(numSlots);
    // Here, we assume all ciphertexts in the cryptoContext will have numSlots slots.
    // We start with a depleted ciphertext that has used up all of its levels.
    // Plaintext ptxt = cryptoContext->MakeCKKSPackedPlaintext(x, 1, depth - 1, nullptr, numSlots);
    Plaintext ptxt = cryptoContext->MakeCKKSPackedPlaintext(x, 1, 0, nullptr, numSlots);
    ptxt->SetLength(numSlots);
    std::cout << "Input: " << ptxt << std::endl;

    // Encrypt the encoded vectors
    Ciphertext<DCRTPoly> ciph = cryptoContext->Encrypt(keyPair.publicKey, ptxt);

    std::cout << "Initial number of levels remaining: " << depth - ciph->GetLevel()<< std::endl;

    // Step 5: Perform the bootstrapping operation. The goal is to increase the number of levels remaining
    // for HE computation.
    auto ciphertextAfter = cryptoContext->EvalBootstrap(ciph);

    std::cout << "Number of levels remaining after bootstrapping: " << depth - ciphertextAfter->GetLevel() << std::endl
              << std::endl;

    // Step 7: Decryption and output
    Plaintext result;
    cryptoContext->Decrypt(keyPair.secretKey, ciphertextAfter, &result);
    result->SetLength(numSlots);
    std::cout << "Output after bootstrapping \n\t" << result << std::endl;

    // Bootstrap_ntime(x)=Bootstrap(x)
    for (size_t i = 0; i < 29; i++)
    {
        // ciphertextAfter = cryptoContext->EvalBootstrap(ciphertextAfter);
        // ciphertextAfter = cryptoContext->EvalAdd(ciphertextAfter,2);
        // ciphertextAfter = cryptoContext->EvalAdd(ciphertextAfter,ciphertextAfter);
        ciphertextAfter = cryptoContext->EvalMult(ciphertextAfter,3);

        // ciphertextAfter = cryptoContext->EvalMult(ciphertextAfter,ciphertextAfter);
        // auto ciphertextTwoIterations = cryptoContext->EvalBootstrap(ciph, numIterations, precision);
        // https://github.com/openfheorg/openfhe-development/blob/7b8346f4eac27121543e36c17237b919e03ec058/src/pke/examples/iterative-ckks-bootstrapping.cpp#L175C5-L175C97
        // ciphertextAfter = cryptoContext->EvalBootstrap(ciphertextAfter,2,30);
        // ciphertextAfter = cryptoContext->EvalBootstrap(ciphertextAfter,2,17);

        ciphertextAfter = cryptoContext->EvalBootstrap(ciphertextAfter);

        std::cout << "Number of levels remaining after bootstrapping: " << depth - ciphertextAfter->GetLevel() << std::endl
                  << std::endl;

        // Step 7: Decryption and output
        // Plaintext result;
        cryptoContext->Decrypt(keyPair.secretKey, ciphertextAfter, &result);
        result->SetLength(numSlots);
        std::cout << "Output after " << i + 2 << "-th bootstrapping \n\t" << result << std::endl;
    }
}

```

The output wrong output I get is,

```Output
	(7.76915e+07, 6.46624e+08, 7.2886e+08, 3.01021e+07, 7.68142e+07, 3.05283e+09, 7.6545e+08, 1.31403e+09, ... ); Estimated precision: 17 bits

Number of levels remaining after bootstrapping: 11

Output after 22-th bootstrapping 
	(-367.7, -2043.2, -32.9019, 456.583, 1033.32, 910.398, 2948.52, 2115.58, ... ); Estimated precision: 16 bits

Number of levels remaining after bootstrapping: 11

Output after 23-th bootstrapping 
	(133.518, -1984.94, -854.728, 301.341, 84.9585, -238.205, 4111.82, -245.682, ... ); Estimated precision: 16 bits

Number of levels remaining after bootstrapping: 11

Output after 24-th bootstrapping 
	(-692.562, -1220.03, 548.961, -843.712, -147.574, -669.181, 4834.61, 1753.75, ... ); Estimated precision: 14 bits

Number of levels remaining after bootstrapping: 11

Output after 25-th bootstrapping 
	(203.905, -2061.53, -29.8208, -366.293, 1063.94, 1538.95, 3953.17, 324.673, ... ); Estimated precision: 13 bits

Number of levels remaining after bootstrapping: 11

Output after 26-th bootstrapping 
	(-1672.04, -1027.19, 550.593, -109.611, 529.485, -525.458, 4442.74, 219.507, ... ); Estimated precision: 12 bits

Number of levels remaining after bootstrapping: 11

Output after 27-th bootstrapping 
	(-133.95, -1354.19, -432.28, -287.268, 1580.93, -558.664, 4627.9, 1683.16, ... ); Estimated precision: 11 bits

Number of levels remaining after bootstrapping: 11

Output after 28-th bootstrapping 
	(-529.766, -2484.34, -293.372, 429.385, -435.883, 615.414, 3950.3, -250.882, ... ); Estimated precision: 9 bits

Number of levels remaining after bootstrapping: 11

Output after 29-th bootstrapping 
	(-1372, -1407.39, 1247.81, -1029.61, 841.563, -853.237, 4307.84, 1104.41, ... ); Estimated precision: 8 bits

Number of levels remaining after bootstrapping: 11

Output after 30-th bootstrapping 
	(-244.028, 2.93125, -958.362, 816.723, 411.217, 43.7412, 4156.76, 975.984, ... ); Estimated precision: 6 bits

```

Thanks very much.  
Bhavin.

---

<div class="post-metadata">

**Author:** ![ypolyakov](https://yyz1.discourse-cdn.com/flex031/user_avatar/openfhe.discourse.group/ypolyakov/32/47_2.png) [@ypolyakov](https://openfhe.discourse.group/u/ypolyakov)\
**Post date:** [November 11, 2024, 10:06pm UTC](https://openfhe.discourse.group/t/ckks-bootstrapping-during-multiplication-gives-weird-output/1696/2 "2024-11-11T22:06:51Z")

</div>

Please see [Questions on CKKS bootstrapping with some computations after bootstrapping - #5 by ypolyakov](https://openfhe.discourse.group/t/questions-on-ckks-bootstrapping-with-some-computations-after-bootstrapping/535/5) The high-level idea is that the message should be close [-1,1] in magnitude for CKKS bootstrapping to achieve good precision. In your case, the messages become too large, and this is why you get bad results.

---

<div class="post-metadata">

**Author:** ![bhavinmoriya](https://avatars.discourse-cdn.com/v4/letter/b/e68b1a/32.png) [@bhavinmoriya](https://openfhe.discourse.group/u/bhavinmoriya)\
**Post date:** [November 12, 2024, 9:58am UTC](https://openfhe.discourse.group/t/ckks-bootstrapping-during-multiplication-gives-weird-output/1696/3 "2024-11-12T09:58:22Z")

</div>

Thank you very much for the detailed clarification. I have few more questions,

1. If |x|,|y| \<1, then I could multiply y forever with x, or things get jumpy near zero.
2. If I have to scale down the number to [-1,1], I have to decrypt the ciphertext and set up an if else statement to do the scaling.

Thanks very much.

---

<div class="post-metadata">

**Author:** ![ypolyakov](https://yyz1.discourse-cdn.com/flex031/user_avatar/openfhe.discourse.group/ypolyakov/32/47_2.png) [@ypolyakov](https://openfhe.discourse.group/u/ypolyakov)\
**Post date:** [November 14, 2024, 3:22am UTC](https://openfhe.discourse.group/t/ckks-bootstrapping-during-multiplication-gives-weird-output/1696/4 "2024-11-14T03:22:59Z")

</div>

> [@bhavinmoriya](#):
>
> If |x|,|y| \<1|x|,|y|\<1|x|,|y| \<1, then I could multiply yyy forever with xxx, or things get jumpy near zero.

If the encrypted value gets close to 0, then the same error will occur.

> [@bhavinmoriya](#):
>
> If I have to scale down the number to [-1,1][−1,1][-1,1], I have to decrypt the ciphertext and set up an if else statement to do the scaling.

You can scale it down homomorphically by multiplying by a double less than 1 if you can estimate the range of encrypted values.