# Weird behavior on Chebyshev

**URL:** <https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630>\
**Category:** Library Questions\
**Created:** [June 13, 2023, 6:07pm UTC](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630 "2023-06-13T18:07:58Z")\
**Posts on this page:** 8\
**Page:** 1

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**Author:** ![marc-alonso](https://avatars.discourse-cdn.com/v4/letter/m/6f9a4e/32.png) [@marc-alonso](https://openfhe.discourse.group/u/marc-alonso)\
**Post date:** [June 13, 2023, 6:07pm UTC](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630/1 "2023-06-13T18:07:58Z")

</div>

Hello, I am using chebyshev approximation to evaluate exp(x). The problem is that, even though values are in the interval, results are messy.

I am using 2^16 ring with security 128 bits, this is the key part of the code:

```auto
vector<double> v = {-5, -4, -3, -2, -1, 0, 0.1, 0.2, 0.3, 1, 2, 3, 4, 5, 6, 7, 8};
Plaintext p = cryptoContext->MakeCKKSPackedPlaintext(v, 1, 0, nullptr, numSlots);
Ciphertext<DCRTPoly> c = crypto_context->Encrypt(key_pair.publicKey, p);
c = crypto_context->EvalChebyshevFunction([](double x) -> double { return exp(x); }, c, -10, 10, 27);

```

When i print the decrypted results, i get something like this:

```auto
(-2.88821e+07, -2.86466e+07, -2.81768e+07, -2.72802e+07, -2.57516e+07, -2.33729e+07, -2.30795e+07, -2.2775e+07, -2.24593e+07, ...

```

Am I doing something wrong?

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<div class="post-metadata">

**Author:** ![iquah](https://yyz1.discourse-cdn.com/flex031/user_avatar/openfhe.discourse.group/iquah/32/8_2.png) [@iquah](https://openfhe.discourse.group/u/iquah)\
**Post date:** [June 13, 2023, 6:58pm UTC](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630/2 "2023-06-13T18:58:42Z")

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For the given range of values (-10,10) using only 27 degrees is insufficient. Try bumping it up by a lot and gradually reduce the degree to trade off between the accuracy and multiplication depth you want.

[openfhe-development/src/pke/examples/FUNCTION\_EVALUATION.md at main · openfheorg/openfhe-development · GitHub](https://github.com/openfheorg/openfhe-development/blob/main/src/pke/examples/FUNCTION_EVALUATION.md) might be useful for you to determine the depth to set your program to

I’m marking this as closed, but feel free to open it up again if your problem isn’t solved.

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<div class="post-metadata">

**Author:** ![marc-alonso](https://avatars.discourse-cdn.com/v4/letter/m/6f9a4e/32.png) [@marc-alonso](https://openfhe.discourse.group/u/marc-alonso)\
**Post date:** [June 13, 2023, 7:23pm UTC](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630/3 "2023-06-13T19:23:41Z")

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Same happens with degree 200 and 288.

Output with 288:

```auto
(-2783.7, -2783.69, -2783.66, -2783.58, -2783.34, -2782.71, -2782.61, -2782.49, -2782.36, -2780.99, -2776.32, -2763.63, -2729.11, -2635.3, -2380.28, -1687.08, 197.247, -2782.71, -2782.71, -2782.71, -2782.71, ...

```

I don’t think is related to the degree, the function is pretty easy to approximate (with gsl\_chebyshev a degree=27 returns an average error of 10^-6)

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<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:** [June 14, 2023, 3:09am UTC](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630/4 "2023-06-14T03:09:42Z")

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@marc-alonso Could you post your complete example here, including the cryptocontext parameters? `exp` grows fast with an argument (i.e., exponentially). It is important to know what multiplicative depth you are using. You may need room (extra levels) not to cause an overflow over your ciphertext/plaintext modulus when computing e^{10}, which is about 22,000.

Overall, it is not the best scenario for Chebyshev interpolation as you have both very small and very large values in this range, i.e., you would probably need a relatively large CKKS scaling factor for this case to achieve desired precision. Depending on your application, I would probably consider working in the logarithmic scale, i.e., switching the problem from exponential to linear.

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**Author:** ![Pro7ech](https://yyz1.discourse-cdn.com/flex031/user_avatar/openfhe.discourse.group/pro7ech/32/301_2.png) [@Pro7ech](https://openfhe.discourse.group/u/Pro7ech)\
**Post date:** [June 14, 2023, 7:18am UTC](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630/5 "2023-06-14T07:18:17Z")

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This is most likely a plaintext overflow. The CKKS plaintext isn’t an element of the ciphertext ring \mathbb{Z}[X]\_{Q\_{\ell}}/(X^{N}+1) (Q\_{\ell} = \prod\_{i=0}^{\ell} q\_{i}), it is an element of \mathbb{R}[X]/(X^{N}+1), so to remain well define its infinity norm must remain small than Q\_{\ell}/2 during all steps of the computation.

Check that the ratio between the remaining modulus Q\_{\ell} after the evaluation and the plaintext scaling factor is greater than ~2\exp(x). If not, then you’ll have a modular reduction by Q\_{\ell} on the plaintext during the evaluation and this will corrupt the plaintext.

Getting a resulting plaintext with large values that all look aline such as

```auto
(-2783.7, -2783.69, -2783.66, -2783.58, -2783.34, -2782.71, -2782.61, -2782.49, -2782.36, -2780.99, -2776.32, -2763.63, -2729.11, -2635.3, -2380.28, -1687.08, 197.247, -2782.71, -2782.71, -2782.71, -2782.71, ...

```

is typical of a plaintext overflow.

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<div class="post-metadata">

**Author:** ![narger](https://yyz1.discourse-cdn.com/flex031/user_avatar/openfhe.discourse.group/narger/32/420_2.png) [@narger](https://openfhe.discourse.group/u/narger)\
**Post date:** [June 14, 2023, 10:36am UTC](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630/6 "2023-06-14T10:36:57Z")

</div>

I suggest you to take advantage of the fact that e^x = (e^{x/10})^{10} and approximate e^{x/10} in [-1, 1] (it might take more levels but it is more stable for sure).

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<div class="post-metadata">

**Author:** ![iquah](https://yyz1.discourse-cdn.com/flex031/user_avatar/openfhe.discourse.group/iquah/32/8_2.png) [@iquah](https://openfhe.discourse.group/u/iquah)\
**Post date:** [June 16, 2023, 9:10pm UTC](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630/7 "2023-06-16T21:10:58Z")

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@marc-alonso

Has the issue been resolved?

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<div class="post-metadata">

**Author:** ![marc-alonso](https://avatars.discourse-cdn.com/v4/letter/m/6f9a4e/32.png) [@marc-alonso](https://openfhe.discourse.group/u/marc-alonso)\
**Post date:** [June 26, 2023, 9:22am UTC](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630/8 "2023-06-26T09:22:28Z")

</div>

Yes, I used this approach:

> [@Weird behavior on Chebyshev](https://openfhe.discourse.group/t/weird-behavior-on-chebyshev/630/6):
>
> I suggest you to take advantage of the fact that e^x = (e^{x/10})^{10} and approximate e^{x/10} in [-1, 1] (it might take more levels but it is more stable for sure).

Thank you