This program aims to find an approximate polynomial of a sine function for the bootstrapping for HEAAN (In fact, we approximate the function , but there is no big difference). Homomorphic evaluation of a sine function is the key part of the bootstrapping for HEAAN, and we suggest the bootstrapping optimized way to approximate a sine function by a polynomial in the paper "Better Bootstrapping for Approximate Homomorphic Encryption" (https://eprint.iacr.org/2019/688.pdf).
It is written in C++ and requires NTL library (with GMP). You can download each library at
We checked that the program properly works on Ubuntu 18.04.1 LTS with the libraries NTL(ver. 11.3.2) and GMP(ver. 6.1.2)
First, you can make "Makefile" by typing "cmake .".
Next, type "make all" and obtain the executable file "find_polynomial".
You can use the executable file by typing "./find_polynomial 'Degree ' 'Error' 'Scaling'".
You should put integer for each argument, and each argument has the following meaning.
- Degree : degree bound of interpolation polynomial
- Error :
of maximum deviation of test values from each i-0.25
- Scaling : the number of scaling
[Refer to "Better Bootstrapping for Approximate Homomorphic Encryption" (https://eprint.iacr.org/2019/688.pdf) for more details]
For example, you can type "./find_polynomial 30 10 2"
In summary, you should type the following commands on terminal.
- Type "cmake .".
- Type "make all".
- Type "./find_polynomial 'degree' 'error' 'scaling'"
[Example]
- Degree of polynomial : degree of an interpolation polynomial of
.
- Degree : Deg[i] is the number of nodes in
.
- Max_Error : log_2 of a maximum error of an interpolation polynomial computed through the BS-GS algorithm.
Errors between and an approximate polynomial on various tested values are stored in the directory "result/error/".
The file is of .csv format, and its name is the form of "DegxxErrxxScalingxx.csv", where xx's are argument values you put in.
[Example]
| Tested Values | Real Values | Approximate Values | Error | log_2 of Error |
|---|---|---|---|---|
| -0.250977 | -0.00613588 | -0.00613588 | 7.01431E-09 | -27.0871 |
| -0.250938 | -0.00589045 | -0.00589045 | 6.73471E-09 | -27.1457 |
| -0.250937 | -0.00564502 | -0.00564501 | 6.45502E-09 | -27.2069 |
| ... | ... | ... | ... | ... |
Coefficients for the Baby-step Giant-step algorithm introduced in the paper, which is needed for implementing the bootstrapping for HEAAN, are stored in the directory "result/coef/".
The file is of .csv format, and its name is the form of "DegxxErrxxScalingxx.csv", where xx's are argument values you put in.
[Example]
| 0.0292182 | -0.131917 | -0.295158 | -0.102132 | 0.857435 | -0.0648289 | 0.384785 | -0.0222439 |
| 0.0864300 | -0.156425 | -0.332357 | -0.110424 | -1.38072 | -0.0652592 | 0.771865 | -0.0215346 |
| 0.451401 | -0.0301473 | -0.906402 | -0.0158559 | 0.394436 | -0.00718066 | -0.138797 | -0.00200036 |
| 0.0281924 | -0.00196015 | -0.0162168 | -0.000480076 | 0.000828514 | -6.78823E-05 | -0.000500756 | 0 |
Let
be a polynomial corresponding to i-th row.
For example, .
Then, the approximate polynomial is given by
, where
is the adjusted chebyshev polynomial of deg i.