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CKKS Bootstrapping

When a ciphertext is created, it has a fixed multiplication count called a level. When two cipehrtexts are multiplied, the resulting ciphertext will have the level reduced by one compared to the input ciphertexts. A ciphertext with level 0 can no longer be multiplied. It is possible to reset this level to a higher value by the bootstrapping operation.

Regular Bootstrap

The Bootstrapping operation requires the bootstrap key. The bootstrap key incorporates every single fixed rotation key that is needed by the bootstrap operation. Because of this, the bootstrap key is quite large (about 12.3GB).

from desilofhe import Engine

engine = Engine(use_bootstrap=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
bootstrap_key = engine.create_bootstrap_key(secret_key, stage_count=3)

message = [-1, 0, 1]
ciphertext = engine.encrypt(message, public_key, level=0)
bootstrapped = engine.bootstrap(
    ciphertext, relinearization_key, conjugation_key, bootstrap_key
)

Small Bootstrap Key

Alternatively, the bootstrapping operation can be performed using the rotation key and the small bootstrap key instead of the bootstrapping key. This approach reduces memory usage to around 3.8GB but makes the operation slower.

from desilofhe import Engine

engine = Engine(use_bootstrap=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
rotation_key = engine.create_rotation_key(secret_key)
small_bootstrap_key = engine.create_small_bootstrap_key(secret_key)

message = [-1, 0, 1]
ciphertext = engine.encrypt(message, public_key, level=0)
bootstrapped = engine.bootstrap(
    ciphertext,
    relinearization_key,
    conjugation_key,
    rotation_key,
    small_bootstrap_key,
)

bootstrapped_stage_count_5 = engine.bootstrap(
    ciphertext,
    relinearization_key,
    conjugation_key,
    rotation_key,
    small_bootstrap_key,
    stage_count=5,
)

Benchmark

Here are the benchmarks of the bootstrapping operation. The experiments were performed on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz for the CPU measurements and an NVIDIA GeForce RTX 5090 for the GPU measurements. These values are the averages of 10 runs.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 104.701 44.156 31.336 1.811 1.108
Small 4 68.009 28.008 19.886 1.102 0.675
Small 5 68.211 27.267 18.952 0.959 0.533
Medium 3 31.161 12.771 9.905 0.382 0.132
Medium 4 24.741 10.258 7.861 0.319 0.109
Medium 5 24.351 9.764 7.205 0.290 0.101
Large 3 28.217 11.966 9.640 0.394 0.130
Large 4 24.558 10.208 8.148 0.336 0.115
Large 5 22.141 9.017 6.947 0.286 0.099

Compact Representation

Here are the benchmarks of the bootstrapping operation using the compact representation. The experiments were performed on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz for the CPU measurements and an NVIDIA GeForce RTX 5090 for the GPU measurements. These values are the averages of 10 runs.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 96.553 42.578 29.691 1.998 1.368
Small 4 63.582 27.218 19.273 1.276 0.850
Small 5 62.344 26.509 18.525 1.142 0.720
Medium 3 29.211 12.533 9.851 0.470 0.208
Medium 4 24.006 10.396 8.072 0.408 0.191
Medium 5 23.116 9.858 7.408 0.398 0.191
Large 3 27.378 12.158 9.892 0.477 0.202
Large 4 24.188 10.593 8.631 0.438 0.197
Large 5 21.825 9.475 7.365 0.393 0.185

Bootstrap To 14 Levels

The original parameter set for bootstrapping leaves 10 levels after 3 stages. To overcome this, a new parameter set has been introduced to have 14 levels after the bootstrapping operation. Due to the limitations of the new parameter set, the only supported stage count is 3.

from desilofhe import Engine

engine = Engine(use_bootstrap_to_14_levels=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
bootstrap_key = engine.create_bootstrap_key(secret_key)

message = [-1, 0, 1]
ciphertext = engine.encrypt(message, public_key, level=0)
bootstrapped = engine.bootstrap(
    ciphertext, relinearization_key, conjugation_key, bootstrap_key
)

Small Bootstrap Key

The small boostrap key can also be used for this parameter set.

from desilofhe import Engine

engine = Engine(use_bootstrap_to_14_levels=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
rotation_key = engine.create_rotation_key(secret_key)
small_bootstrap_key = engine.create_small_bootstrap_key(secret_key)

message = [-1, 0, 1]
ciphertext = engine.encrypt(message, public_key, level=0)
bootstrapped = engine.bootstrap(
    ciphertext,
    relinearization_key,
    conjugation_key,
    rotation_key,
    small_bootstrap_key,
)

Benchmark

Here are the benchmarks of the bootstrapping operation to 14 levels. The experiments were performed on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz for the CPU measurements and an NVIDIA GeForce RTX 5090 for the GPU measurements. These values are the averages of 10 runs.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 127.894 54.147 38.694 1.799 1.167
Medium 3 37.282 15.802 12.275 0.433 0.157
Large 3 34.474 14.705 11.863 0.428 0.154

Compact Representation

Here are the benchmarks of the bootstrapping operation to 14 levels using the compact representation. The experiments were performed on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz for the CPU measurements and an NVIDIA GeForce RTX 5090 for the GPU measurements. These values are the averages of 10 runs.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 104.195 44.009 30.703 1.872 1.257
Medium 3 30.067 12.713 9.921 0.443 0.154
Large 3 27.908 12.056 9.772 0.435 0.148

Bootstrap To 17 Levels

A new parameter set has been introduced to have 17 levels after the bootstrapping operation. Due to the limitations of the new parameter set, the only supported stage count is 3.

from desilofhe import Engine

engine = Engine(use_bootstrap_to_17_levels=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
bootstrap_key = engine.create_bootstrap_key(secret_key)

message = [-1, 0, 1]
ciphertext = engine.encrypt(message, public_key, level=0)
bootstrapped = engine.bootstrap(
    ciphertext, relinearization_key, conjugation_key, bootstrap_key
)

Small Bootstrap Key

The small bootstrap key can also be used for this parameter set.

from desilofhe import Engine

engine = Engine(use_bootstrap_to_17_levels=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
rotation_key = engine.create_rotation_key(secret_key)
small_bootstrap_key = engine.create_small_bootstrap_key(secret_key)

message = [-1, 0, 1]
ciphertext = engine.encrypt(message, public_key, level=0)
bootstrapped = engine.bootstrap(
    ciphertext,
    relinearization_key,
    conjugation_key,
    rotation_key,
    small_bootstrap_key,
)

Benchmark

Here are the benchmarks of the bootstrapping operation to 17 levels. The experiments were performed on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz for the CPU measurements and an NVIDIA GeForce RTX 5090 for the GPU measurements. These values are the averages of 10 runs.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 151.952 68.810 48.566 1.961 1.241
Medium 3 46.149 19.665 15.286 0.487 0.184
Large 3 41.509 18.188 14.736 0.484 0.179

Compact Representation

Here are the benchmarks of the bootstrapping operation to 17 levels using the compact representation. The experiments were performed on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz for the CPU measurements and an NVIDIA GeForce RTX 5090 for the GPU measurements. These values are the averages of 10 runs.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 121.410 52.936 37.071 1.951 1.304
Medium 3 35.540 15.341 12.027 0.462 0.172
Large 3 33.510 14.494 11.756 0.445 0.167

Sparse Bootstrap

The bootstrapping operation is faster with a reduced slot_count. The sparse bootstrapping operation can be done with either a bootstrap key or a small bootstrap key.

Bootstrap Key

from desilofhe import Engine

engine = Engine(slot_count=1024, use_bootstrap=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
bootstrap_key = engine.create_bootstrap_key(secret_key, stage_count=3)

message = [-1, 0, 1, 0] * 256
ciphertext = engine.encrypt(message, public_key, level=0)
bootstrapped = engine.bootstrap(
    ciphertext, relinearization_key, conjugation_key, bootstrap_key
)

Small Bootstrap Key

from desilofhe import Engine

engine = Engine(slot_count=1024, use_bootstrap=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
rotation_key = engine.create_rotation_key(secret_key)
small_bootstrap_key = engine.create_small_bootstrap_key(secret_key)

message = [-1, 0, 1, 0] * 256
ciphertext = engine.encrypt(message, public_key, level=0)
bootstrapped = engine.bootstrap(
    ciphertext,
    relinearization_key,
    conjugation_key,
    rotation_key,
    small_bootstrap_key,
    stage_count=1,
)

Benchmark

Here are the benchmarks of the sparse bootstrapping operation. The experiments were performed on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz for the CPU measurements and an NVIDIA GeForce RTX 5090 for the GPU measurements. These values are the averages of 10 runs.

Slot
Count		 Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
32 Small 1 41.807 15.935 11.247 0.388 0.154
32 Small 2 30.554 12.425 8.827 0.297 0.119
32 Medium 1 27.144 11.007 8.003 0.274 0.102
32 Medium 2 25.642 10.124 7.331 0.245 0.095
32 Large 1 26.390 10.838 8.136 0.283 0.105
1024 Small 2 66.343 26.318 18.263 0.667 0.269
1024 Small 3 47.166 18.208 12.920 0.462 0.187
1024 Small 4 42.810 16.901 12.023 0.442 0.175
1024 Medium 2 29.439 12.086 9.075 0.341 0.118
1024 Medium 3 25.668 10.331 7.640 0.280 0.102
1024 Medium 4 24.269 9.675 7.109 0.266 0.097
1024 Large 2 27.871 11.549 8.977 0.332 0.119
1024 Large 3 24.264 10.138 7.726 0.281 0.104

Lossy Bootstap

The lossy bootstrap operation is faster while sacrificing some precision. The resulting significant figures below the decimal is about halved. In general, the regular bootstrapping is the recommended method, but depending on the precision requirements this method could also be useful. The lossy bootstrapping operation can be done with either a lossy bootstrap key or a small bootstrap key. The lossy bootstrapping operation is not supported for the bootstrap to 14 levels or bootstrap to 17 levels parameter sets.

Lossy Bootstrap Key

from desilofhe import Engine

engine = Engine(use_bootstrap=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
lossy_bootstrap_key = engine.create_lossy_bootstrap_key(
    secret_key, stage_count=3
)

message = [-1, 0, 1]
ciphertext = engine.encrypt(message, public_key, level=3)
bootstrapped = engine.lossy_bootstrap(
    ciphertext, relinearization_key, conjugation_key, lossy_bootstrap_key
)

Small Bootstrap Key

from desilofhe import Engine

engine = Engine(use_bootstrap=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
rotation_key = engine.create_rotation_key(secret_key)
small_bootstrap_key = engine.create_small_bootstrap_key(secret_key)

message = [-1, 0, 1]
ciphertext_level_3 = engine.encrypt(message, public_key, level=3)
bootstrapped = engine.lossy_bootstrap(
    ciphertext_level_3,
    relinearization_key,
    conjugation_key,
    rotation_key,
    small_bootstrap_key,
)

ciphertext_level_5 = engine.encrypt(message, public_key, level=5)
bootstrapped_stage_count_5 = engine.lossy_bootstrap(
    ciphertext_level_5,
    relinearization_key,
    conjugation_key,
    rotation_key,
    small_bootstrap_key,
    stage_count=5,
)

Benchmark

Here are the benchmarks of the lossy bootstrapping operation. The experiments were performed on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz for the CPU measurements and an NVIDIA GeForce RTX 5090 for the GPU measurements. These values are the averages of 10 runs.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 78.765 34.013 24.618 1.574 1.042
Small 4 54.581 22.787 16.286 1.023 0.636
Small 5 56.500 22.972 16.304 0.863 0.497
Medium 3 24.215 10.152 7.751 0.327 0.107
Medium 4 20.844 8.500 6.440 0.269 0.090
Medium 5 20.603 8.172 6.072 0.247 0.086
Large 3 22.593 9.528 7.605 0.333 0.107
Large 4 20.196 8.446 6.668 0.286 0.096
Large 5 18.667 7.607 5.838 0.243 0.084

Lossy bootstrapping is optimized for ciphertexts that contain only real values.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 73.834 31.791 22.962 1.498 1.020
Small 4 48.913 20.259 14.753 0.935 0.614
Small 5 50.502 20.878 14.786 0.789 0.480
Medium 3 18.382 7.756 5.976 0.257 0.084
Medium 4 15.042 6.190 4.747 0.194 0.069
Medium 5 14.737 6.026 4.521 0.178 0.065
Large 3 16.090 7.103 5.819 0.256 0.084
Large 4 14.142 6.127 5.018 0.218 0.075
Large 5 13.195 5.463 4.279 0.178 0.063

Sign Bootstrap

The sign bootstrapping operation is a variant of the lossy bootstrap that enables bootstrapping of sign values, i.e., -1 and 1, with higher precision. Although this is applicable only to sign values, the resulting ciphertext achieves roughly three times more significant digits. By efficiently reducing the noise of sign values, this operation facilitates fast and high-precision comparison functions such as min and max. The sign bootstrapping can be performed with either a small bootstrap key or a lossy bootstrap key. The sign bootstrapping operation is not supported for the bootstrap to 14 levels or bootstrap to 17 levels parameter sets.

Sign Bootstrap with Lossy Bootstrap Key

from desilofhe import Engine

engine = Engine(use_bootstrap=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
lossy_bootstrap_key = engine.create_lossy_bootstrap_key(
    secret_key, stage_count=3
)

message = [-1, 1]
ciphertext = engine.encrypt(message, public_key, level=3)
bootstrapped = engine.sign_bootstrap(
    ciphertext, relinearization_key, conjugation_key, lossy_bootstrap_key
)

Small Bootstrap Key

from desilofhe import Engine

engine = Engine(use_bootstrap=True)
secret_key = engine.create_secret_key()
public_key = engine.create_public_key(secret_key)
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
rotation_key = engine.create_rotation_key(secret_key)
small_bootstrap_key = engine.create_small_bootstrap_key(secret_key)

message = [-1, 1]
ciphertext_level_3 = engine.encrypt(message, public_key, level=3)
bootstrapped = engine.sign_bootstrap(
    ciphertext_level_3,
    relinearization_key,
    conjugation_key,
    rotation_key,
    small_bootstrap_key,
)

ciphertext_level_5 = engine.encrypt(message, public_key, level=5)
bootstrapped_stage_count_5 = engine.sign_bootstrap(
    ciphertext_level_5,
    relinearization_key,
    conjugation_key,
    rotation_key,
    small_bootstrap_key,
    stage_count=5,
)

Benchmark

Here are the benchmarks of the sign bootstrapping operation. The experiments were performed on an Intel(R) Core(TM) i7-10700K CPU @ 3.80GHz for the CPU measurements and an NVIDIA GeForce RTX 5090 for the GPU measurements. These values are the averages of 10 runs.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 80.301 34.065 25.102 1.554 1.038
Small 4 53.714 22.838 16.278 1.024 0.635
Small 5 56.007 22.980 16.256 0.873 0.499
Medium 3 24.672 10.255 7.749 0.326 0.107
Medium 4 20.785 8.491 6.419 0.260 0.090
Medium 5 20.055 8.245 6.059 0.248 0.086
Large 3 22.629 9.558 7.582 0.322 0.107
Large 4 19.777 8.480 6.697 0.288 0.096
Large 5 18.573 7.627 5.839 0.243 0.084

Sign bootstrapping is optimized for ciphertexts that contain only real values.

Key
Size		 Stage
Count		 Runtime (s)
1 Thread
4 Threads
16 Threads		 Sync
GPU		 Async
GPU
Small 3 72.609 35.043 23.200 1.489 1.016
Small 4 47.554 20.534 14.700 0.933 0.613
Small 5 49.238 20.838 14.762 0.788 0.480
Medium 3 18.408 7.758 5.954 0.250 0.084
Medium 4 14.733 6.204 4.757 0.193 0.069
Medium 5 14.747 6.010 4.514 0.181 0.065
Large 3 16.352 7.121 5.795 0.253 0.084
Large 4 14.097 6.185 5.017 0.216 0.074
Large 5 13.177 5.456 4.267 0.182 0.063

Comparison of Bootstrapping Error for Sign Messages

The following shows the average error observed when bootstrapping sign messages (i.e., -1 and 1) using the regular, lossy, and sign bootstrapping methods, respectively.

Regular
Bootstrap		 Lossy
Bootstrap		 Sign
Bootstrap
Average error 8.03008e-4 1.60562e-3 1.98124e-08
Precision 10.3 bits 9.3 bits 25.6 bits

Precision of Bootstrapping

The following represents the precision of the bootstrapping operation. It is the average of the bootstrapping error measured over messages uniformly distributed in the interval [-1, 1]. When evaluating the precision of the sign bootstrapping, we use only the sign values (i.e., -1 and 1).

Average error Precision Compact
Average error		 Compact
Precision
Bootstrap 6.72878e-07 20.5 bits 6.67666e-07 20.5 bits
Bootstrap to 14 Levels 7.29874e-06 17.1 bits 2.67900e-05 15.2 bits
Bootstrap to 17 Levels 1.08748e-06 19.8 bits 7.02514e-07 20.4 bits
Sparse Bootstrap 2.59681e-07 21.9 bits 2.53088e-07 21.9 bits
Lossy Bootstrap 4.01552e-4 11.3 bits 3.80117e-4 11.4 bits
Sign Bootstrap 1.98124e-08 25.6 bits 2.20190e-08 25.4 bits

The errors were measured with the following code.

Bootstrap 
import numpy as np
from desilofhe import Engine

engine = Engine(use_bootstrap=True)

secret_key = engine.create_secret_key()
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
bootstrap_key = engine.create_bootstrap_key(secret_key)

message = np.linspace(-1, 1, engine.slot_count)

ciphertext = engine.encrypt(message, secret_key)
bootstrapped = engine.bootstrap(
    ciphertext, relinearization_key, conjugation_key, bootstrap_key
)
decrypted = engine.decrypt(bootstrapped, secret_key)

average_noise = np.mean(abs(message - decrypted))
print(f"Bootstrap Average Noise :{average_noise}")
print(f"Bootstrap Average Noise (Bits) :{np.log2(average_noise)}")
Bootstrap to 14 Levels 
import numpy as np
from desilofhe import Engine

engine = Engine(use_bootstrap_to_14_levels=True)

secret_key = engine.create_secret_key()
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
bootstrap_key = engine.create_bootstrap_key(secret_key)

message = np.linspace(-1, 1, engine.slot_count)

ciphertext = engine.encrypt(message, secret_key)
bootstrapped = engine.bootstrap(
    ciphertext, relinearization_key, conjugation_key, bootstrap_key
)
decrypted = engine.decrypt(bootstrapped, secret_key)

average_noise = np.mean(abs(message - decrypted))
print(f"BootstrapTo14Levels Average Noise :{average_noise}")
print(f"BootstrapTo14Levels Average Noise (Bits) :{np.log2(average_noise)}")
Bootstrap to 17 Levels 
import numpy as np
from desilofhe import Engine

engine = Engine(use_bootstrap_to_17_levels=True)

secret_key = engine.create_secret_key()
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
bootstrap_key = engine.create_bootstrap_key(secret_key)

message = np.linspace(-1, 1, engine.slot_count)

ciphertext = engine.encrypt(message, secret_key)
bootstrapped = engine.bootstrap(
    ciphertext, relinearization_key, conjugation_key, bootstrap_key
)
decrypted = engine.decrypt(bootstrapped, secret_key)

average_noise = np.mean(abs(message - decrypted))
print(f"BootstrapTo17Levels Average Noise :{average_noise}")
print(f"BootstrapTo17Levels Average Noise (Bits) :{np.log2(average_noise)}")
Sparse Bootstrap 
import numpy as np
from desilofhe import Engine

engine = Engine(slot_count=1024, use_bootstrap=True)

secret_key = engine.create_secret_key()
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
bootstrap_key = engine.create_bootstrap_key(secret_key)

message = np.linspace(-1, 1, engine.slot_count)

ciphertext = engine.encrypt(message, secret_key)
bootstrapped = engine.bootstrap(
    ciphertext, relinearization_key, conjugation_key, bootstrap_key
)
decrypted = engine.decrypt(bootstrapped, secret_key)

average_noise = np.mean(abs(message - decrypted))
print(f"SparseBootstrap Average Noise :{average_noise}")
print(f"SparseBootstrap Average Noise (Bits) :{np.log2(average_noise)}")
Lossy Bootstrap 
import numpy as np
from desilofhe import Engine

engine = Engine(use_bootstrap=True)

secret_key = engine.create_secret_key()
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
lossy_bootstrap_key = engine.create_lossy_bootstrap_key(secret_key)

message = np.linspace(-1, 1, engine.slot_count)

ciphertext = engine.encrypt(message, secret_key)
bootstrapped = engine.lossy_bootstrap(
    ciphertext, relinearization_key, conjugation_key, lossy_bootstrap_key
)
decrypted = engine.decrypt(bootstrapped, secret_key)

average_noise = np.mean(abs(message - decrypted))
print(f"Lossy Bootstrap Average Noise :{average_noise}")
print(f"Lossy Bootstrap Average Noise (Bits) :{np.log2(average_noise)}")
Sign Bootstrap 
import numpy as np
from desilofhe import Engine

engine = Engine(use_bootstrap=True)

secret_key = engine.create_secret_key()
relinearization_key = engine.create_relinearization_key(secret_key)
conjugation_key = engine.create_conjugation_key(secret_key)
lossy_bootstrap_key = engine.create_lossy_bootstrap_key(secret_key)

message = [-1, 1] * (engine.slot_count // 2)

ciphertext = engine.encrypt(message, secret_key)
bootstrapped = engine.sign_bootstrap(
    ciphertext, relinearization_key, conjugation_key, lossy_bootstrap_key
)
decrypted = engine.decrypt(bootstrapped, secret_key)

average_noise = np.mean(abs(message - decrypted))
print(f"Sign Bootstrap Average Noise :{average_noise}")
print(f"Sign Bootstrap Average Noise (Bits) :{np.log2(average_noise)}")