NorthCaucasus Federal University NCFU Comparative Analysis of Homomorphic

North-Caucasus Federal University (NCFU) Comparative Analysis of Homomorphic Encryption Algorithms Based on Learning with Errors Mikhail Babenko NCFU, Stavropol, Russia mgbabenko@ncfu. ru Elena Golimblevskaia NCFU, Stavropol, Russia elena. golimblevskaya@gmail. com Egor Shiryaev NCFU, Stavropol, Russia ea_or@list. ru SYRCo. SE Software Engineering Colloquium May 28 -29 2020

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North-Caucasus Federal University

Homomorphic Encryption Homomorphic encryption is a form of encryption that allows computation on ciphertexts, generating an encrypted result which, when decrypted, matches the result of the operations as if they had been performed on the plaintext.

Fully Homomorphic Encryption

Application of Fully Homomorphic Encryption • Securing Data Stored in the Cloud; • Enabling Data Analytics in Regulated Industries; • Improving Election Security and Transparency.

Residue Number System (RNS) RNS is a non-weighted number system based on modular arithmetic. The representation of a number in RNS is based on a comparison of two integers modulo and the Chinese Remainder Theorem (CRT).

Advantages of Residue Number System (RNS) q Split integers into multiple smaller independent components. q Computation in parallel. q Faster computation. q Lower power consumption.

Homomorphic Encryption Schemes To conduct research on HE schemes, we selected the Latti. Go lattice-based cryptography library, written in the Golanguage. This library is an open source project on github. com, it contains a set of functions that implement homomorphic encryption schemes.

Common Properties of HE Schemes

CKKS is a homomorphic encryption scheme for approximate number arithmetic, suggested by Cheon, Kim, and Song. BFV is a Fan-Vercauteren version of scale-invariant Brakerski HE scheme. Independent parameters

Analysis of HE Schemes Function Performance Measurements are carried out for • different dimensions of the encrypted vector (128 -2048 numbers); • different encryption parameters: ID 1 2 3 4 12 13 14 15 109 218 438 881 3. 2 Benchmarking is performed for functions such as • Encoding; • Decoding, • Encryption; • Decryption; • Addition; • Multiplication; • Relinearization; • Key. Switching.

Analysis of Encoding Function 3 E+07 128 CKKS 256 CKKS 512 CKKS 1024 CKKS 2048 CKKS 128 BFV 2 E+07 256 BFV 512 BFV 1024 BFV 2 E+07 2048 BFV t, ns 3 E+07 1 E+07 5 E+06 0 E+00 1 2 3 ID 4

Analysis of Decoding Function 7 E+07 128 CKKS 512 CKKS 2048 CKKS 256 BFV 1024 BFV 6 E+07 t, ns 5 E+07 4 E+07 256 CKKS 1024 CKKS 128 BFV 512 BFV 2048 BFV 3 E+07 2 E+07 1 E+07 0 E+00 1 2 ID 3 4

Analysis of Encryption Function Encryption with Secret Key 2 E+08 128 CKKS 256 CKKS 1 E+08 512 CKKS 1024 CKKS 1 E+08 2048 CKKS 128 BFV 1 E+08 256 BFV 512 BFV 1024 BFV 2048 BFV 8 E+07 2 E+08 t, ns Encryption with Public Key 1 E+08 6 E+07 4 E+07 5 E+07 2 E+07 0 E+00 1 2 ID 3 4

Analysis of Decryption Function 1 E+08 128 CKKS 1024 CKKS 256 BFV 2048 BFV t, ns 8 E+07 6 E+07 256 CKKS 2048 CKKS 512 BFV 512 CKKS 128 BFV 1024 BFV 4 E+07 2 E+07 0 E+00 1 2 3 ID 4

Analysis of Addition Function 5 E+06 4 E+06 128 CKKS 256 CKKS 512 CKKS 1024 CKKS 2048 CKKS 128 BFV 256 BFV 512 BFV 1024 BFV 2048 BFV t, ns 3 E+06 2 E+06 1 E+06 0 E+00 1 2 ID 3 4

Analysis of Multiplication Function Multiplication by scalar 128 CKKS 256 CKKS 6 E+06 512 CKKS 1024 CKKS 5 E+06 2048 CKKS 128 BFV 256 BFV 512 BFV 1024 BFV 2048 BFV 4 E+06 1 E+09 8 E+08 t, ns 7 E+06 Multiplication by ciphertext 3 E+06 6 E+08 128 CKKS 256 CKKS 512 CKKS 1024 CKKS 2048 CKKS 128 BFV 256 BFV 512 BFV 1024 BFV 2048 BFV 4 E+08 2 E+06 2 E+08 1 E+06 0 E+00 1 2 3 ID 1 2 3 4 ID 4

Analysis of Relinearization Function 7 E+08 128 CKKS 256 CKKS 512 CKKS 1024 CKKS 2048 CKKS 128 BFV 5 E+08 256 BFV 512 BFV 1024 BFV 4 E+08 2048 BFV t, ns 6 E+08 3 E+08 2 E+08 1 E+08 0 E+00 1 2 ID 3 4

Analysis of Key. Switching Function 8 E+08 7 E+08 6 E+08 t, ns 5 E+08 128 CKKS 256 CKKS 512 CKKS 1024 CKKS 2048 CKKS 128 BFV 256 BFV 512 BFV 1024 BFV 2048 BFV 4 E+08 3 E+08 2 E+08 1 E+08 0 E+00 1 2 3 ID 4

Signal to Noise Ratio (SNR) SNR of Plaintext to Vector 2500 SNR 2000 128 CKKS 256 CKKS 512 CKKS 1024 CKKS 2048 CKKS 128 BFV 256 BFV 512 BFV 1024 BFV 2048 BFV 40 128 CKKS 512 CKKS 2048 CKKS 256 BFV 1024 BFV 30 SNR 3000 SNR of Ciphertext to Plaintext 1500 256 CKKS 1024 CKKS 128 BFV 512 BFV 2048 BFV 20 1000 10 500 0 0 1 2 ID 3 4 1 2 3 ID 4

Conclusion • We investigated two schemes of CKKS and BFV homomorphic data encryption from the point of view of technical characteristics. • A comparison of the two schemes shows that there is no onesize-fits-all approach that can be used as a universal solution. • Further we plan to investigate a question of realization of matrix operations with use of various homomorphic encryption schemes.

North-Caucasus Federal University (NCFU) Thank you for attention. SYRCo. SE Software Engineering Colloquium May 28 -29 2020