a slide of Hanaoka's talk in ECC2018(https://cy2sec.comm.eng.osaka-u.ac.jp/ecc2018/program.html)

1. Practical Two-level
Homomorphic Encryption in
Prime-order Bilinear Groups
Goichiro Hanaoka*1
Joint-work-with: Nuttapong Attrapadung*1,
Shigeo Mitsunari*2, Yusuke Sakai*1,
Tadanori Teruya*1
*1 AIST, *2 Cybozu labs
2018/11/21 ECC 2018 1

2. Outline
• Background
• Two-level Homomorphic encryption
• An efficient construction
• Security
• Implementation
• Conclusion
2018/11/21 ECC 2018 2

3. Background
2018/11/21 ECC 2018 3

4. 2018/11/21 ECC 2018 4
Computing on encrypted data
• Data analysis with taking care of sensitive data
Disease Risk
70%
If X2>∑Y then
◯◯70%
F : Diagnosis
Y : Database

5. Homomorphic Encryption (HE)
• Allows computation on encrypted data
• Many applications related to privacy-preserving
schemes
• Types of HE
• Additively HE (ex. Goldwasser-Micali, Okamoto-
Uchiyama, Paillier, Lifted-ElGamal)
• Enc 𝑚 + Enc 𝑚′ = Enc(𝑚 + 𝑚′)
• Multiplicatively HE (ex. RSA, ElGamal)
• Enc 𝑚 × Enc 𝑚′ = Enc 𝑚𝑚′
• Fully HE (ex. Gentry, BGV, BV, GSW, …)
• Can do homomorphic add. and mult.
2018/11/21 ECC 2018 5

6. Pros and Cons
• Add. HE, Mult. HE
• Applications are restricted
• Fully HE (FHE)
• Any computations possible, but inefficient
• Security relies on less standard assumptions
• Leveled HE
• The number of homomorphic mult. is restricted.
• An intermediate notion between A/M HE and FHE.
2018/11/21 ECC 2018 6
A/M HE Leveled HE FHE
Efficiency very good medium bad
Functionality medium good very good

7. Two-level HE
• HE that allows one homomorphic multiplication
• Allows degree-2 polynomial homomorphic evaluations
• Allows inner product of two vectors
• 𝑥 = 𝑥1, 𝑥2, … , 𝑦 = 𝑦1, 𝑦2, …
• σ𝑖 Enc1 𝑥𝑖 × Enc1 𝑦𝑖 = Enc2 σ𝑖 𝑥𝑖 × 𝑦𝑖
2018/11/21 ECC 2018 7
×1 2
3
3 4
12
12 13
25
++
: Level-1 : Level-2

8. Applications
• Secure 2-DNF formula evaluation
• Delegated secure inner-product on encrypted data
• Efficient (symmetric) private information retrieval
• Cross tabulation on encrypted data
• Efficient election protocol
• …
2018/11/21 ECC 2018 8

9. Existing Two-level HE
• Boneh, Goh, Nissim (TCC 2005)
• Based on Composite-order pairings, hence much less efficient
• Freeman (EUROCRYPT 2010)
• Composite-to-prime-order transformation framework, applied to BGN
• Herold, Hesse, Hofheinz, Rafols, Rupp (CRYPTO 2014)
• Improving Freeman’s frameworks
• Only Type 1 pairings, inefficient
• Catalano, Fiore (ACM CCS 2015)
• Transformation from d-Level HE to (2d)-level
• Instantiations are not necessarily efficient
• AHM+ (AsiaCCS 2018): This talk
• Efficient construction based on the lifted-ElGamal encryption
• Portable high-speed implementations
• Note:
• Decryption in all these schemes requires discrete log (DL)
• Hence plaintext space should be sufficiently small (up to 32-bit)
2018/11/21 ECC 2018 9

10. An Efficient Construction of
Two-level HE
2018/11/21 ECC 2018 10

11. Basic Idea
•Existing schemes
• Establish a “broader fundamental &
theoretical framework”
• Then, construct L2HE as an “application”
•Our scheme
• Concentrate on “L2HE-dedicated design”
• Start from “promising tools” for fast HE,
i.e. Type-3 pairing and ElGamal
• Not general but fully tuned for L2HE
2018/11/21 ECC 2018 11

12. An Efficient Construction
• Combine the lifted-ElGamal encryption scheme
with Type 3 pairings
• First, straightforwardly construct two-level HE
• Then, consider “simpler” construction
• While Freeman considered a conversion of composite-to-
prime order
• Level-1 (L1) ciphertext (CT) is same as lifted-ElGamal
• Format of level-2 (L2) CT is same as Freeman’s scheme
• Note: Type 3 pairings
• Cyclic groups 𝔾1, 𝔾2, 𝔾T of order prime 𝑝 with
bilinear map 𝑒: 𝔾1 × 𝔾2 → 𝔾T
• 𝑒 𝑎𝑃, 𝑏𝑄 = 𝑒 𝑃, 𝑄 𝑎𝑏 for 𝑎, 𝑏 ∈ ℤ 𝑝, 𝑃 ∈ 𝔾1, 𝑄 ∈ 𝔾2
• 𝔾1 ≠ 𝔾2 and no efficient map between 𝔾1 and 𝔾22018/11/21 ECC 2018 12

13. Summary of Constructions
2018/11/21 ECC 2018 13
Freeman (EUROCRYPT 2018)
AHM+ (AsiaCCS 2018, this talk)
BGN scheme
based on
composite order
BGN (Freeman)
scheme based on
prime order
(includes 2-level HE)
Construction by
converting
Lifted-ElGamal
Enc 𝑚 = 𝑔 𝑚ℎ 𝑟, 𝑔 𝑟
Type 3 pairing
𝑒 𝑎𝑃1, 𝑏𝑃2 = 𝑔T
𝑎𝑏
AHM+
2-level HE scheme
Construction by
combining
algebraic structures

14. Setup and Key Generation
• Setup
• Cyclic group 𝔾𝑖 = ⟨𝑃𝑖⟩ over an elliptic curve with
prime order 𝑝 for 𝑖 = 1, 2
• 𝔾T = 𝑔T , where 𝑔T = 𝑒 𝑃1, 𝑃2
• Key generation
• Secret key 𝑠1, 𝑠2 ∈ ℤ 𝑝 is generated at random
• Public key 𝑄1 = 𝑠1 𝑃1, 𝑄2 = 𝑠2 𝑃2
(with optional precomputation
𝑧1 = 𝑔T, 𝑧2 = 𝑔T
𝑠1, 𝑧3 = 𝑔T
𝑠2, 𝑧4 = 𝑔T
𝑠1 𝑠2)
• Note: Colors
• Green: Public part
• Blue: Secret and hidden part
2018/11/21 ECC 2018 14

15. Level-1 CT and Enc./Dec.
• Encrypt
• Plaintext 𝑚 and randomness 𝑟
• Enc 𝔾 𝑖
𝑚 = (𝑚𝑃𝑖 + 𝑟𝑄𝑖, 𝑟𝑃𝑖) for 𝑖 = 1, 2
• Duplicated form:
Enc1 𝑚 ≔ Enc 𝔾1
𝑚 , Enc 𝔾2
𝑚
• Note: 𝔾1 can be mult. with 𝔾2 only, vice versa, so that
duplicated form is needed for general usage
• Decrypt
• For 𝑖 = 1, 2, decrypt Enc 𝔾 𝑖
𝑚 = (𝑆, 𝑇) by
𝑆 − 𝑠𝑖 𝑇 = 𝑚𝑃𝑖 + 𝑟𝑄𝑖 − 𝑠𝑖 𝑟𝑃𝑖 = 𝑚𝑃𝑖
and then, to obtain 𝑚, solve DL
• Almost same as lifted-ElGamal
2018/11/21 ECC 2018 15

16. Homomorphic Addition on L1 CT
• For 𝑖 = 1, 2,
Enc 𝔾 𝑖
𝑚1 + Enc 𝔾 𝑖
𝑚2
= 𝑚1 𝑃𝑖 + 𝑟1 𝑄𝑖, 𝑟1 𝑃𝑖 + 𝑚2 𝑃𝑖 + 𝑟2 𝑄𝑖, 𝑟2 𝑃𝑖
= 𝑚1 + 𝑚2 𝑃𝑖 + 𝑟1 + 𝑟2 𝑄𝑖, 𝑟1 + 𝑟2 𝑃𝑖
= Enc 𝔾 𝑖
(𝑚1 + 𝑚2)
• Also, same as lifted-ElGamal
2018/11/21 ECC 2018 16
1 2
3
+
: Level-1

17. Homomorphic Multiplication
• 𝐶1 = 𝑆1, 𝑇1 = 𝑚1 𝑃1 + 𝑟1 𝑄1, 𝑟1 𝑃1 = Enc 𝔾1
𝑚1 ∈ 𝔾1
2
• 𝐶2 = 𝑆2, 𝑇2 = 𝑚2 𝑃2 + 𝑟2 𝑄2, 𝑟2 𝑃2 = Enc 𝔾2
𝑚2 ∈ 𝔾2
2
• 𝐶1 × 𝐶2 ≔ 𝑒 𝑆1, 𝑆2 , 𝑒 𝑆1, 𝑇2 , 𝑒 𝑇1, 𝑆2 , 𝑒 𝑇1, 𝑇2
= 𝑧1
𝑚1 𝑚2 𝑧4
𝜏′, 𝑧2
𝜎′, 𝑧3
𝜌′, 𝑧1
𝜎′+𝜌′−𝜏′
= Enc2 𝑚1 𝑚2 ∈ 𝔾T
4
• 𝑧1 = 𝑔T, 𝑧2 = 𝑔T
𝑠1, 𝑧3 = 𝑔T
𝑠2, 𝑧4 = 𝑔T
𝑠1 𝑠2
• Tensor product of 𝐶1, 𝐶2
• Its result is an level-2 ciphertext
2018/11/21 ECC 2018 17
×3 4
12
: Level-1
: Level-2

18. Homomorphic Addition on L2 CT
• Enc2 𝑚1 + Enc2 𝑚2
= 𝑧1
𝑚1 𝑧4
𝜏1, 𝑧2
𝜎1, 𝑧3
𝜌1, 𝑧1
𝜎1+𝜌1−𝜏1
+ 𝑧1
𝑚2 𝑧4
𝜏2, 𝑧2
𝜎2, 𝑧3
𝜌2, 𝑧1
𝜎2+𝜌2−𝜏2
= ( 𝑧1
𝑚1+𝑚2 𝑧4
𝜏1+𝜏2, 𝑧2
𝜎1+𝜎2,
൯𝑧3
𝜌1+𝜌2, 𝑧1
(𝜎1+𝜎2)+(𝜌1+𝜌2)−(𝜏1+𝜏2)
= Enc2(𝑚1 + 𝑚2)
• Usual vector addition
2018/11/21 ECC 2018 18
12 13
25
+
: Level-2

19. Decryption for Level-2 CT
• Decrypting an level-2 ciphertext 𝑐1, 𝑐2, 𝑐3, 𝑐4
Dec2 c1, c2, 𝑐3, 𝑐4 ≔
𝑐1 𝑐4
𝑠1 𝑠2
𝑐2
𝑠2
𝑐3
𝑠1
=
𝑒 𝑆1, 𝑆2 𝑒 𝑠1 𝑇1, 𝑠2 𝑇2
𝑒 𝑆1, 𝑠2 𝑇2 𝑒 𝑠1 𝑇1, 𝑆2
= 𝑒 𝑆1 − 𝑠1 𝑇1, 𝑆2 − 𝑠2 𝑇2
= 𝑒 𝑚𝑃1 , 𝑚′
𝑃2 = 𝑒 𝑃1, 𝑃2
𝑚𝑚′
then solve DLP to obtain 𝑚𝑚′
• Note: 𝑐1, 𝑐2, 𝑐3, 𝑐4 =
𝑧1
𝑚𝑚′
𝑧4
𝜏
, 𝑧2
𝜎
, 𝑧3
𝜌
, 𝑧1
𝜎+𝜌−𝜏
∈ 𝔾T
4
,
where 𝑧1 = 𝑔T, 𝑧2 = 𝑔T
𝑠1, 𝑧3 = 𝑔T
𝑠2, 𝑧4 = 𝑔T
𝑠1 𝑠2
2018/11/21 ECC 2018 19

20. Size and Benchmark on BN462
• Note:
• Use x64 Linux on Core i7-6700
• Without compressed form
• Use lookup tables for decryption
(20-bit plaintext)
2018/11/21 ECC 2018 20
Calc. time
in msec
Enc1 0.452
Enc2 1.14
Dec1 9.01
Dec2 10.01
ReRand1 0.447
ReRand2 1.14
Add1 0.0109
Add2 0.0231
Mult 8.47
Bit size
Secret key 924
Public key 27720
Dup. L1 CT 5544
L2 CT 22176

21. Comparison of Size
• Fre10: Freemen’s scheme (EUROCRYPT 2010)
• Compare bit size on a 462-bit Barreto-Naehrig
(BN) curve
2018/11/21 ECC 2018 21

22. Comparison of Time
• CT: Ciphertext
• Fre10: Freemen’s scheme in EUROCRYPT 2010
• Compare calculation time on a 462-bit BN curve
2018/11/21 ECC 2018 22

23. Proving the Knowledge of Plaintexts
• Zero-knowledge proof protocols can be applied
• Example 1: Duplicated form of L1 CT
• Dup. L1 CT is Enc 𝔾1
𝑚 , Enc 𝔾2
𝑚′
• Attach a proof of “𝑚 = 𝑚′”
• Example 2: Proving a CT encrypts a bit
• Attach a proof of “encrypted plaintext is 0 or 1”
• Applications: Voting, two-party computation
2018/11/21 ECC 2018 23

24. Proof of Equality
• Duplicated L1 CT:
• Enc 𝔾1
𝑚 , Enc 𝔾2
𝑚′ = 𝐶1, 𝐶2 , 𝐶3, 𝐶4
= (𝑚𝑃1 + 𝜌𝑄1, 𝜌𝑃1), (𝑚′𝑃2 + 𝜎𝑄2, 𝜎𝑃2)
where 𝜌, 𝜎 ← ℤ 𝑝 are randomly chosen
• Should be “𝑚 = 𝑚′”
• Equality can be proved in the same way of NIZK
DH-tuple proof
2018/11/21 ECC 2018 24

25. NIZK Proof of Equality
• L1 CT: 𝐶1, 𝐶2 , 𝐶3, 𝐶4
= (𝑚𝑃1 + 𝜌𝑄1, 𝜌𝑃1), (𝑚′𝑃2 + 𝜎𝑄2, 𝜎𝑃2)
• Prove:
• Randomly choose: 𝑟𝜌, 𝑟𝜎, 𝑟 𝑚 ← ℤ 𝑝
• 𝑅1, 𝑅2, 𝑅3, 𝑅4 ← 𝑟 𝑚 𝑃1 + 𝑟𝜌 𝑄1, 𝑟𝜌 𝑃1, 𝑟 𝑚 𝑃2 + 𝑟𝜎 𝑄2, 𝑟𝜎 𝑃2
• 𝑐 ← 𝐻 public param, 𝐶1, 𝐶2, 𝐶3, 𝐶4, 𝑅1, 𝑅2, 𝑅3, 𝑅4
• 𝑠𝜌, 𝑠 𝜎, 𝑠 𝑚 ← 𝑟𝜌 + 𝑐𝜌, 𝑟𝜎 + 𝑐𝜎, 𝑟 𝑚 + 𝑐𝑚
• Proof 𝜋 = 𝑐, 𝑠𝜌, 𝑠 𝜎, 𝑠 𝑚
• Verify:
• 𝑐 = 𝐻 public param, 𝐶1, 𝐶2, 𝐶3, 𝐶4, 𝑅1
′
, 𝑅2
′
, 𝑅3
′
, 𝑅4
′
where
𝑅1
′
, 𝑅2
′
, 𝑅3
′
, 𝑅4
′
← 𝑠 𝑚 𝑃1 + 𝑠𝜌 𝑄1 − 𝑐𝐶1, 𝑠𝜌 𝑃1 − 𝑐𝐶2, 𝑠 𝑚 𝑃2 + 𝑠 𝜎 𝑄2 − 𝑐𝐶3, 𝑠 𝜎 𝑃2 − 𝑐𝐶4
2018/11/21 ECC 2018 25

26. Security
2018/11/21 ECC 2018 26

27. Confidentiality
• Shown scheme is IND-CPA secure under the SXDH
assumption
• Note1: IND-CPA (INDistinguishability against
Chosen Plaintext Attack)
• Hidden plaintext from ciphertext
• Standard base-line security notion
• Note2: SXDH (Symmetric eXternal Diffie-Hellman)
assumption
• 𝑃1 ∈ 𝔾1, 𝑃2 ∈ 𝔾2, for random 𝛼, 𝛽, 𝛾,
𝑃1, 𝛼𝑃1, 𝛽𝑃1, 𝛼𝛽𝑃1 ≈ 𝑃1, 𝛼𝑃1, 𝛽𝑃1, 𝛾𝑃1 and
𝑃2, 𝛼𝑃2, 𝛽𝑃2, 𝛼𝛽𝑃2 ≈ 𝑃2, 𝛼𝑃2, 𝛽𝑃2, 𝛾𝑃2
are computationally indistinguishable
2018/11/21 ECC 2018 27

28. Circuit Privacy
• Shown scheme is circuit private
• Namely, ReRand𝑖 𝑐 ≈ Enc𝑖(Dec𝑖 𝑐 )
• Rerandomization: ReRand𝑖 𝑐 ≔ 𝑐 + Enc𝑖(0)
• ReRand𝑖 𝑐 removes a trace of circuit from 𝑐
• Note: Arithmetic circuit depends on secret
• E.g., for 𝑖 = 1, 2, and for a secret integer 𝑛,
𝑛 × Enc𝑖 𝑚 = ෍
𝑗=1
𝑛
Enc𝑖 𝑚 = Enc𝑖 𝑛𝑚
• Should be Enc𝑖 𝑚 + Enc𝑖 𝑚′ ≈ Enc𝑖 𝑚 + 𝑚′ and
Enc1 𝑚 × Enc1 𝑚′ ≈ Enc2 𝑚𝑚′
• Note: It is obvious that CTs are in which group
𝔾1, 𝔾2, 𝔾T
2018/11/21 ECC 2018 28

29. Implementation
2018/11/21 ECC 2018 29

30. Practical Two-level
Homomorphic Encryption in
Prime-order Bilinear Groups
Goichiro Hanaoka*1
Joint-work-with: Nuttapong Attrapadung*1,
Shigeo Mitsunari*2, Yusuke Sakai*1,
Tadanori Teruya*1
*1 AIST, *2 Cybozu labs
2018/11/21 ECC 2018 30

31. Our Implementation
• Available in “mcl”: A library for pairings
• BN254, 381, 462, BLS12-381
• C++: https://github.com/herumi/mcl
• Web browser/Node.js:
https://github.com/herumi/she-wasm
• High-performance implementation for
x64/ARM64
• WebAssembly (wasm)
• Runs on Microsoft Edge, Firefox, Chrome, Safari
without any plug-ins
• Open source: BSD 3-clause
2018/11/21 ECC 2018 31

32. Benchmarks on wasm
• Calculation times in msec
• Use BN254
• Use lookup tables for decryption (20-bit plaintext)
2018/11/21 ECC 2018 32
Native (x64) JavaScritpt with wasm
x64 Linux on
Core i7-7700
Firefox on
Core i7-7700
Safari on
iPhone 7
Enc 𝔾1
0.018 0.3 0.96
Enc 𝔾2
0.048 0.82 1.72
Add 𝔾1
0.00062 0.016 0.016
Add 𝔾2
0.002 0.036 0.048
Mult 1.17 15.6 24.3
Dec2 0.66 7.8 12.6

33. Demo
2018/11/21 ECC 2018 33

34. Importance of WebAssembly (wasm)
Implementation
• Large deployment advantages
• wasm is a portable and fast binary instruction format
• Runs on many modern browser
• Microsoft Edge, Safari, Google Chrome, and Mozilla Firefox on
Windows, Linux, macOS, iPhone, Android, and so on…
• Requires no plugins
• Being developed as a web standard via the W3C
• Distribution is easy
2018/11/21 ECC 2018 34

35. Demonstrations of wasm
• Inner product:
https://herumi.github.i
o/she-wasm/she-
demo.html
• Oblivious transfer:
https://ppdm.jp/ot/
2018/11/21 ECC 2018 35

36. Conclusion
• Practical efficient two-level homomorphic
encryption
• Many times add. and one-time mult. on encrypted data
• Based on Type 3 (asymmetric) pairing
• Combine the lifted-ElGamal encryption scheme
• Faster than Freeman’s scheme (EUROCRYPT 2010)
• Portable high-performance implementation
• C++/asm/WebAssembly
• https://github.com/herumi/mcl
• https://github.com/herumi/she-wasm
• Open source: BSD 3-clause
2018/11/21 ECC 2018 36
Thank you!