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Multiparty Computation with Low Communication, Computation and Interaction via Threshold FHE Bar-Ilan University Gilad Asharov UCLA Abhishek Jain NYU Adriana López-Alt Tel-Aviv University Eran Tromer University of Toronto Vinod Vaikuntanathan IBM Research Daniel Wichs

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2-Party Computation Using FHE (semi-honest) y a b y = f(a,b) Y A =Encrypt(a) Y=Eval(f,A,B) Charlie Sally

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Advantages Low round complexity Low communication complexity Independent of the function f Independent of Sally’s input b Low computation Charlie’s work is independent of f A simple template Can we get all these advantages in the multiparty case?

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Threshold Key Generation Key Generation

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Threshold Key Generation Key Generation

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Input Encryption A B CD a c b d A=Enc(a) B=Enc(b) C=Enc(c) D=Enc(d)

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Homomorphic Evaluation ABCD Y ABCD Y ABCD YABCD Y

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Delegate to a Cloud ABCD Homomorphic Evaluation Y

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Threshold Decryption Dec Y Y Y Y

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Threshold Decryption Dec m m m m

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MPC with Threshold FHE Threshold Key Gen Encrypt and Evaluate Threshold Decryption

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MPC with TFHE Threshold KeyGen and Threshold Dec can be implemented using generic MPC Advantages: Low communication complexity (even in malicious) The homomorphic evaluation can be delegated / only one party Disadvantages: Needs generic MPC techniques Round complexity can be high Threshold Key Gen Encrypt and Evaluate Threshold Decryption Threshold Key Gen Encrypt and Evaluate Threshold Decryption

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Our Main Results Threshold KeyGen and Threshold Dec algebraically [BV11b, BGV12] (based on LWE) Advantages: Low communication complexity (even in malicious) The homomorphic evaluation can be delegated / only one party Simple: there is no need for generic MPC protocol Extremely low round complexity Only 3 broadcast rounds (CRS model) 2 rounds reusable PKI – optimal(!) Threshold Key Gen Encrypt and Evaluate Threshold Decryption Threshold Key Gen Encrypt and Evaluate Threshold Decryption

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Our Main Results (malicious) Threshold KeyGen and Threshold Dec algebraically [BV11b, BGV12] (based on LWE) Advantages: Low communication complexity (even in malicious) The homomorphic evaluation can be delegated / only one party (assuming cs poofs / SNARGs) Simple: there is no need for generic MPC protocol Extremely low round complexity Only 3 broadcast rounds (CRS model) 2 rounds reusable PKI – optimal(!) UC security (assuming UC-NIZK) Threshold Key Gen Encrypt and Evaluate Threshold Decryption Threshold Key Gen Encrypt and Evaluate Threshold Decryption

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Related Work [C ramer D amgard N ielsen 01]– MPC using threshold HE [G entry 09] – MPC using threshold FHE [B endlin D amgard 10] – threshold version for LWE [K atz O strovsky 04] – lower bound of 5 rounds for MPC in the plain model [M yers S ergi s helat 11] – threshold version of [vDGHV10]

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The LWE Assumption [Regev05] Distribution 1Distribution 2 also secure if q is odd and we choose noise to be small and even (2e instead e)

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Basic LWE-Based Encryption Symmetric KeyPublic Key

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Key-Homomorphic Properties of the Basic Scheme Two public keys, same “coefficient” A A new public key with secret key: s 1 +s 2, coefficient A (almost the same as El-Gammal)

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Threshold Key Generation A s1s1 s3s3 (A,p 1 ) = As 1 +2e 1 (A,p 3 ) = As 3 +2e 3 (A,p 2 ) = As 2 +2e 2 (A,p 4 ) = As 4 +2e 4 s2s2 s4s4

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Threshold Key Generation A s1s1 s3s3 (A,p 1 ) = As 1 +2e 1 (A,p 3 ) = As 3 +2e 3 (A,p 2 ) = As 2 +2e 2 (A,p 4 ) = As 4 +2e 4 s2s2 s4s4

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Threshold Key Generation A s2s2 s4s4 (A,p 1 ) = As 1 +2e 1 (A,p 3 ) = As 3 +2e 3 (A,p 2 ) = As 2 +2e 2 (A,p 4 ) = As 4 +2e 4 (A,p*) = As*+2e* (A,p * ) Joint secret key: s * =s 1 +s 2 +s 3 +s 4 Joint public key: p * =p 1 +p 2 +p 3 +p 4 s1s1 s3s3

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Threshold Decryption s1s1 s3s3 s2s2 s4s4

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s1s1 s3s3 s2s2 s4s4

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s1s1 s3s3 s2s2 s4s4

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Basic LWE-Based Encryption – Homomorphism

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FHE From LWE [BV11b],[BGV12]

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Evaluation Key

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Threshold KeyGen –Round 2 s2s2 s4s4 s1s1 s3s3 … … … …

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Threshold KeyGen – End Of Round 2 s2s2 s4s4 s1s1 s3s3 … … … …

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… … … … Threshold KeyGen – Round 3 s2s2 s4s4 s1s1 s3s3 …………

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Threshold KeyGen – End Of Round 3 s2s2 s4s4 s1s1 s3s3

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Threshold FHE - KeyGen one round!

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The MPC Protocol Threshold KeyGen (2 rounds) – Round 1: Creates public key – Round 2: Creates evaluation key The parties encrypt their inputs (sent concurrently with round 2 of KeyGen) Threshold Dec (1 round)

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Malicious

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Conclusion TFHE based on LWE – In the paper: Ring – LWE 3 Rounds MPC 2 Rounds in reusable PKI - optimal(!) Low Communication Complexity Easy to delegate Thank You!

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