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FULLY HOMOMORPHIC ENCRYPTION from the Integers Marten van Dijk (RSA labs) Craig Gentry (IBM T J Watson) Shai Halevi (IBM T J Watson) Vinod Vaikuntanathan (IBM T J Watson)

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Computing on Encrypted Data ClientServer/Cloud (Input: x)(Program: P) I want to delegate the computation to the cloud I want to delegate the computation to the cloud, but the cloud shouldnt see my input Enc [ P(x) ] Enc(x) P (An Example)

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Enc [ P(x) ] Enc(x) P Definition: [ KeyGen, Enc, Dec, Eval ] (as in regular encryption) Compactness: Size of Evaled ciphertext independent of P Fully Homomorphic Encryption (FHE) = Security: Semantic Security [GM82] Definition: [ KeyGen, Enc, Dec ] Eval

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– BGN05 & GHV10a: quadratic formulas Fully Homomorphic Encryption First Defined: Privacy homomorphism [RAD78] Limited Variants: – GM & Paillier: additively homomorphic – RSA & El Gamal: multiplicatively homomorphic – their motivation: searching encrypted data NON-COMPACT homomorphic encryption [CCKM00, SYY99, IP07,MGH08,GHV10b,…]

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Fully Homomorphic Encryption First Defined: Privacy homomorphism [RAD78] – using just integer addition and multiplication – their motivation: searching encrypted data Is there an elementary Construction of FHE? Big Breakthrough : [Gentry09] First Construction of Fully Homomorphic Encryption using algebraic number theory / ideal lattices – easier to understand, implement and improve No ideal lattices

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OUR RESULT Theorem [DGHV10]: We have a fully homomorphic public-key encryption scheme – which uses only add and mult over the integers, – which is secure based on the approximate GCD problem & the sparse subset sum problem.

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Construction

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A Roadmap 1. Secret-key Somewhat Homomorphic Encryption (under the approximate GCD assumption) 2. Public-key Somewhat Homomorphic Encryption (under the approximate GCD assumption) 3. Public-key FULLY Homomorphic Encryption (under approx GCD + sparse subset sum) (a simple transformation) (borrows from Gentrys techniques)

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Secret-key Homomorphic Encryption Secret key: an n 2 -bit odd number p To Encrypt a bit b: – pick a random large multiple of p, say q·p – pick a random small even number 2·r – Ciphertext c = q·p+2·r+b To Decrypt a ciphertext c: – c (mod p) = 2·r+b (mod p) = 2·r+b – read off the least significant bit (q ~ n 5 bits) (r ~ n bits) noise (sec. param = n)

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Secret-key Homomorphic Encryption How to Add and Multiply Encrypted Bits: – Add/Mult two near-multiples of p gives a near-multiple of p. – c 1 = q 1 ·p + (2·r 1 + b 1 ), c 2 = q 2 ·p + (2·r 2 + b 2 ) – c 1 +c 2 = p·(q 1 + q 2 ) + 2·(r 1 +r 2 ) + (b 1 +b 2 )« p – c 1 c 2 = p·(c 2 ·q 1 +c 1 ·q 2 -q 1 ·q 2 ) + 2·(r 1 r 2 +r 1 b 2 +r 2 b 1 ) + b 1 b 2 « p LSB = b 1 XOR b 2 LSB = b 1 AND b 2

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Two Issues Ciphertext grows with each operation Noise grows with each operation Useless for many applications (cloud computing, searching encrypted ) – Consider c = qp+2r+b Enc(b) (q-1)pqp(q+1)p(q+2)p 2r+b – c (mod p) = r 2r+b r

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Two Issues Ciphertext grows with each operation Useless for many applications (cloud computing, searching encrypted ) After some operations, the ciphertext becomes undecryptable Noise grows with each operation

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Solving the Two Issues Ciphertext grows with each operation – Publish x 0 = q 0 p (*) – Take ciphertext (mod x 0 ) after each op (Add/Mult) (*) More complex way using x 0 = a near-multiple of p

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Solving the Two Issues Ciphertext grows with each operation – Publish x 0 = q 0 p – Take ciphertext (mod x 0 ) after each op (Add/Mult) Ciphertext stays less than x 0 always The encrypted bit remains same Noise does not increase at all

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Solving the Two Issues Ciphertext grows with each operation Can perform limited number of hom. operations (Somewhat Homomorphic Encryption) – Somewhat Fully: (A variant of) Squashing + Bootstrapping [G09] Noise grows with each operation – Publish x 0 = q 0 p – Take ciphertext (mod x 0 ) after each op (Add/Mult) *

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Security (of the secret-key somewhat homomorphic scheme)

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The Approximate GCD Assumption q0pq0p p? p q 0 [0…Q] odd p [0…P] (q 0 p,q 1 p+r 1,…, q t p+r t ) Assumption: no PPT adversary can guess the number p Parameters of the Problem: Three numbers P,Q and R q [0…Q] r [-R…R] q 1 p+r 1 (name coined by Howgrave-Graham)

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p? p Assumption: no PPT adversary can guess the number p Semantic Security : no PPT adversary can guess the bit b (q 0 p,q 1 p+2r 1 +b,…,q k p+2r k +b) = (proof of security) (q 0 p,q 1 p+r 1,…, q t p+r t )

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A Taste of the Security Proof Encryption breaker Adv A lsb(q) q 0 p,{q i p+r i } p c=qp+r Claim: On every c=qp+r, A predicts lsb(q) correctly w.p. 1-1/poly Proof: Lots of details ( Worst-case to average case over c, computing lsb(q) = computing lsb(r), a hybrid argument.) Approx GCD solver Adv B q p c=qp+r q 0 p,{q i p+r i }

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A Taste of the Security Proof Main Idea: Use lsb(q) to make q successively smaller – If lsb(q) = 0, c [c/2] ( new-c = q/2*p+[r/2] ) – If lsb(q) = 1, c [(c-p)/2] ( new-c = (q-1)/2*p+[r/2] ) Adv A lsb(q) q 0 p,{q i p+r i } p c=qp+r Adv B q p c=qp+r q 0 p,{q i p+r i } First Try:

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A Taste of the Security Proof Main Idea: Use lsb(q) to make q successively smaller – If lsb(q) = 0, c [c/2] ( new-c = q/2*p+[r/2] ) – If lsb(q) = 1, c [(c-p)/2] ( new-c = (q-1)/2*p+[r/2] ) Lemma: Given two near-multiples z 1 =q 1 p+r 1 and z 2 =q 2 p+r 2 (+lsb oracle), can compute z=gcd(q 1,q 2 ) · p+r for small r W.h.p. gcd(q 1,q 2 )=1 – If lsb(q) = 1, c [(c-z)/2] ( new-c = (q-1)/2*p+[(r-r)/2] ) – Learn q bit by bit

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A Taste of the Security Proof Main Idea: Use lsb(q) to make q successively smaller Lemma: Given two near-multiples z 1 =q 1 p+r 1 and z 2 =q 2 p+r 2 (+lsb oracle), can compute z=gcd(q 1,q 2 ) · p+r for small r Observation: the Binary GCD algo. is noise-tolerant, given lsb oracle. (similar to the RSA hardcore bit proof of [ACGS88])

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– Lagarias algorithm Studied by [HG01]; also [Lag82,Cop97,NS01] (equivalent to simultaneous Diophantine approximation) Lattice-based Attacks – Coppersmiths algo. for finding small polynomial roots – Nguyen/Stern and Regevs orthogonal lattice method All run out of steam when log Q > (log P) 2 (our setting of parameters: log Q = n 5, log P = n 2 ) How Hard is Approximate GCD?

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Future Directions Efficient fully homomorphic encryption (Currently: n 5 size ciphertexts; n 10 running-time blowup) Security of the approx GCD assumption (e.g., a worst-case to average-case reduction to regular – non-ideal – lattice problems?) Some recent improvements [SV10,SS10]

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Questions?

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Delegate computation on data without giving away access to it The Goal

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Public-key Homomorphic Encryption Secret key: an n 2 -bit odd number p To Decrypt a ciphertext c: – c (mod p) = 2·r+b (mod p) = 2·r+b – read off the least significant bit Eval (as before) Public key: [ q 0 p+2r 0,q 1 p+2r 1,…,q t p+2r t ] = (x 0,x 1,…,x t ) = t+1 random encryptions of 0 Δ

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c = + b (mod x 0 ) Public-key Homomorphic Encryption Secret key: an n 2 -bit odd number p To Decrypt a ciphertext c: – c (mod p) = 2·r+b (mod p) = 2·r+b – read off the least significant bit Eval (as before) Public key: [ q 0 p+2r 0,q 1 p+2r 1,…,q t p+2r t ] = (x 0,x 1,…,x t ) To Encrypt a bit b: pick random subset S [1…t] Δ

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Parameter Regimes R log Q/(log P) 2 (very rough diagram) min=0max=P

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Public-key Homomorphic Encryption Secret key: an n 2 -bit odd number p To Decrypt a ciphertext c: – c (mod p) = 2·r+b (mod p) = 2·r+b – read off the least significant bit Eval (as before) Public key: [ q 0 p+2r 0,q 1 p+2r 1,…,q t p+2r t ] = (x 0,x 1,…,x t ) – t+1 encryptions of 0 – W.l.o.g., x 0 = q 0 p+2r 0 is the largest Δ

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c = + b (mod x 0 ) Public-key Homomorphic Encryption Secret key: an n 2 -bit odd number p To Decrypt a ciphertext c: – c (mod p) = 2·r+b (mod p) = 2·r+b – read off the least significant bit Eval (as before) Public key: [ q 0 p+2r 0,q 1 p+2r 1,…,q t p+2r t ] = (x 0,x 1,…,x t ) To Encrypt a bit b: pick random subset S [1…t] Δ c = p[ ] + 2[ ] + b (mod x 0 ) c = p[ ] + 2[ ] + b – kx 0 (for a small k) = p[ ] + 2[ ] + b (mult. of p) + (small even noise) + b

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c = + b (mod x 0 ) Public-key Homomorphic Encryption Secret key: an n 2 -bit odd number p To Decrypt a ciphertext c: – c (mod p) = 2·r+b (mod p) = 2·r+b – read off the least significant bit Eval: Reduce mod x 0 after each operation To Encrypt a bit b: pick random subset S [1…t] Ciphertext Size Reduction – Resulting ciphertext < x 0 – Underlying bit is the same (since x 0 has even noise) – Noise does not increase by much (*) Public key: [ q 0 p+2r 0,q 1 p+2r 1,…,q t p+2r t ] = (x 0,x 1,…,x t ) Δ (*) additional tricks for mult

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A Roadmap Secret-key Somewhat Homomorphic Encryption Public-key Somewhat Homomorphic Encryption 3. Public-key FULLY Homomorphic Encryption

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How Somewhat Homomorphic is this? Can evaluate (multi-variate) polynomials with m terms, and maximum degree d if: f(x 1, …, x t ) = x 1 ·x 2 ·x d + … + x t-d+1 ·x t-d+2 ·x t Final Noise ~ (2 n ) d +…+(2 n ) d = m(2 n ) d Say, noise in Enc(x i ) < 2 n or

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Somewhat HEBootstrappable From Somewhat to Fully FHE = Can eval all fns. Theorem [Gentry09]: Convert bootstrappable FHE. Augmented Decryption ckt. Dec NAND c1c1 sk c2c2

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Is our Scheme Bootstrappable? What functions can the scheme EVAL? Complexity of the (aug.) Decryption Circuit (?) Can be made bootstrappable – Similar to Gentry09 Caveat: Assume Hardness of Sparse Subset Sum (polynomials of degree < n) (degree ~ n 1.73 polynomial)

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