FULLY HOMOMORPHIC ENCRYPTION

IBM T. J. Watson

Vinod Vaikuntanathan

from the Integers

Joint Work with M. van Dijk (MIT & RSA labs), C. Gentry (IBM), S. Halevi (IBM)

= Modern Application: Computing on Encrypted Data

Public-key Encryption

Client Server/Cloud

(Input: x) (Program: P)

Enc(P(x))

Enc(x)

= Modern Application: Computing on Encrypted Data

Public-key Encryption

Client Server/Cloud

(Input: x) (Program: P)

Enc(P(x))

Enc(x)

Fully HomomorphicEncryption (FHE) =

=Definition: (KeyGen, Enc, Dec)

Fully HomomorphicEncryption (FHE) =

(as in regular public-key encryption)Eval(c,F) = c’

– If c = Enc(x), then Dec(c’) = F(x)

Definition: (KeyGen, Enc, Dec, Eval)

ANDXOR

AND

= Fully HomomorphicEncryption (FHE) =

Two Important Properties

Definition: (KeyGen, Enc, Dec, Eval)

Compactness: Length of c’ is independent of the size of F

Circuit/Function Privacy: c’ does not reveal “any more information about F than the output F(x)”

Theorem [GHV’10]: “Circuit privacy comes for free”.Any (compact) FHE → (Compact) circuit-private FHE

FHE = Compact FHE

Fully Homomorphic Encryption

► First Defined: “Privacy homomorphism” [RAD’78]

– their motivation: searching encrypted data

– BGN’05 & GHV’10: quadratic formulas

Fully Homomorphic Encryption

► First Defined: “Privacy homomorphism” [RAD’78]

► Limited Variants:

– GM & Paillier: additively homomorphic

– RSA & El Gamal: multiplicatively homomorphic

– their motivation: searching encrypted data

c1 = m1e c2 = m2

e cn = mne

Xc* = c1c2…cn = (m1m2…mn)e mod N

► NON-COMPACT homomorphic encryption:

– SYY’99 & MGH’08: c* grows exp. with degree/depth

– IP’07 works for branching programs

Fully Homomorphic Encryption

► First Defined: “Privacy homomorphism” [RAD’78]

– 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

Our Result

Theorem [DGHV’10]: There exists 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.

Construction

A Roadmap

1. Secret-key “Somewhat” Homomorphic Encryption

2. Public-key “Somewhat” Homomorphic Encryption

3. Public-key FULLY Homomorphic Encryption

(fairly generic transformation)

(borrows from Gentry’s techniques)

Secret-key Homomorphic Encryption

Secret key: an n2-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 ~ n5 bits)

(r ~ n bits)

“noise”

(sec. param = n)

Secret-key Homomorphic Encryption

How to Add and Multiply Encrypted Bits:

– Add/Mult two near-multiples of p gives a near-multiple of p.

– c1 = q1·p + (2·r1 + b1), c2 = q2·p + (2·r2 + b2)

– c1+c2 = p·(q1 + q2) + 2·(r1+r2) + (b1+b2) « p

– c1c2 = p·(c2·q1+c1·q2-q1·q2) + 2·(r1r2+r1b2+r2b1) + b1b2 « p

LSB = b1 XOR b2

LSB = b1 AND b2

Problems

Ciphertext grows with each operation

Noise grows with each operation

Useless for many applications (cloud computing, searching encrypted e-mail)

– Consider c = qp+2r+b ← Enc(b)

(q-1)p qp (q+1)p (q+2)p

2r+b

– c (mod p) = r’ ≠ 2r+b

r’– lsb(r’) ≠ b

Problems

Ciphertext grows with each operation

Noise grows with each operation

Useless for many applications (cloud computing, searching encrypted e-mail)

Can perform “limited” number of hom. operations

What we have: “Somewhat Homomorphic” Encryption

Public-key Homomorphic Encryption

Secret key: an n2-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: [q0p+2r0,q1p+2r1,…,qtp+2rt] = (x0,x1,…,xt)

– t+1 “near-multiples” of p, with even noise

– W.l.o.g., x0 = q0p+2r0 is the largest

Δ

c = + b (mod x0)

Public-key Homomorphic Encryption

rxSi

i 2

Secret key: an n2-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: [q0p+2r0,q1p+2r1,…,qtp+2rt] = (x0,x1,…,xt)

To Encrypt a bit b: pick random subset S [1…t]

Δ

c = p[ ] + 2[ ] + b (mod x0) Si

iq

Siirrc = p[ ] + 2[ ] + b – kx0 (for a small k)

Siiq

Siirr

= p[ ] + 2[ ] + b 0kqqSi

i

0krrrSii

(mult. of p) + (“small” even noise) + b

c = + b (mod x0)rxSi

i 2

Public-key Homomorphic Encryption

Secret key: an n2-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 x0 after each operation

To Encrypt a bit b: pick random subset S [1…t]

Ciphertext Size Reduction

– Resulting ciphertext < x0

– Underlying bit is the same (since x0 has even noise)

– Noise does not increase by much(*)

Public key: [q0p+2r0,q1p+2r1,…,qtp+2rt] = (x0,x1,…,xt)Δ

(*) additional tricks for mult

A Roadmap

Secret-key “Somewhat” Homomorphic Encryption

Public-key “Somewhat” Homomorphic Encryption

3. Public-key FULLY Homomorphic Encryption

How “Somewhat” Homomorphic is this?

Can evaluate (multi-variate) polynomials with m terms, and maximum degree d if:

f(x1, …, xt) = x1·x2·xd + … + xt-d+1·xt-d+2·xt

Final Noise ~ (2n)d+…+(2n)d = m•(2n)d

Say, noise in Enc(xi) < 2n

2/22/22nnd pm or nd ~

“Somewhat” HE“Bootstrappable”

From “Somewhat” to “Fully”

FHE = Can eval all fns.

Theorem [Gentry’09]: Convert “bootstrappable” → FHE.

Augmented Decryption ckt.

Dec Dec

NAND

c1 sk skc2

Is our Scheme “Bootstrappable”?

What functions can the scheme EVAL?

Complexity of the (aug.) Decryption Circuit

(?)

Can be made bootstrappable– Similar to Gentry’09

Caveat: Assume Hardness of “Sparse Subset Sum”

(polynomials of degree < n)

(degree ~ n2 polynomial)

Security(of the “somewhat” homomorphic scheme)

The Approximate GCD Assumption

q1p+r1

p?

p

q1 ← [0…Q]r1 ← [-R…R]

odd p ← [0…P]

(q1p+r1,…, qtp+rt)

Assumption: no PPT adversary can guess the number p

Parameters of the Problem: Three numbers P,Q and R

(coined by Howgrave-Graham)

p?

p

Assumption: no PPT adversary can guess the number p

Semantic Security [GM’82]: no PPT adversary can guess the bit b

PK =(q0p+2r0,{qip+ri})

Enc(b) =(qp+2r+b)

=(proof of security)

(q1p+r1,…, qtp+rt)

A “Taste” of the Security Proof

Adv

B

{qip+ri}

p

Approx GCD solver

Adv

A

PK, Enc(b)

b

Encryption breaker

(w.p. 1/2+ε)

p

– PK = {qip+2ri},

– Enc(b) = qp+r, where lsb(r)=b

A “Taste” of the Security Proof

Adv

B

{qip+ri}

p

Approx GCD solver

Adv

Alsb(r)

Encryption breaker

(w.p. 1/2+ε)

p

Technical Details:– Random noise vs. even noise

{qip+ri}pc=qp+r

– Distribution of ciphertext: qp+r vs. c = + b (mod x0)rxSi

i 2

(statistically close distributions by Leftover Hash Lemma)

A “Taste” of the Security Proof

Adv

B

{qip+ri}

p

Approx GCD solver

Adv

Alsb(r)

Encryption breaker

p

Success Amplification– Boost from 1/2+ε to 1-1/poly(n)

{qip+ri}pc=qp+r

(w.p. 1/2+ε)

A “Taste” of the Security Proof

Adv

B

{qip+ri}

p

Approx GCD solver

Adv

Alsb(r)

Encryption breaker

p

Computing lsb(q) and lsb(r) are equivalent

{qip+ri}pc=qp+r

– lsb(q) = lsb(r) lsb(c), since p is odd

lsb(q)

A “Taste” of the Security Proof

Adv

B

{qip+ri}

p

Approx GCD solver

Adv

A

Encryption breaker

p

Computing q and p are equivalent

{qip+ri}pc=qp+r

–

lsb(q)

c=qp+r

q

cp

q

A “Taste” of the Security Proof

Adv

B

{qip+ri}

Approx GCD solver

Adv

A

Encryption breaker

p

The Idea: Use lsb(q) to make q successively smaller

{qip+ri}pc=qp+r

lsb(q)

c=qp+r

q

– 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])

A “Taste” of the Security Proof

Adv

B

{qip+ri}

Approx GCD solver

Adv

A

Encryption breaker

p

The Idea: Use lsb(q) to make q successively smaller

{qip+ri}pc=qp+r

lsb(q)

c=qp+r

q

A New Trick: inspired by the binary GCD algorithm

A “Taste” of the Security Proof

Given two near-multiples z1=q1p+r1 and z2=q2p+r2

– Get bi := lsb(qi); if z1<z2 swap them

– If b1=b2=1, set z1←z1-z2 and b1=b1-b2 (mod 2)

(At least one of the bi’s must be zero now!)

– For any bi=0, set zi←[zi/2]

(The new qi is half the old qi!)

– Repeat until one zi is zero, and let z* be the other

At the end, z*= gcd(q1,q2)·p+r* = 1·p+r*

Bin

ary

-GC

D

(for random q1 and q2, gcd(q1,q2)=1 with constant prob.) Run Binary-gcd(z*,zi); the intermediate bits spell out the qi

– Lagarias’ algorithm

Studied by [Lag82,Cop97,HG01,NS01](equivalent to “simultaneous Diophantine approximation”)

Lattice-based Attacks

– Coppersmith’s algo. for finding small polynomial roots

– Nguyen/Stern and Regev’s orthogonal lattice method

All run out of steam when log Q > (log P)2

(our setting of parameters: log Q = n5, log P = n2)

How Hard is Approximate GCD?

Caveat: In Crypto, we need average-case hardness

Algorithms for Approx GCD: A Template

B =

R x1 x2 … xt

-x0

-x0

… -x0

b1

b2

b3

bt+1

(follows Lagarias’82)

Create a Lattice basis

– Rows of B span a (t+1)-dimensional lattice 1)( tZBL

Algorithms for Approx GCD: A Template

R x1 x2 … xt

-x0

-x0

… -x0

(follows Lagarias’82)

Create a Lattice basis

– Length of the basis vectors ~ QP (or more)

– (q0,q1,…,qt) • B is a short lattice vector

1st entry = q0R < QR

ith entry (i>1) = q0(qip+ri) – qi(q0p+r0) = q0ri – qir0

< QR (in abs. value)

Algorithms for Approx GCD: A Template

R x1 x2 … xt

-x0

-x0

… -x0

(follows Lagarias’82)

Create a Lattice basis

– Length of the basis vectors ~ QP (or more)

– ||(q0,q1,…,qt) • B||2 < QR t

Try to find this short vector using (say) LLL

When will this Algorithm Succeed?

Need t to be large (> log Q/log P) – If t is small, then (q0,q1,…,qt) is not the shortest vector

– Need [(QPt)R]1/t+1 > QR, or t > log Q/log P

If t is large, LLL (&co.) perform badly

– Minkowski: lattice vector shorter than 1/1)det( tBt

(exponentially many vectors shorter than ) 1/1)det()( tBtpoly

Set log Q = ω(log2 P). Then, t = ω(log P)

– Only finds vectors of size 2O(t)•det(B)1/t+1 >> QR

– Contemporary lattice reduction is not strong enough

R x1 x2 … xt

-x0

-x0

…

-x0

Future Directions

Efficient fully homomorphic encryption

(Currently: n8 factor blow-up in running time)(Currently: 1000000000000000000000000 factorblowup)

Efficiency for special-purpose tasks

Questions?

[1] “Fully Homomorphic Encryption from the Integers”,http://eprint.iacr.org/2009/616, To Appear in Eurocrypt 2010.

Parameter Regimes

tlogQ/logP

size

(lo

g s

cale

)

the solution weare seeking

auxiliary solutions(Minkowski’s bound)converges to ~ logQ+logP

What LLL can findmin(blue,purple)+t

blue lineremains abovepurple line

log Q