Secure Computation of Linear Algebraic Functions

Enav Weinreb – CWI, Amsterdam

Joint work with: Matt Franklin, Eike Kiltz, Payman Mohassel

and Kobbi Nissim

Talk Overview

Secure Computation in General Secure Linear Algebra Based on

“Oblivious Gaussian Elimination” Secure Linear Algebra Based on Linearly

Recurrent Sequences Recent Developments and Open

Problems

Secure Computation

Alice has an input x Bob has an input y Let f:{0,1}2n{0,1} be a Boolean function. Alice and Bob wish to compute f(x,y) without

leaking any further information on their private inputs.

The players cooperate but do not trust each other.

Secure Computation - Example

yx

x > y ?

The Millionaires’ Problem

The Millionaires’ Problem

1,000,000,000$

Secure Computation - Example

x

x > y ?

Answer: x < y

x = 100$ ???

x = 999,999,999$ ???

Real WorldIdeal World

xxy y

f(x,y)

f(x,y)f(x,y)

h(x)h(x)

Levels of security:Computational - adversary is computationally limitedInformation theoretic - adversary is computationally unbounded.“Leak no further information”

Complexity Measures and Adversary Model

Important complexity measures:• Communication complexity

• Round complexity

• Computational complexity Adversary models:

• Honest but curious – adversary follows the protocol but tries to learn more information

• Malicious – adversary arbitrarily deviates from the protocol

Boolean Circuit Complexity

Let f:{0,1}2n {0,1} We consider digital circuits with the

gates {AND, OR, NOT} that compute f in the natural way.

circuit size – number of gates circuit depth – max distance from

an input wire to output 00 00 1 11 1

0 0 1 1

10

01

0

x1 x2 x3 x4 x5 x6 x7 x8

General Result – two-party [Yao]

Boolean circuit that computes f(x,y) with size s(n)

impliessecure two party protocol for computing f(x,y)

with: communication complexity linear in s(n) 2 rounds.computational security.

General Result – Multi-Party [BGW, CCD]

Boolean circuit that computes f(x1,...,xk) with size s(n) and depth d(n)

impliesA secure k-party protocol for computing f(x1,...,xk)

with: communication complexity linear in s(n) round complexity d(n) Information theoretic security against:

• Less than k/2 adversarial players – honest but curious• Less than k/3 adversarial players – malicious

Talk Overview

Secure Computation in General Secure Linear Algebra Based on

“Oblivious Gaussian Elimination” Secure Linear Algebra Based on Linearly

Recurrent Sequences Recent Developments and Open

Problems

Linear Algebraic Functions

Matrix singularity: Alice and Bob hold A ∊ Fnxn and B ∊ Fnxn respectively,

where F is a finite field They wish to (securely) compute whether M=A+B is

singularEfficient secure protocol for singularity leads to efficient

protocols for:• solving a joint system of equations (linear constraints may

contain private information!)• computing det(M), char.poly(M), min.poly(M)• computing subspaces intersection• more...

Applying General Results

Circuit complexity of matrix singularity is similar to number of multiplications in matrix product.• Best known result O(n2.38) [Coppersmith Winograd]

Input size is only n2 - trivial non-cryptographic protocol has complexity n2

Can we achieve this in a secure protocol? Can we achieve this keeping the round complexity

low?

A previous result

“Secure linear algebra in a constant number of rounds.” [Cramer Damgård]

Information theoretic security constant round complexity communication complexity O(n3)

Our results

Secure protocol for singularity(A+B) in the computational two party setting with:

• communication complexity O(n2log n)

• round complexity O(log n)

Recent improvements [Mohassel W]

• constant round

• information theoretical security

Oblivious Gaussian Elimination

Protocol from [Nissim W] Achieves:

• communication complexity O(n2log n)

• round complexity O(n0.275) Cryptographic assumption: public key

homomorphic encryption

Tool: Homomorphic Encryption Public key encryption scheme

• Public key PK is published – everybody can encrypt• Secret key SK is private – only one can decrypt

For

Corollary:

Example: [Goldwasser Micali] (QR) for F=GF(2).

Fcba ,,

)(E ba )(E a )(E b)(E ac )(E ca

)(E vc )(E vc

)(E 2M1M )(E 21MM

(with PK only)

Initial Step

),( SKPKGenerates

)(E APK

)(E MPK

A ∊ Fnxn B ∊ Fnxn

)(E BPK+ =

Is M singular?

PK

Algorithms on Encrypted Data

Bob can locally compute:

What about multiplication?

)(E ba )(E a )(E b

)(E ac )(E ca

)(E vc )(E vc

)(E 2M1M )(E 21MM

Use Alice!

? )(E ab)(E a )(E b

Multiplication

)(E

)(E

b

a

PK

PK

ba rr ,Chooses random

)))(((E baPK rbra PK

PK

E

E

)(

)(

b

a

rb

ra

)))(((E baPK rbra )(E bPK ar

)(E braPK)(E abPK)(E baPK rr

),( SKPK

Multiplying a Vector by a Scalar

)(E

)(E

v

a

PK

PK),( SKPK

Communication complexity is O(n).

)(E vaPK

Encrypted Matrix Singularity (reminder)

),( SKPK

Is singular?M

)(E MPK

Find a row that “starts” with a 1.

Swap this row and the top row.

“Eliminate” the leftmost column.

Continue recursively.

0111

1110

Gaussian Elimination0010

1001

1010

1001

Oblivious Gaussian Elimination

)(E)(E

)(E)(E

)(E

PK1PK

1PK11PK

kkk

k

PK

MM

MM

M

“Find a row that starts with a 1.” “Swap this row and the top row.”

),( SKPK

Use Alice!

STEP 1: Randomization Bob multiplies E(M) by a random full rank matrix

R.

E(M) R E(M) Set m = log2n

RM

Finding a row starting with a 1

M

1

1m

w.h.p

Finding a row that starts with a 1

STEP 2: Moving the 1 to the top row.

m1

M

m1

M

Moving the 1 to the top row.

Bob computes E(M[1,1]M1)

• If M[1,1]=0 Bob gets E(0).

• If M[1,1]=1 Bob gets E(M1). For every 2 ≤ j ≤ m, Bob computes

E(Mj) E(Mj – M[j,1]M[1,1]M1) Same with E(M2), E(M3), ..., E(Mm)

Update E(M1) = E(Mi) Eliminate leftmost column.

0011010

)0(E

)(E 3M)0(E

0

0

m

i 1

1

m

M

Moving the 1 to the top row.

Continue recursively on the lower right submatrix Finally, multiply all diagonal elements.

M is singular if and only if the product of the diagonal entries is 1.

0

0

0

1

M0

11

1

m

)(nO )( 2nO )( 2nO

Communication Complexity

)]1,[(E ]1,[ jMrjM )))(]1,[((E

11]1,[ MjM rMrjM

Single row One column

Alice Bob

Alice Bob

)(nO

)(nO )( 2nO

)(E11 MrM

Overall

)( 3nO

)( 3nO

)( 2nO)(nO

Lazy Evaluation

Single row One column

Alice Bob

Alice Bob

)(nO

)(nO )( 2nO

)(nO

Overall

)( 3nO

)( 2nO

Memory

Send data “on demand”

Talk Overview

Secure Computation in General Secure Linear Algebra Based on

“Oblivious Gaussian Elimination” Secure Linear Algebra Based on Linearly

Recurrent Sequences Recent Developments and Open

Problems

Improved Round Complexity

Protocol from [Kiltz Mohassel W Franklin] Achieves:

• communication complexity O(n2log n)

• round complexity O(log n) Setting:

• Two party with computational security Computational assumption – homomorphic

encryption

Linearly Recurrent Sequences

General idea: apply algorithms designed for sparse matrices for secure computation on general matrices.

Assumption – the underlying field is large |F| > nlog n

(otherwise – use field extension)

A Simple Reduction

Randomized approach:

To check if M is singular:

• Pick a random vector v.

• Check whether the system Mx = v is solvable.

Not solvable – M is singular.Solvable – with high prob. (1 – 1/|F|), M is non-singular

Deciding if Mx = v is Solvable [Wiedemann]

Consider the n+1 vectors:

v, Mv, M2v, ..., Mnv There are a=(a0, ..., an) such that

∑aiMiv = 0 Linearly recurrent sequences:

If ∑aiMiv =0 then for all j:

∑aiMi+jv = Mj(∑aiMiv) = Mj0 = 0

Deciding if Mx = v is Solvable [Wiedemann86]

For every b=(b0, ..., bn) such that ∑biMiv = 0, consider the polynomial pb(x) = ∑bixi

The set of such polynomials forms an ideal in F[x] – the annihilator ideal

Minimal polynomial m(x) – the generator of the ideal

The annihilator ideal Let fM(x) be the characteristic polynomial of M.

[Cayley Hamilton]: fM(M)=0 → fM(M)v = 0 → fM(x) is in the annihilator ideal → m(x) | fM(x)

We will be interested in the constant coefficient of m(x).

The Constant Coefficient of m(x)

Claim:

(i) If m(0) ≠ 0 then Mx = v is solvable.

(ii) If m(0) = 0 then Mx = v is not solvable

The Constant Coefficient of m(x)

Claim:

(i) If m(0) ≠ 0 then Mx = v is solvable.

(ii) If m(0) = 0 then Det(M) = 0.

Conclusion:

With probability (1 – 1/|F|):

m(0) = 0 if and only if det(M)=0

Proof of the Claim (i)

(i) If m(0)≠0 then Mx=v is solvable. m(x) = cnxn+...+c1x+c0

• where c0=m(0) ≠ 0 m(M)v = 0 (m(x) is in the ideal)

• cnMnv+...+c1Mv+c0v = 0

• M(cnMn-1v+...+c1v) = -c0v

set x = -c0-1(cnMnv+...+c1Mv)

• Mx = v the system is solvable.

Proof of the Claim (ii)

(ii) If m(0)=0 then Det(M) = 0.

fM(0) = Det(M)

We saw before that m(x) | fM(x).

Hence fM(0)=0 and thus Det(M) = 0 □

Berlekamp/Massey Algorithm

We are interested in computing m(0). Berlekamp/Massey algorithm:

computes m(x) in O(n log n) operations, given v, Mv, ..., M2n-1v.

• General idea: the algorithm uses an intermediate result of the extended Euclidean algorithm executed on:• x2n

• a polynomial whose coefficients are the elements uTM0v, uTM1v, ..., uTM2n-1v for some random vector u.

And now: the protocol

Multiplying two matrices

)(E

)(E

B

A),( SKPK

Communication complexity is O(n2)

)(E AB

Secure Two-Party Algorithm (sketch)

E (M)(PK,SK)

E(Miv)i=0,1,…,2n-1

E(m(x))

m(0) =? 0

Yao’s general method applied on Berlekamp/Massey algorithm: O(1) rounds, O(n logn) communication

Yao’s general method applied on Berlekamp/Massey algorithm: O(1) rounds, O(n logn) communication

Decryption of E(m(0)r) where r is a random number.

Decryption of E(m(0)r) where r is a random number.

Next slide: O(log n) rounds,

O(n2 log n) communication

Next slide: O(log n) rounds,

O(n2 log n) communication

Computing the Sequence EPK(Miv)

1. Bob is given E(M) and computes E(v)

2. Bob computes E(M2^i), i=1...log n• log n rounds, n2 log n communication

3. Bob computes:• E(Mv)

• E(M3v|M2v) = E(M2) · E(Mv|v)

• E(M7v|M6v|M5v|M4v) = E(M4) ·E(M3v|M2v|Mv|v)

4. Finally: E(v), E(Mv), …, E(M2n-1v)

• O(log n) rounds, O(n2 log n) communication

Talk Overview

Secure Computation in General Secure Linear Algebra Based on

“Oblivious Gaussian Elimination” Secure Linear Algebra Based on Linearly

Recurrent Sequences Recent Developments and Open

Problems

Recent Developements

Protocol from [Mohassel W] For every constant t:

• communication complexity O(n2+1/t)

• round complexity t Gives information theoretic security. Based on a reduction to deciding the singularity of

Toeplitz matrices.

Open Problem

Secure Linear Algebra

• Malicious case for two party computation General Secure Computation

• Understand the relation between circuit complexity and secure protocol complexity of problem.

• Is linear communication complexity always possible?