On the Cryptographic Complexity of
the Worst Functions
Amos Beimel (BGU)
Yuval Ishai (Technion)
Ranjit Kumaresan (Technion)
Eyal Kushilevitz (Technion)
How Bad are the Worst Functions?
Function class FN of all functions f : [N][N] {0,1}
Standard Complexity
Theoretic Measures
• Circuit complexity
• (N2/log N)
[Sha48,Lup58]
• 2-party communication
complexity
• (log N) [Yao79]
Information-theoretic
Cryptography
• Communication complexity
• Randomness complexity
This work: Cryptographic complexity of the
worst functions
Model
Security Model
Crypto Primitives
• Information-theoretic
• Secure Computation
• Unbounded adversaries
• Statistical/perfect security
• Semi-honest adversary
• Various models
• Communication/randomness
• Secret Sharing
• No deviation from protocol
• Share complexity
Functions
• Function class FN : Class of
all two argument functions f
: [N]  [N]  {0,1}
• Interested in worst f  FN
Secure Computation
What is Known?
Information Theoretic Security
• Honest majority [RB89,BGW88]
• 2-party in the OT-hybrid or
preprocessing model [Kil88,Bea95]
• Impossible in plain model [Kus89]
• Private Simultaneous Messages [FKN94]
y
x
f1(x,y)
f2(x,y)
Can communication complexity be made logarithmic in N?
• Best upper bounds linear in N
– Sublinear if big honest majority [BFKR90,IK04]
• Counting arguments yield weak lower bounds
2-Party Secure Computation (2PC)
What is Known?
Information Theoretic Security
• Impossible in plain model [Kus89]
• OT-hybrid/preprocessing model
• Popular protocols [GMW87, Y86]
GMW [GMW87]
• Gate-by-gate evaluation
of given circuit
• #OTs required:
Twice #AND gates
• Communication cost:
Twice #AND gates
x
f1(x,y)
y
f2(x,y)
Information-theoretic
garbled circuits [Yao86]
• Depends on circuit structure
• Quadratic in formula
depth
• Exponential in depth
overhead for circuits
OT-Hybrid Model
Oblivious Transfer [Rab81,EGL85]
b
x0 , x1
x0 , x1
???
xb
xb
Complete
• Given ideal OT oracle, can
get information theoretic
2-party secure
computation [Kil88,GV88]
Pre-computation
• Random OT correlations
can be “corrected” [Bea95]
b
OT Extension
• Impossible in information
theoretic setting [Bea97]
• OT as an “atomic currency”
y0 , y1
x0 , x1
d=cb
z0 = x0yd
z1 = x1y1-d
c, yc
b
zbyc
OT Complexity
OT Complexity of a function f
Number of (bit) OTs required to securely evaluate f
• Let FN be the class of all 2-party f : [N]  [N]  {0,1}
• What is the OT complexity of the worst function in FN?
Circuit based 2PC:
• O(N2/log N) [GMW87]
Truth-table based 2PC:
• O(N) via1-out-of-N OT
• 1-out-of-N OT from O(N)
1-out-of-2 OTs [BCR86]
f(x,1)
f(x,2)
.
.
f(x,N)
x
y
???
f(x,y)
This work: O(N2/3) OT complexity
y
Preprocessing Model
Correlated Randomness
Offline Phase
Correlated Randomness
• Independent of inputs
• May depend on f
rA
rB
Online Phase
x
OT Correlations
y
rA
rB
f(x,y)
f(x,y)
• Special case
• Pre-computed OTs
• “Simpler” correlations
• Indep. of function
Correlated Randomness Complexity
Correlated Randomness Complexity of a function f
Size of correlated randomness required to securely evaluate f
• Let FN be the class of all 2-party f : [N]  [N]  {0,1}
• Correlated randomness complexity of the worst function in FN?
O(log N) online communication [IKMOP13]
• Correlated randomness: O(N2)
Truth-table based 2PC: O(N)
• Via 1-out-of-N OT [BCR86]
This work: 2O(log N) correlated randomness
Private Simultaneous Messages (PSM)
What is Known?
f (x,y)
x
r
Model [FKN94]
r
y
• Multiple clients
• Share randomness
• Single referee
• Non-interactive
• Referee learns only f(x,y)
• No collusion
Why PSM?
• Minimal model of secure computation [FKN94]
• Applications in round-efficient protocol design [IKP10]
• Connections to secret sharing! [BI05]
PSM Complexity
PSM Complexity of a function f
Communication complexity of PSM protocol for f
• What is the PSM complexity of the worst function in FN?
[FKN94,IK97]
• Efficient for f with
small formulas,
branching programs
• Worst case f : O(N)
• Lower bound: 3logN-4
f(x,y)
f(x,1+s)f(x,1)
+ r1
f(x,2+s)f(x,2)
+ r2
. .
. .
f(x,N+s)f(x,N)
+ rN
x
y-s, ry-s
r
r
r = s, (r1, …, rN)
This work: O(N) PSM complexity
y
Secret Sharing
What is Known?
Model
Share Complexity
Size of each share
• External dealer + n parties
• Dealer has input secret s
• Sends “shares” to parties
• Then, inactive
• Access structure
• Set of “authorized” subsets
• Secret hidden from unauth. subsets
• Any auth. subset can reconstruct s
Poly(n) share complexity for every n-party access structure?
• Best upper bound: 2O(n) [BL90,Bri89,KW93]
• Best lower bound: (n/log n) [Csi97]
Share Complexity
Forbidden Graph Access Structures
Forbidden Graph [SS97]
• Graph G = (V,E) with |V| = N
• Authorized subsets:
• Sets {u,v} with (u,v)  E
• Any set of size 3
• What is the share complexity of the worst N-vertex graph?
• Naïve solution: O(N) [SS97,BL90]
• O(N/log N) share complexity [BDGV96,EP97,Bub86]
This work: O(N) share complexity
Talk Outline
• Main Technical Tool – PIR
• OT Complexity
• Correlated Randomness Complexity
• PSM Complexity
• Share Complexity for Forbidden Graphs
Private Information Retrieval
DB
DB
q1
q2
a1
a2
z
aq11
qa22
r
i
Query generation
• (q1, q2)  Q(i , r)
Answer generation
• ak  A( k, qk , DB)
Reconstruction
• z  R(i , r, a1, a2)
Model [CGKS95]
• Single client
• Multiple servers
• Each server has same DB
• Size of DB = N (bits)
• DB unknown to client
• Client input: index i  [N]
• Privately retrieve DB[ i ]
• No collusion among servers
• Goal: min. communication
Best Known PIR Schemes
2-server: O(N1/3) [CGKS95]
3-server: 2O(log N) [Yek07,Efr09]
Talk Outline
• Main Technical Tool – PIR
• OT Complexity
– Upper bound: O(N2/3)
2-server PIR
• Correlated Randomness Complexity
• PSM Complexity
• Share Complexity for Forbidden Graphs
OT-Hybrid Model (Recap)
x0 , x 1
b
xb
• OT is “complete”
• Pre-computation
• No OT extension
OT Complexity of a function f
Number of (bit) OTs required to securely evaluate f
• Let FN be the class of all 2-party f : [N]  [N]  {0,1}
• What is the OT complexity of the worst function in FN?
• Circuit based 2PC for worst f :
• O(N2/log N) [GMW87]
• Truth-table based 2PC for worst f :
• O(N), 1-out-of-N OT [BCR86]
O(N2/3) Upper Bound on OT Complexity
Via 2-server PIR
Q’ = Q(x||y, r1r2)
x
r1
r2
y
GMW(C(Q’))
q1
• DB = truth table of f
• Client query = x||y
q2
a1 = A(1, q1, f )
a2 = A(2, q2, f )
R’ = R(x||y, r1r2, a1, a2)
x
r1
a1
a2
r2
GMW(C(R’))
f(x,y)
High-level idea
Use 2 party secure
computation to emulate
client + 2 PIR servers
f(x,y)
y
Notation
• PIR Algorithms: Q, A, R
• (q1, q2)  Q(i , r)
• ak  A( k, qk , DB)
• z  R(i , r, a1, a2)
• Circuit for alg. B: C(B)
• |C(B)|= #ANDs in C(B)
O(N2/3) Upper Bound on OT Complexity
Via 2-server PIR
Q’ = Q(x||y, r1r2)
x
r1
r2
y
GMW(C(Q’))
q1
• Query does not leak
additional info
q2
a1 = A(1, q1, f )
a2 = A(2, q2, f )
R’ = R(x||y, r1r2, a1, a2)
x
r1
a1
a2
r2
y
Efficiency
• 2-server PIR [CGKS95]
• |C(Q)|=|C(R)|= O(N2/3)
• By property of GMW:
• O(N2/3) OT comp.
• O(N2/3) communication
GMW(C(R’))
f(x,y)
Privacy
• Privacy of GMW
• Privacy of 2-server PIR
f(x,y)
More Applications
• Honest majority secure computation
– Efficient in circuit size [RB89,BGW88]
– Specific setting: n = 3 parties with at most 1 corruption
– Communication 2O(log N) via 3-server PIR
• “ - Secure Sampling” from joint distribution D [PP12]
– Protocol lets Alice & Bob to sample (x,y) from D
• Alice knows nothing about y (over what is implied by D)
• Bob knows nothing about x (over what is implied by D)
– Rate of secure sampling D  [N]  [N] from OT
– New upper bound: O(N2/3 poly(log N, 1/))
Talk Outline
• Main Technical Tool – PIR
• OT Complexity
– Upper bound: O(N2/3)
2-server PIR
• Correlated Randomness Complexity
– Upper bound: 2O( log N)
3-server PIR
• PSM Complexity
• Share Complexity for Forbidden Graphs
Preprocessing Model (Recap)
Correlated Randomness
Offline Phase
Correlated Randomness
• Independent of inputs
• May depend on f
• OT correlations special case
rA
rB
Online Phase
x
rA
f(x,y)
Correlated Randomness
Complexity of a function f
Size of correlated randomness
required to securely evaluate f
Correlated randomness complexity
of the worst function in FN?
y
rB
f(x,y)
Truth-table based 2PC: O(N)
• Via 1-out-of-N OT [BCR86]
Correlated Randomness Complexity:
2O(log N) Upper Bound
Via 3-server PIR
Offline Phase
r1
r2
a3 = A(3, q3, f )
a3,1
r1
a3,1
a3,2
OTA
q3=Q3(r1  r2)
a3 = a3,1a3,2
OTA
OTB
OTB
a3,2
r2
High-level idea
Use 2 party secure
computation to emulate
client + 3 PIR servers
• DB = truth table of f
• Client query = x||y
Key Observation
• Individual PIR query
independent of input
• Q = (Q1,2 , Q3)
• (q1, q2)  Q1,2(i, r)
• q3  Q3 (r)
Correlated Randomness Complexity:
2O(log N) Upper Bound
Q’ = Q1,2(x||y, r1r2)
x
r1
r2
y
GMW(C(Q’))
q1
Correlated Randomness
q2
a1 = A(1, q1, f )
a2 = A(2, q2, f )
R’ = R(x||y, r1r2, a1, a2, a3,1a3,1)
x r1
a1
a3,1
a3,2
a2
GMW(C(R’))
f(x,y)
Online Phase
f(x,y)
r2 y
• Shares of randomness for
PIR query generation alg.
• Shares of answer to third
PIR query
• OT correlations for GMW
Notation
• PIR Algorithms: Q, A, R
• Circuit for alg. B: C(B)
• |C(B)|= #ANDs in C(B)
Correlated Randomness Complexity:
2O(log N) Upper Bound
Q’ = Q1,2(x||y, r1r2)
x
r1
a3,1
a3,2
r2
Privacy
• Additive secret sharing
• Privacy of GMW
• Privacy of 3-server PIR
y
GMW(C(Q’))
q1
• Query does not leak
additional info
q2
a1 = A(1, q1, f )
a2 = A(2, q2, f )
R’ = R(x||y, r1r2, a1, a2, a3,1a3,1)
x r1
a1
a3,1
a3,2
a2
• 2O(log N) OT correlations
• 2O(log N) communication
GMW(C(R’))
f(x,y)
r2
•
•
y•
Efficiency
3-server PIR [Efr09]
|C(Q)|=|C(R)|=2O(log N)
By property of GMW:
f(x,y)
• Correlated rand.: 2O(log N)
Improving the Bounds?
• (OT + communication) complexity of 2PC
– Bounded by communication complexity of 2-server PIR
• Client shares its input, then acts as OT oracle
• (Cor. Rand. + communication) complexity of 2PC
– Bounded by communication comp. of 3-server PIR [IKM+13]
• 3rd server provides correlated randomness to servers 1 & 2
Summary
• Main Technical Tool – PIR
• OT Complexity
– Upper bound: O(N2/3)
2-server PIR
• Correlated Randomness Complexity
– Upper bound: 2O( log N)
3-server PIR
• PSM Complexity
– Upper bound: O(N)
4-server PIR
• Share Complexity for Forbidden Graphs
– Upper bound: O(N)
Using PSM above
Thank You!
Preliminary Version: www.cs.umd.edu/~ranjit/BIKK.pdf
Slides: www.cs.umd.edu/~ranjit/BIKK.pptx
Talk Outline
• Main Technical Tool – PIR
• OT Complexity
– Upper bound: O(N2/3)
2-server PIR
• Correlated Randomness Complexity
– Upper bound: 2O( log N)
3-server PIR
• PSM Complexity
– Upper bound: O(N)
4-server PIR
• Share Complexity for Forbidden Graphs
– Upper bound: O(N)
Using PSM above
Share Complexity (Recap)
Forbidden Graph Access Structures
Model
• External dealer + n parties
• Dealer inactive after sending “shares”
• Access structure: “authorized” subsets
Forbidden Graph [SS97]
Share Complexity
Size of each share
• Graph G = (V,E) with |V| = N
• Authorized subsets:
• Sets {u,v} with (u,v)  E
• Any set of size 3
• What is the share complexity of the worst N-vertex graph?
• O(N/log N) share complexity [DPGV96,EP97,B86]
Bipartite Case
Forbidden Bipartite Graph
• Graph G = (L,R,E) with |L| = |R| = N
• Authorized subsets:
• {x,y} with x  L, y  R, (x,y)  E
• Any set of size 3
• G associated with f :[N][N] {0,1}
Secret Sharing
• Share s using 3-out-of-2N
Shamir secret sharing
• Also secret share s = sL sR s’
• Send sL to x  L
• Send sR to y  R
• How to share s’ ?
PSM & Secret Sharing
r
xL
Af (x,r)
PSM Notation
Shared rand. : r
Alice with input x
• Message: Af (x,r)
Bob with input y
• Message: Bf (y,r)
yR
Bf (y,r)
Good for s’ = 1
For s’ = 0
Pick some x0, y0
s.t f (x0 , y0) = 0
High-level Idea
Shares :
• PSM messages
Reconstruction :
• PSM reconstruction
Secret Sharing
Scheme for s’
If dealer input s’ = 0
• x  L : Af (x0,r)
• y  R : Bf (y0,r)
If dealer input s’ = 1
• x  L : Af (x ,r)
• y  R : Bf (y ,r)
Forbidden Graph Access Structures
• From Bipartite to General Graphs
– Decomposed into log N bipartite graphs
– Apply standard techniques [BL90,Sti94]
• Forbidden graph access structures
– O(N) share complexity
– Via O(N) PSM
• Scheme is non-linear (?)
– Matches best known lower bound for linear schemes:
(N) [Min12]
Summary
• Cryptographic complexity of worst functions
– Main Technical Tool - PIR
• OT Complexity
– Upper bound: O(N2/3)
2-server PIR
• Correlated Randomness Complexity
– Upper bound: 2O( log N)
• PSM Complexity
– Upper bound: O(N)
3-server PIR
4-server PIR
• Share Complexity for Forbidden Graphs
– Upper bound: O(N)
Using PSM above
Thank You!
Preliminary Version: www.cs.umd.edu/~ranjit/BIKK.pdf
Slides: www.cs.umd.edu/~ranjit/BIKK.pptx
Talk Outline
• Main Technical Tool – PIR
• OT Complexity
– Upper bound: O(N2/3)
2-server PIR
• Correlated Randomness Complexity
– Upper bound: 2O( log N)
3-server PIR
• PSM Complexity
– Upper bound: O(N)
4-server PIR
• Share Complexity for Forbidden Graphs
PIR Examples [CGKS95]
2d server PIR with O(N1/d) communication
Tc
T{c}, if c  T
T \{c}, if c  T
PIR Queries
• T1 R [N]
• T2 = T 1  i
PIR Answers

jT
DB[ j ]
DB
T1
A(1,T1)
T1
DB
T

2
A(2,T2)
T2 i
z = A(1,T1)  A(2,T2)
Efficiency
• Client  Server j : O(N) bits
• Server j  Client : 1 bit
PIR Examples [CGKS95]
2d server PIR with O(N1/d) communication
DB as d-dim. hypercube
Index i  (i1, … , id)
• Binary rep of (i -1)
PIR Queries
Pick (T1 , … , Td) R [N1/d]d
Server k : Query Tk1 , … , kd
• (T1(k1 i1), … ,Td(kd id))
where k  (k1,…, kd)
PIR Answers

DB[k1,…, kd]
k1T1’,…,kdTd’
DB
DB
T 
T00...0
A(1, T00...0)
S1
d
11…1

A(2d,T11…1)
S2d i
z = A(1,T00..0)   A(2d,T11..1 )
Efficiency
• Client  Server j : O(dN1/d) bits
• Server j  Client : 1 bit
Reducing the #Servers [CGKS95]
Key Observation
Any server can emulate d other
servers with cost O(N1/d)
Query Tk1 , … , kd for Server k
(T1(k1 i1), … ,Td(kd id))
where k  ( k1,…, kd)
Example: 2-server O(N1/3) PIR
Server 1: Query T000 = (T1 , T2 , T3)
List “potential” queries for T100: (T1t, T2 , T3) for t  [N1/3]
Similarly for T010: (T1, T2t, T3) & T001: (T1, T2, T3t)
Answer query & 3N1/3 “potential” queries
Server 2: Query T111 =(T1  i1, T2  i2, T3  i3)
List “potential” queries for T011 ,T101 , T110
Answer query & 3N1/3 “potential” queries
Client picks correct answer in each answer list and XORs them
Private Simultaneous Messages (Recap)
Model [FKN94]
f(x,y)
x
Single referee
Two (or more) clients
Non-interactive
Referee learns only f(x,y)
Clients share randomness
• Unknown to referee
• All parties know f
• No collusion
•
•
•
•
•
r
r
y
PSM Complexity of a function f
Communication complexity of PSM protocol for f
• What is the PSM complexity of the worst function in FN?
Efficient for small-depth formulae
Worst case f : O(N) [FKN94]
O(N) Upper Bound on PSM Complexity
Via 4-server PIR
High-level idea
Clients use shared
randomness & referee’s
help to emulate client + 3
PIR servers in 4-server PIR
scheme of [CGKS95]
f(x,y)
x
r
r
y
4-server PIR [CGKS95]
Obtained by collapsing basic
16-server O(N1/4) PIR scheme
• DB = truth table of f
• Client query i = x||y
•
•
•
•
Key Observation
Index i  (i1 , i2 , i3 , i4)
Input x specifies i1, i2
Input y specifies i3, i4
15 of 16 servers
emulated by clients
O(N) Upper Bound on PSM Complexity
Via 4-server PIR
Query Tk1 , … , kd for Server k
(T1(k1 i1), … ,T4(k4 i4))
where k  ( k1,…, k4)
Key Observation
• i  (i1 , i2 , i3 , i4)
• x specifies i1, i2
• y specifies i3, i4
i1
T1 i1
ix2
T2 i2
T**01
T1111
T0000=(T1,…,T4)
T**00
T**10
Query + Answer Generation
Alice knows T1 i1 , T2 i2
• Answers for T**00
• “Potential” answers for T**01, T**10
Bob knows T3 i3 , T4 i4
• Answers for T00**
• “Potential” answers for T01**, T10**
Missing query T1111 equals
• (T1 i1 , T2 i2, T3 i3 , T4 i4)
Answer to T1111 computed by referee
i3y i4
T00**
T3 i3
T01**
T10**
T4 i4
O(N) Upper Bound on PSM Complexity
Via 4-server PIR
Query + Answer Generation
• Answers for T**00,T00**
• “Potential” answers for
T**01, T**10 , T01**, T10**
• Referee answers T1111
Reconstruction
Selecting from “potential” answer list
• Use known PSM (small-depth circuit)
• PSM outputs XOR of these 15 answers
Remaining answer computed by referee
• Finally, XORs this with PSM output
Referee’s reconstruction
function is “non-universal”
Summary
• Cryptographic complexity of worst functions
– Main Technical Tool - PIR
• OT Complexity
– Upper bound: O(N2/3)
2-server PIR
• Correlated Randomness Complexity
– Upper bound: 2O( log N)
• PSM Complexity
– Upper bound: O(N)
3-server PIR
4-server PIR
• Share Complexity for Forbidden Graphs
– Upper bound: O(N)
Using PSM above
Thank You!
Preliminary Version: www.cs.umd.edu/~ranjit/BIKK.pdf
Slides: www.cs.umd.edu/~ranjit/BIKK.pptx
The research leading to these results has received
funding from the European Union's Seventh Framework
Programme (FP7/2007-2013) under grant agreement
no. 259426 – ERC – Cryptography and Complexity