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Alice
Bob
private vector: y
random number: v = r1 + r2 + r3 + r4
private vector: x
v1
v2
v3
v4
x = v1+v2+v3+v4
v1
v3
y+r1, v2
y+r3, v4
Alice gets:
x
y+r2,
y+r4
v4
hiding v1,v2,v3,v4
among random vectors
v2
v1
v3
1-out-of-n
Oblivious Transfer
y + v = (v1 y + r1) + (v2
y + r2) + (v3
y + r3) + (v4 y + r4)
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Alice (A’)
Bob (B’)
(2a2+1)+R4
q4 q4’
2b2+R4
p4 p4’
2a2+R2
q2 q2’
(2b2+1)+R2
p2 p2’
−2a1+R5
q5 q5’
−(2b1+1)+R5
p5 p5’
−(2a2+1)+R8
q8 q8’
−2b2+R8
p8 p8’
2a1+R1
q1 q1’
(2b1+1)+R1
p1 p1’
−(2a1+1)+R7
q7 q7’
−2b1+R7
p7 p7’
−2a2+R6
q6 q6’
−(2b2+1)+R6
p6 p6’
(2a1+1)+R3
q3 q3’
2b1+R3
p3 p3’
: A dominates B
: B dominates A
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M
X=b
(a) Normal Linear Equations
(without cooperation)
M1 M2
X=b
(c) Heterogeneous Cooperation
M1
M2
b1
X=
b2
(b) Homogeneous Cooperation
M1 + M2
X=
b1 + b1
(d) Hybrid Cooperation
Legend: M1: Alice’s private matrix, M2: Bob’s private matrix
b1: Alice’s private vector, b2: Bob’s private vector
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´k
Alice
Bob
private input: M1
private input M2=Y1+...+Y4
X4
X2
X1
X3
X4
M1=X1+X2+X3+X4
hiding X1,...,X4
among random matrices
X2
X1
X3
P(X1+Y1)Q+R1, ...,
P(X4+Y4)Q+R4
Alice gets:
1−out−of−N
Oblivious Transfer
P(M1+M2)Q =P(X1+Y1)Q+R1 + ...+P(X4+Y4)Q+R4− (R1+...+R4)
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Input 1
Input
Computation
Computation
D
Result
Result
Input 2
Single−Input Computation Model
Multi−Input Computation Model
Input 1
Computation
D1
Computation
Alice
Alice
Result
Result
Input 2
D2
Bob
Bob
Secure Multiparty Computation Model
(A) Transformation of a Multi−Input Computation Model
to a Secure Multiparty Computation Model
Legend:
D = union(D1, D2)
Secure Multiparty Computation Model
(B) Transformation of a Single−Input Computation Model
to a Secure Multiparty Computation Model
Privacy Requirements
Conputation
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