Abdullah Talayhan - 26.11.2025
CS470/519
Introduction to Applied Cryptography
Week10c
1
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Abdullah Talayhan - 26.11.
CS470/ Introduction to Applied Cryptography Week10c
Secure Multi-Party Computation
(MPC)
Goal x 1 x 2 x 3 x 4 x 5 f ( x 1 , x 2 , x 3 , x 4 , x 5 ) = y
5 x 1 x 2 x 3 x 4 x 5 f ( x 1 , x 2 , x 3 , x 4 , x 5 ) = y The goal is to compute f (i.e. learn the output y ) without disclosing any x i Goal
x 1 x 2 x 3 x 4 x 5 Example: Total Salary x Salary i : f ( x 1 , x 2 , …, x 5 ) = ∑ i x i =?
Core Technique behind MPC Secret Sharing x 1 x 2 x 3 x 4 x 5 f ( x 1 , x 2 , x 3 , x 4 , x 5 ) = y Split the secret Split the secret Split the secret Split the secret Split the secret
Linear Secret Sharing Secret Sharing Goal: Distributed a secret , to parties. Such that without parties coming together, you can not recover. We work in. s n n s 𝔽 p r 1 , r 2 , …, r n − 1 $ ← 𝔽 p r n = s − n − 1 ∑ i = 1 r i Notice: s = r 1
- r 2
- r 3
- … + r n
x 1 x 2 x 3 x 4 x 5 Example: Total Salary x Salary i : f ( x 1 , x 2 , …, x 5 ) = ∑ i x i =?
x 1 x 2 x 3 x 4 x 5 Example: Total Salary x Salary i : x 2 = r 2,1 + r 2,2 + r 2,3 + r 2,4 + r 2,
x
3
= r
3,
+ r
3,
+ r
3,
+ r
3,
+ r
3,
x 4 = r 4,1 + r 4,2 + r 4,3 + r 4,4 + r 4,
x 5 = r 5,
- r 5,
- r 5,
- r 5,
- r 5,
x
1
= r
1,
+ r
1,
+ r
1,
+ r
1,
+ r
1,
x 1 x 2 x 3 x 4 x 5 Example: Total Salary x Salary i : r 1, r 1, r 1, r 1,
x
1
= r
1,
+ r
1,
+ r
1,
+ r
1,
+ r
1, x 2 = r 2,1 + r 2,2 + r 2,3 + r 2,4 + r 2,
x
3
= r
3,
+ r
3,
+ r
3,
+ r
3,
+ r
3,
x 4 = r 4,1 + r 4,2 + r 4,3 + r 4,4 + r 4,
x 5 = r 5,
- r 5,
- r 5,
- r 5,
- r 5,
x 1 x 2 x 3 x 4 x 5 Example: Total Salary x Salary i : Party 3,4,5 do the same
x
1
= r
1,
+ r
1,
+ r
1,
+ r
1,
+ r
1, x 2 = r 2,1 + r 2,2 + r 2,3 + r 2,4 + r 2,
x
3
= r
3,
+ r
3,
+ r
3,
+ r
3,
+ r
3,
x 4 = r 4,1 + r 4,2 + r 4,3 + r 4,4 + r 4,
x 5 = r 5,
- r 5,
- r 5,
- r 5,
- r 5,
17 x 1 x 2 x 3 x 4 x 5 Example: Total Salary x Salary i : r 2, , r 3, , r 4, , r 5, r 1, , r 3, , r 4, , r 5,2 r 1,3,^ r 2,3,^ r 4,3,^ r 5, r 1, , r 2, , r 3, , r 5, r 1, , r 2, , r 3, , r 4,
x 1 = r 1,1 + r 1,2 + r 1,3 + r 1,4 + r 1,
x 2 = r 2,1 + r 2,2 + r 2,3 + r 2,4 + r 2,
x
3
= r
3,
+ r
3,
+ r
3,
+ r
3,
+ r
3,
x 4 = r 4,1 + r 4,2 + r 4,3 + r 4,4 + r 4,
x 5 = r 5,
- r 5,
- r 5,
- r 5,
- r 5,
19 x 1 x 2 x 3 x 4 x 5 Example: Total Salary x Salary i : r 2, , r 3, , r 4, , r 5, r 1, , r 3, , r 4, , r 5, r 1, , r 2, , r 4, , r 5, r 1, , r 2, , r 3, , r 5, r 1, , r 2, , r 3, , r 4,
y
1
= r
1,
+ r
2,
+ r
3,
+ r
4,
+ r
5,
x 1 = r 1,1 + r 1,2 + r 1,3 + r 1,4 + r 1,
x
3
= r
3,
+ r
3,
+ r
3,
+ r
3,
+ r
x 3, 2 =^ r 2,1 +^ r 2,2 +^ r 2,3 +^ r 2,4 +^ r 2,
x 4 = r 4,1 + r 4,2 + r 4,3 + r 4,4 + r 4,
x 5 = r 5,
- r 5,
- r 5,
- r 5,
- r 5, y 2 = r 1,
- r 2,
- r 3,
- r 4,
- r 5, y 3 = r 1,
- r 2,
- r 3,
- r 4,
- r 5, y 4 = r 1,4 + r 2,4 + r 3,4 + r 4,4 + r 5, y 4 = r 1,5 + r 2,5 + r 3,5 + r 4,5 + r 5, ∑ i x i = ∑ i y i
20 x 1 x 2 x 3 x 4 x 5 f ( x 1 , x 4 , x 5 ) = y
The goal is to compute (i.e. learn the output ) without disclosing any and a subset
of size shares is enough to learn the outcome.
f y x i t Threshold MPC Goal t = 3