seminar of A Threshold Cryptosystem without a Trusted Party in Advances in Cryptology — EUROCRYPT ’91. EUROCRYPT 1991 by Torben Pryds Pedersen

1. A Threshold Cryptosystem without a Trusted Party
Advances in Cryptology — EUROCRYPT ’91. EUROCRYPT 1991
Torben Pryds Pedersen
Aarhus University, Computer Science Department
林彥賓
2021/1/23

2. 1. chooses at random and computes and brocast
2. each player opnen , the public key is
3. choose random polynomial of degree such that .
f (z) =
i f +
i,0 f z +
i,1 ... + f z
i,k−1
k−1
where
4. compute for and brocast
5. send secretely to for
6. verifies share received by by checking
Pi xi h =
i gxi
C =
i Com(h )
i
Ci h = h
∏i=1
n
i
Pi f (z)
i k − 1 f (0) =
i xi
f =
i,0 xi
Pi F =
i,j gfi,j
j = 0, ..., k − 1 Fi,j
Pi s =
i,j f (j)
i Pj j = 0, ..., k − 1
Pi Pj g =
si,j
F
∏ℓ=0
k−1
j,ℓ
iℓ
2

3. Set
Let be polynomid over
we know that for , thus is a share of
f (0), f (1), f (2), ..., f (k −
1 1 1 1 1)
+
...
+
f (0), f (1), f (2), ..., f (k −
k−1 k−1 k−1 k−1 1)
x = f(0), s =
1 f(1), ..., s =
k−1 f(k − 1)
s =
i s
∑j=1
n
j,i
f Z :
q f(z) = f (z) +
1 ... + f (z)
n
s =
i f(i) i = 1, ..., n si f(0) = x
3

4. For share of , can compute public key
σ =
i g =
j=1
∏
n
sj,i
(h F )
j=1
∏
n
j
ℓ=1
∏
k−1
j,ℓ
iℓ
si Pi
4