Abstract by anuja a sonalker on Asymmetric Key Distribution

3.5 Proof of correct Signature formation: 
If there are a total of t+1 shares required to make a valid signature coming from t Share 
Servers and one Special Server, the following is true: 
1
 
2
 
3
 
4 … 
 
t+1 
| 
| 
| 
|
| 
S
1 
= M
d
1 
mod N S
2 
= M
d
2 
mod N S
3 
= M
d
3 
mod N S
4 
= M
d
4 
mod N …S
t+1 
= M
d
c 
mod N 
Then, S = 
∏
+
=
1
1
t
i
i
S  mod N i.e., 
S = [S
1
x S
2
x S
3
x S
4…
x S
t+1 
] mod N 

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25 
$ 
S
= [(M
d
1 
mod N) x (M
d
2 
mod N) x (M
d
3 
mod N) x (M
d
4 
mod N) … x
(M
d
t
mod N) x (M
dc
mod N) ] mod N
$ 
S = (M
dc 
mod N) x
t
Π
i=1
S
i 
mod N
$ 
S = S
c
x
∏
+
=
1
1
t
i
i
S  mod N 
Further, we see that 
$ 
S = M
d
1 
+ 
d
2 
+
d
3 
+ 
d
4 
+…+ 
d
t 
+ 
d
c
mod N 
$ 
S = M
d
mod N 
$ 
S is the complete signed signature equivalent to the complete key d being used 
to sign the CSR. 
It can also be verified thus: 
S
e
mod N = M. 
If the signature was computed using the right shares, using the public exponent e on 
the final signature results in getting the original message back again. 
3.6 Key Generation by the Trusted Dealer. 
The Trusted Dealer is responsible for the generation and distribution of the secret 
components and the private-key shares to the concerned players in the system 
appropriately. 
He generates the modulus N, the public exponent e, the complete private exponent d. He 
then chops up the private exponent d into the required number of private shares for 
distribution. The secret components are generated in a technique similar to that used in 
the RSA algorithm, which was described earlier.

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26 
In order to generate a modulus N, the Trusted Dealer starts by generating a very strong, 
large prime number p. The emphasis on strong and large has been highlighted earlier. On 
successful verification of its prime nature, it generates another prime number q of the 
same order of magnitude as p. By order of magnitude of the secret components, we mean 
the size of the numbers in bits. The two numbers p and q are known as the prime 
derivatives of N. The TD computes N = p · q, which is of the order of magnitude of p + 
q. He selects the public key e such that e is relatively prime when compared to p and q. 
He computes the complete private-key d
 
from e as per the RSA scheme. Once the private-
key is computed, he sets himself to the task of splitting up the key into the required 
number of secret key shares.