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Complexity and Fast Algorithms for MultiexponentiationsPowerPoint Presentation

Complexity and Fast Algorithms for Multiexponentiations

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### Complexity and Fast Algorithms for Multiexponentiations

Source: IEEE Transactions on Computers

Vol. 49 pp.141-147 2000

Author: Vassil S. Dimitrov, Graham A. Jullien,

and William C. Miller

Speaker: Lai, Yi-Peng

Date: 04/25/2002

Authentication

- Symmetric
verifier知 the secret (secret key) or an image of the secret (password)

- Asymmetric
verifier知 a public key

Symmetric Authentication

- One-way function without challenge 1981
1st round: Image = fk (r),

input i = fk-1 (r), compute f(i), verify f(i) ?= image,

replace image with i.

……

n-th round: Image = fk-n+1 (r)

input i = fk-n (r), compute f(i), verify f(i) ?= image,

replace image with i.

- Dynamic authentication

Asymmetric Authentication

- Static :
- Dynamic :

Dynamic Asymmetric Authentication

- generic equation: GQv=1 mod n
- public number deduced from id: G
- public verification key: (v,n)
- private number: Q
- non-zero random number: r

Dynamic Asymmetric GQ1

Verification key: (v, n)

verifier

claimant

id

Format Mechanism

r{1,2,…,n-1}

R=rv mod n

G

d{0,1,…,v-1}

d

Secret Q

D=rQd mod n

注:因為於id訂定時已藏入相關於該id對應的public number G 並算出符合generic equation(GQv=1 mod n)的secret Q

Dynamic Asymmetric GQ2

Verification key: (v, n), where v=2k

verifier

claimant

id

Format Mechanism

r{1,2,…,n-1}

R=rv mod n

G1, G2,…, Gm

d1 ~ dm{0,1,…,2k-1–1}

d1 ~ dm

Secret g1, g2,…, gm

注: Gi = gi2 mod n, where i= 1~m

Conclusion

- Computation引入中國餘數定理
- NetWare 4.11 and 5.0 based on GQ1 challenge 32bits v=216+1
- Smart card (ST 16601 3.57MHz):
(1)14sec for RSA – 512bits, CRT, n=p1p2p3

(2)14sec for GQ1 – 768bits, v=216+1

(3)1 sec for GQ2 – 512bits,k=5,m=3,n=p1p2p3

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