Complexity and Fast Algorithms for Multiexponentiations

# Complexity and Fast Algorithms for Multiexponentiations

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## Complexity and Fast Algorithms for Multiexponentiations

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1. 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

2. Authentication • Symmetric verifier知 the secret (secret key) or an image of the secret (password) • Asymmetric verifier知 a public key

3. 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

4. Asymmetric Authentication • Static : • Dynamic :

5. 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

6. 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=rQd 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

7. 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

8. 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