1. 1 An Average-case Analysis of the Gaussian Algorithm for Lattice Reduction An Average-case Analysis of the Gaussian Algorithm for Lattice Reduction Combinatorics, Probability and Computing, 1997 H. Daudé, P. Flajolet, B. Vallée, 田錦燕 95/3/10

2. 2 Abstract Continuous model - average-case → constant - probability distribution - dynamics - functional operator Discrete model

3. 3 Lattice reduction in dimension 2 給兩個在歐氏空間 ( Euclidean vector space ) ，線 性獨立 ( linearly independent ) 的向量 ( vector )u 與 v ， 求 u 與 v 最短非零的線性組合 ( Z-linear combination )

4. 4 Define

5. 5 1 B D F 是 minimal basis 在 D 之內是肯定不是最小 B\D 是 Quasi-minimal basis z = v/u, basis of the form (1,z) 若 (u,v) 為 Quasi-minimal, 則 minimal 必為下六者其中之一 R(z) 0

6. 6 R(z)=1/2 z=1

7. 7 Continuous model vs Discrete model Continuous model input 自 Domain D Discrete model

8. 8 Average-case average-case → constant SGA Algo. 執行 k 次的機率

9. 9 Probability distribution 10 8 Simulations SGA Algo. 執行 k 次的機率

10. 10 Dynamic analysis 做 k 次的 case, 分佈並非平均分佈 密度