课程名称︰椭圆曲线密码学 课程性质︰数学系选修 课程教师︰陈君明 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2015/10/06 考试时限（分钟）：5:30~6:30 试题 : 1. A group is a set G and an operation * for which the following axioms hold: ‧For any a,b∈G, the element a*b∈G. ‧......(please complete the definition) (6%) 2. (a) Show that (Z*,× mod 7) is a cyclic group. (6%) 7 (b) Show that (Z*,× mod 9) is a cyclic group. (6%) 9 (c) Show that (Z*,× mod 8) is not a cyclic group. (6%) 8 3. Suppose m is the multiplicative inverse of 52 in the Galois field F (GF ) 313 313 (a) Use the Extended Euclidean Algorithm to find m. (6%) (b) Describe how to use the Fermat Little Thorem to find m. (6%) 3 4. (a) How many elements are there in the quotient ring F [x]/<x + x + 1>?(6%) 2 2 3 (b) Find the multiplicative inverse of x + x in F [x]/<x + x + 1>. (6%) 2 2015 5. (a) Evaluate 3 mod 109. (6%) 2015 (b) Evaluate 3 mod 108. (6%) 3 6. (a) Factor x + 1 over R,C,F ,F ,F respectively. (10%) 2 3 7 16 (b) Factor x + x over F . (6%) 2 (c) How many irreducible polynomials of degree 5 over F ? (6%) 2 (d) How many irreducible polynomials of degree 6 over F ? (6%) 2 7. (a) Find an irreducible polynomial of degree 3 over F . (6%) 5 (b) Construct the Galois field F and explain its operations. (6%) 125 --

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