[試題] 98下 陳君明 密碼學 第一次小考
課程名稱︰密碼學
課程性質︰
課程教師︰陳君明
開課學院：
開課系所︰數學系
考試日期（年月日）︰2010/03/16
考試時限（分鐘）：30
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Student ID:__________ Name:__________
s=____= 2 + "the last digit of your ID", 2≦s≦11
1) Consider the group G = (Z13*, ×mod13)
a) s^-1(the multiplicative inverse of s) is __________
b) o(s)(the order of s) = __________
c) The index [G : <s>] = __________
d) Explain why G is a cyclic group
2) Consider the homomorphism f:(Z12, + mod 12) → (Z13*, ×mod 13)
defineed by f(1) = s
a) f(3) = __________
b) Is f an isomorphism? Explain
3) Prove that the inverse of any element g in a group G is unique
4) |GL2(Z13)| = __________, |SL2(Z13) = __________|
5) Consider the symmetric group S6:
a) ┌1 2 3 4 5 6 ┐ ┌1 2 3 4 5 6 ┐
│ │。│ │ = __________
└3 6 2 4 1 5 ┘ └6 4 1 2 5 3 ┘
b) ┌1 2 3 4 5 6 ┐-1
│ │ = __________
└3 6 2 4 1 5 ┘
6) Show that the quotientring Z[x]/<x^2 -4> is NOT an integral domain
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03/12 16:46, , 1F
03/12 16:46, 1F