信息安全数学基础-陈恭亮-ch9-群.pdf

文档介绍：?ˉì?IK?81?JJIIJI11??48??￡??w?'4ò?1lù+&ES?ê??:?????/??)??t°????&ES?ó§??******@sjtu.?ˉì?IK?81?JJIIJI12??48??￡??w?'4ò?+?(??ì?+?k?)¤??+?ì?+?A^?ˉì?IK?81?JJIIJI13??48??￡??w?'4ò?ì?+????nnn1\+Z?z?f+H′ì?+.kH={0}?H=< m >=mZ,ù￥m′H￥?????ê.XJH6=<0>,KH′???.yyyXJH={0},(?¤á.XJH6={0},K3?"?êa∈H.??H′f+,¤±?a∈H.ù`2H￥k??ê.?H￥?????ê???kH=< m>=￠t,é???a∈H,?a?Ap??{,3?êq, r|?a=qm+r,0≤r < = 0,Kr=a?qm∈H,ù?m???5g?.?d,r= 0, a=qm∈mZ.?H?mZ.?w,kmZ?H.?d,H=..?ˉì?IK?81?JJIIJI14??48??￡??w?'4ò?????nnn2z???ì?+ó?u\+???m?k?ì?+ó?u\+Z/mZ.?yyy?ì?+G=< a >={an|n∈Z}.?N?f:Z?→Gn7?→an??f(n+m) =an+m=anam=f(n)f(m),¤±f′Z?G?ó?,′÷?.?a§?n9,+Gó?uZ/ker(f).?a?n1,ker(f) =<0>?ker(f) =?éAu??ì?+,??éAum?k?ì?+.y..???????1?G′??+,a∈+< a >??????a??,P?ord(a).?ˉì?IK?81?JJIIJI15??48??￡??w?'4ò?????nnn3?G′??+,a∈′???,K(i)ak=e?k= 0.(ii)??ak(k∈Z)üü?ó.XJa′k??m>0,K(iii)m′|?am=e?????ê.(iv)ak=e?m|k.?yyy?Z?+GN?f:n7?→an. f′ó?.kZ/kerf～=< a > .??a′?????dukerf={0},??`2f′?é??.?d, (i)ú(ii)¤á.XJa′k??m,Kkerf=mZ.?d,·?k(iii)m′|?am=e?????ê.(iv)ak=e?duk∈kerf,?dum|k.?ˉì?IK?81?JJIIJI16??48??￡??w?'4ò?????nnn3(Y)?G′+,a∈′???,K(v)ar=ak?r≡k(modm).(vi)ak(k∈Z/mZ)üü?ó.(vii)< a >={a, a2, . . . , am?1, am=e}.(viii)ord(ad) =m(m,d).?yyy(v)ar=ak??r?k∈kerf??r≡k(modm).(vi)??akéAuZ/kerf￥?ó??,üü?ó.(vii)< a >={a, a2, am?1, am=e}?Z/kerf￥????{X?éA.(viii)é???ê1≤d≤m,kord(ad) =m(m,d).(ad)k=e??dk∈kerf??m|dk??m(m,d)|d(m,d)k??m(m,d)|k.?ord(ad) =m(m,d).?ˉì?IK?81?JJIIJI17??48??￡??w?'4ò?????nnn4ì?+?f+′ì?+.?yyy?N?Z?ì?+G=< a >?N?f:f:n7?→′ó?N?.?a§?n1,éuG?f+H,·?kK=f?1(H)′Z?f+.?a?n1,K′ì?+,¤±H=f(K)′ì?+.y..?ˉì?IK?81?JJIIJI18??48??￡??w?'4ò?????nnn5?G′ì?+.XJG′???,KG?)¤??aúa?′k??m,Kak′G?)¤???=?(k, m) = 1.?yyy?N?Z?ì?+G?N?f:f:n7?→′ó?N?.?a§?n9,·?kZ/kerf～=G.??G￥?)¤?éAuZ/kerf￥?)¤?.?kerf= 0?,Z/kerf?)¤?′1ú?1;kerf=mZ, m >0?,Z/kerf?)¤?′k,(k, m) = 1.?d,?n¤á.y..?ˉì?IK?81?JJIIJI19??48??￡??w?'4ò??úúúnnn1?G′k???+.é????a, b∈G,e(ord(a),ord(b)) = 1,Kord(ab) = ord(a)ord(b).?yyy??aord(ab)ord(b)= (ab)ord(ab)ord(b)= 1,¤±ord(a)|ord(ab)ord(b).?o