信息安全数学基础 (裴定一 徐祥 著) 人民邮电出版社 课后答案 18页

'课后答案网您最真诚的朋友www.hackshp.cn网团队竭诚为学生服务，免费提供各门课后答案，不用积分，甚至不用注册，旨在为广大学生提供自主学习的平台！课后答案网：www.hackshp.cn视频教程网：www.efanjy.comPPT课件网：www.ppthouse.com 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！1DB75FA?8EG1.11111111101,3FD1.221.33741332801.4510×11766−737×8142=6.t1a,b=pt2b,1.5=a=p11%"(p,a1)=(p,b1)=1,ordp(a)=t1,ordp(b)=t2.=t1t2(pt1−t2a+b)≥t2,a+b=p11ordp(a+b)≥t2=min{ordp(a),ordp(b)},(7t>t,t1−t2t1−t212pa1+b16=0,((p,pa1h+bi1)=1,"::
,Wni,i1.61,2,···,n"p++Gbpi":1,2,···,n"p:hihinni+1−,p++Gbpipi+1Xλhnihniordp(n!)=i(−)pipi+1i=1Xλhni=pii=11.7j22|n(n+1)(2n+1)课后答案网=n=3s+r,0≤r<3,n+1=3s+(r+1),2n+1=3(2s+r)−(r−1),r−1,r,r+1~G33|n(n+1)(2n+1)":6|n(n+1)(2n+1)1.8j2(a,b)www.hackshp.cnm|am,(a,b)m|bm,((a,b)m>0=x,y∈ZGCax+by=(a,b)+amx+bmy=(a,b)m.7c1c|am,c|bm,:E{!c|(a,b)m(am,bm)=(a,b)m1.9=(a1,a2,···,an)=d,(a1,a2,···,as)=d1,(as+1,···,an)=d2,(d1,d2)=d3,j2d|d3~?d3|d1,d3|d2d3|ai,i=1,2···,nd3|dd,d32bd=d31.10=[a1,a2,···,an]=m,[a1,a2,···,as]=m1,[as+1,···,an]=m2,[m1,m2]=m3,j2m1|m3,m2|m3":ai|m3,i=1,2···,nm|m3~?ai|m,i=1,2···,n!m1|m,m2|m":m3|mm=m31.11(a,c)=1!#x,yC*ax+cy=1abx+cby=b7c|abc|b1.12(1)3k+1b&kb!=b2h":3k+1=6h+1 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！2(2)3k−1b&kb!=b2h":3k−1=6h−1nn1.137G==p1p2,···,pt,t≥2p+3rv{!n=pp1p2···ptpp111>n3·n3···n3≥n8|{z}t+1n−m1.142=n>mFn=(Fm−1)+17d|Fn,d|Fmd|2d%G&8Fnd*9GG&;~G-+Fn+-d*9G1.15p=2k+122b&p=(k+1)−k1.16=a=2u,b=2vu,vba+b=2(u+v)=4s(u+vb!)(a+b,4)=41.17100,10,10;100,20,10;100,50,10;50,20,10;100,50,20;50,20,20;50,50,20;100,100,10.hihj−1ina0+a1p+···+aj−1pk−j1.18pj=pj+aj+aj+1p+···+akp=a+ap+···+apk−jhijhj+1iknn−p=ajpjpj+189G2.1x≡27(mod30)课后答案网2.2x≡157(mod315)2.3x≡67(mod90)2.4(1)1,3,5,7,9,11,13,15,17,19,21www.hackshp.cn(2)0,2,4,6,8,10,12,14,16,18,202.5=t0,t1,···,tm−1bm+~G`0?h%"ti≡i(modm)ti=mqi+i,i=0,1,···,m−1mb!{!t0,t1,···,tm−1"&!H2.616+h1357911131518+h1571113172.7mϕ(n)≡1(modn)ϕ(m)n≡1(modm)n|mϕ(n)−1,m|nϕ(m)−1.n|mϕ(n)+nϕ(m)−1,m|mϕ(n)+nϕ(m)−1.ϕ(n)ϕ(m)ϕ(n)ϕ(m)(m,n)=1mn|m+n−1fm+n≡1(modmn).2.9t1trk1klk1kjv1vl=m=p1···prq1···ql,n=u1···ujq1···ql 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！3s=q1···ql,ϕ(s)=(q1−1)···(ql−1),1111ϕ(m)=m(1−)···(1−)(1−)···(1−),p1prq1ql1111ϕ(n)=n(1−)···(1−)(1−)···(1−),u1ujq1qlϕ(mn)1q1···qls===ϕ(m)ϕ(n)(1−1)···(1−1)(q1−1)···(ql−1)ϕ(s)q1qlt1tr112.10(1)=n=p1···pr,p1"7n≡an(modp)6n≡0(modpk)sk)kk7n6≡0(modpp|n(n−a)⇒p|n−ap6|a⇒p6|n":p|n−a 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！4kfn≡a(modp)2.2022j≡(m−j)(modm)2.21(1)73k−1ztE+-3k−1+-Ct6itEQs3k−1=(3k+1)viii=1Qsvi−1≡1(mod3),f2≡0(mod3)8i=174k−1ztE+-4k−1b&{!%-Ct6itEQs4k−1=(4k−3)uiii=1−1≡1(mod4)876k−1ztE+-6k−1+-CttEbQs6k−1=(6k−5)uiii=1−1≡1(mod6)8(2)=4k−1+-utG=b4ki−1,i=1,···,tQt(4ki−1)tb!;un=i=1Qt(4ki−1)·(4k1−1)tb&课后答案网i=1u≡1(mod4)u=4s+1www.hackshp.cnw=u−2=4s−1(1)wt64v−1+-74v−1|u4v−1|2{4v−14ki−1,i=1,2,···,t8(3)=6k−1+-utG=b6ki−1,i=1,···,tQt(6ki−1)tb!;un=i=1Qt(6ki−1)·(6k1−1)tb&i=1u≡1(mod6)u=6s+1w=u−2=6s−1(1)wt66v−1+-76v−1|u6v−1|2{6v−16ki−1,i=1,2,···,t88=G3.11,1,1,1,1,−1,151111−63.3i)=−1,dtii)=−1,dtiii)91=7·13,==1,2275117711−65−14==−1tiv)6193=11·563,=1,=−1dt13131111 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！53.4i)p=±1(mod5)ii)p=1,3,7,9(mod20)3.5p+B0?
"~Gji−ja=e6.5a,b∈Gaabb=abab=eq-ab=ba−10106.6a2=,a4=,a+rb40−101−1−110b2=,b3=,b+rb31001111nab=,(ab)n,=ab+rb∞01016.7(1)−1=o(a)=m,o(a)=n,e=(aa−1)m=(a−1)me=(aa−1)n=(a)nm≥n,n≥m":m=n 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！8(2)mm−1mn=o(ab)=m,o(ba)=ne=(ab)=a(ba)a⇒(ba)=e(ab)=em≥nen≥m":m=n(3)−1−1mm−1−1nn−1=o(a)=m,o(cac)=n(cac)=cac=eee=(cac)=cacn":a=em≥n,n≥mfm=n.6.8mnq+rr=m=nq+r0≤raϕ(n)G6.28G=(a),H⊳GG/H=(aH)6.29=G/C(G)=(aC(G))∀x,y∈G#c1,c2∈C(G)er1,r2C*x=ar1c,y=ar2cr1r2r2r112xy=(ac1)(ac2)=(ac2)(ac1)=yx6.30|G|=|G/N|·|N|<∞6.316G-1vuly6.32ϕ:4Z→Z5fπ51b4Z→Z→Z5ffπ+FY%"f(4k)=k,π(k)=[k]Kerϕ=Ker(π·f)={20k|k∈Z}6.33f−1&j2Ker(f)={e}7Ker(f)={e}=f(x)=f(y)f(xy)=f(x)f(y)−1=e⇒xy∈Kerf⇒x=yf&8"7ZnbÆ6Rn=p1p20=n=p1·p2ZnbÆ87.11∗=pGF+Up|4"pbp=2FGÆFG~G3rw322℄1x6=0,1x=1f(x−1)(x+x+1)=0"x−16=0x+x+1=02=x+1fx(CharF=2)7.12Zn"G={a|a∈Zn,(a,n)=1}Zn+=1|G|=ϕ(n) 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！10aϕ(n)≡1(modn)7.137.12{*7.14I1={0},I2={0,3},I3={0,2,4},I4=Z67.15(x2)={x2f(x)|f(x)∈F[x]}2F[x]/(x)={a0+a1x|a0,a1∈F}7.16(2,x)={f(x)|f(x)+ob!}(2,x)G*n7.17(p)∩(q)=(pq)7.18R/I∼=Z27.19x1y7.20(1)=a,b∈S(ab∈f(P)fG<{=f(x)=a,f(y)=bf(xy)=ab∈f(P)xy∈P⇒x`y∈P⇒f(x)`f(y)∈f(P)(2)−1=a,b∈R(ab∈f(Q)f(a)f(b)∈Q⇒f(a)`f(b)∈Q⇒a−1−1`b∈f(Q)j2K⊆f(Q)7.21f(a)=a+I7.22x1y7.23x1y课后答案网7.24(1)I6=(4)(2)R/(4)GÆ7.25f:I→I+J/Jf(a)=a+JfG℄SKerf=I∩J℄Sa∼=I+J/J1I/Iwww.hackshp.cn∩J7.267f&j2Ker(f)={0}>"7Ker(f)={0}f(x)=f(y)f(x−y)=0⇒x−y∈Ker(f)⇒x=y3+16=037.27R[x]"x+16=0x"x+1·x+1=09℄-{GÆ7.28f(3)=4nn−1n−17.29n=1ppppsj=n−1x−x+na=(x−x+a)+x−x+(n−1)aEns|7.30xEh7.31i)x=y−12Z,p=2ii),p=386G8.1[E:F][F(α):F]([F(α):F]>1{![F(α):F]=pE=F(α)8.2[F(α):F][E:F]fm|n8.3RGQ+dl~CGR+2!~RGQ+%~CGR+%~ 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！11CGQ+%~8.4[F(u2):F]|[F(u):F]22uF:+!|G&[F(u):F]=[F(u):F][F(u):22)]≤222F(u)]([F(u):F(u[F(u):F]=[F(u):F]f*F(u)=F(u)Pn8.5i=αE:+r9oEf(x)=aix,a0,a1,···,an∈EαF(a0,a1,···,an)i=0:%a0,a1,···,anF:%{*[F(a0,a1,···,an):F]<∞":[F(a0,a1,···,an)(α):F]=[F(a0,a1,···,an)(α):F(a0,a1,···,an)][F(a0,a1,···,an)(α):F]<∞αF:%11113)(i)3338.6Ei=Q(2[E1:Q(2)]=2[E1:Q]=[E1:Q(2)][Q(2):Q]=3×2=6;1√11√1√1E2=Q(23)(3)(i)[E2:Q(23)]=[E2:Q(23)(3)][Q(23)(3):Q(23)]=2×2=4;11[E2:Q]=[E2:Q(23)][Q(23):Q]=4×3=128.7[K(α):F]=[K(α):K][K:F]=[K(α):F(α)][F(α):F]n|m[K(α):K](m,n)=1n|[K(α):K]:n≤[K(α):K]n=[K(α):K]8.8nkmk=x−aF:+CÆ"~Gαm=kn(α)−a=0=αFmknn:+r9oEbf(x)f(x)|x−a⇒f(x)|x−ax−a{⇒degf(x)=mm":f(x)=x−ab{8.9αF:+r9oEb课后答案网x−a52i+18.10x2+42∼=Q()x−x+2Q(2i)i−18.11~1gV8.10wG>√8.12ϕ(a+bwww.hackshp.cn3)=|a2−3b2|√8.13ϕ(a+b−2)=|a2+2b2|8.14x8!?++E+t√√2214+i8.15x+1=(x−α)(x−α)(x+α)(x+α)α=22α=αQ(α,α,−α,−α)=Q(α)8.16=E=F(α1,···,αn)F⊂K⊂EK(α1,···,αn)=EfEGf(x)K:+CÆ8.17∆=−4p3−27q2=−27a2<033x−a~64FX~GA7Ebx−a+CÆ[E:Q]=6:[Q(α):Q]=3Q(α)6=E8.18pppppF:x−a=x−α=(x−α)p(α)Gx−aP:+CÆ√√√√√8.19(1)1,3,5,15;(2)1,2,i,2i;(3)1,ω,ω2,i,ωi,ω2i;8.20GÆ8.21222x+1,x+x+42GsbFq""Xs=aas=a=i=0i=1qP−1ai=s(a6=1(q6=2)s=0j=09.2F3577+3,5F969.7F%bζ,ζ,ζ,ζF17+b3,10,5,11,14,7,12,6a2aa29.37b=0a=0D()++1=0fGF2:?x+x+1=0+bbba2x+x+1=0F2:{∈F22F2:nb&F226⊆F2n8b9.4∗m=GGF+~Gnr?1=G"+3$rbm∀b∈Gb=1mmG"1?x=1:F"1?x=1+G≤mfn≤m⇒m=nnnn−19.5pppppp7a∈Fqa=ax−a=x−a=(x−a)u~Gq−1q−1∗22q−19.67a∈Fq~G#?xx=a⇒a2=(x)2=x=1>"7q−1q−1q−1t·t2=122a=ξ∈FqG(a=ξa=ξ=1⇒tb!=t=2ss2b=ξb=afaFq:#?kq−1k·(q−1)q−1kq−1∗d=xdd=1d9.77a=xa=(x)>"7a=1=ξ∈FqGtq−1q−1·t课后答案网a=ξad=ξd=1⇒d|t=t=dsk=dk1%"(k1,q−1)=1=u∈Zusk(us)kuk1dsdsq−1C*uk1≡1(modq−1)b=ξb=ξ=ξ=ξ=a9.8
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⇒+#v-$V⇒(q−1,k)=19.9n=CharFq=pq=pgV7.2{*9.10=f(x)GFq[x]"~GG10.1(1)x1∧x2,(2)x1∨x2,(3)(x1∧x2)∨x3,(4)((x1∧x2)∨x2∧x3)∧x310.2(1)0,(2)1,(3)1,(4)110.3#y10.4x110.4y课后答案网10.5
110.3+10.6(1)xy+x·0=x·(x+y)·y(2)x·y=www.hackshp.cnx+y(3)6Rxy=xz(xy=xz,y=z(4)x+y=1((u(x+y=x(5)7xy=0x=y=1(6)x=1((u(6+yy=(z+y)(x+y)(7)x(x+y·0)=x10.7.uy10.8[x+(y+z)][(x+y)+z]=x[(x+y)+z]+(y+z)[(x+y)+z]=x[(x+y)+z]+(y[(x+y)+z]+z[(x+y)+z])=x+(y+z)[x+(y+z)][(x+y)+z]=[(x+y)+z](x+y)+[(x+y)+z]z=([(x+(y+z)]x+[(x+(y+z)]y)+[(x+(y+z)]z=(x+y)+zx+(y+z)=(x+y)+z(:
,E")fHx(x+y)=x%sY) 课后答案网：www.hackshp.cn若侵犯了您的版权利益，敬请来信告知！14v1ttv4HHHHHHHtHtv2v3^1:10.9110.3e110.4nx+x=1⇒x,=nx⇒n10.10n
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