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|00000110| 20 55 72 62 61 6e 61 0a | 44 61 74 65 3a 20 54 75 | Urbana.|Date: Tu|
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|00003670| 75 6d 62 65 72 2c 20 24 | 5c 63 6f 20 7b 61 72 7d |umber, $|\co {ar}|
|00003680| 7b 61 73 7d 70 24 20 61 | 6e 64 20 0a 24 61 5c 6e |{as}p$ a|nd .$a\n|
|00003690| 6f 74 5c 65 71 75 69 76 | 20 30 24 2c 20 74 68 65 |ot\equiv| 0$, the|
|000036a0| 6e 20 24 72 5c 65 71 75 | 69 76 20 73 24 2e 5c 65 |n $r\equ|iv s$.\e|
|000036b0| 6e 64 7b 43 6f 7d 0a 50 | 72 6f 6f 66 3a 20 53 69 |nd{Co}.P|roof: Si|
|000036c0| 6e 63 65 20 24 70 24 20 | 69 73 20 61 20 70 72 69 |nce $p$ |is a pri|
|000036d0| 6d 65 2c 0a 24 28 61 2c | 70 29 3d 31 24 2c 20 73 |me,.$(a,|p)=1$, s|
|000036e0| 6f 20 54 68 65 6f 72 65 | 6d 7e 5c 72 65 66 7b 54 |o Theore|m~\ref{T|
|000036f0| 31 7d 20 73 61 79 73 20 | 74 68 65 72 65 20 61 72 |1} says |there ar|
|00003700| 65 20 69 6e 74 65 67 65 | 72 20 24 78 2c 79 24 20 |e intege|r $x,y$ |
|00003710| 77 69 74 68 20 24 61 78 | 2b 70 79 3d 31 24 2e 0a |with $ax|+py=1$..|
|00003720| 48 65 6e 63 65 20 24 24 | 5c 63 6f 20 7b 61 78 7d |Hence $$|\co {ax}|
|00003730| 31 70 5c 71 75 61 64 5c | 68 62 6f 78 7b 61 6e 64 |1p\quad\|hbox{and|
|00003740| 7d 5c 71 75 61 64 20 72 | 5c 65 71 75 69 76 28 31 |}\quad r|\equiv(1|
|00003750| 29 72 5c 65 71 75 69 76 | 20 61 78 72 5c 65 71 75 |)r\equiv| axr\equ|
|00003760| 69 76 20 78 61 72 0a 5c | 65 71 75 69 76 5c 63 6f |iv xar.\|equiv\co|
|00003770| 20 7b 78 61 73 7d 73 70 | 24 24 0a 5c 62 65 67 69 | {xas}sp|$$.\begi|
|00003780| 6e 7b 43 6f 7d 49 66 20 | 70 20 69 73 20 61 20 70 |n{Co}If |p is a p|
|00003790| 72 69 6d 65 20 6e 75 6d | 62 65 72 20 61 6e 64 20 |rime num|ber and |
|000037a0| 24 61 5c 6e 6f 74 5c 65 | 71 75 69 76 30 24 7e 6d |$a\not\e|quiv0$~m|
|000037b0| 6f 64 7e 24 70 24 2c 20 | 74 68 65 6e 0a 66 6f 72 |od~$p$, |then.for|
|000037c0| 20 61 6e 79 20 24 62 24 | 2c 20 74 68 65 72 65 20 | any $b$|, there |
|000037d0| 69 73 20 24 79 24 20 77 | 69 74 68 20 24 5c 63 6f |is $y$ w|ith $\co|
|000037e0| 7b 61 79 7d 62 70 24 2e | 5c 65 6e 64 7b 43 6f 7d |{ay}bp$.|\end{Co}|
|000037f0| 0a 50 72 6f 6f 66 3a 20 | 57 65 20 73 68 6f 77 65 |.Proof: |We showe|
|00003800| 64 20 69 6e 20 74 68 65 | 20 70 72 65 63 65 64 69 |d in the| precedi|
|00003810| 6e 67 20 70 72 6f 6f 66 | 20 74 68 61 74 20 74 68 |ng proof| that th|
|00003820| 65 72 65 20 69 73 20 24 | 78 24 20 77 69 74 68 0a |ere is $|x$ with.|
|00003830| 24 5c 63 6f 20 7b 61 78 | 7d 31 70 24 2e 20 20 4c |$\co {ax|}1p$.  L|
|00003840| 65 74 20 24 79 3d 62 78 | 24 2e 0a 5c 62 65 67 69 |et $y=bx|$..\begi|
|00003850| 6e 7b 43 6f 7d 5b 54 68 | 65 20 60 60 43 68 69 6e |n{Co}[Th|e ``Chin|
|00003860| 65 73 65 20 52 65 6d 61 | 69 6e 64 65 72 20 54 68 |ese Rema|inder Th|
|00003870| 65 6f 72 65 6d 27 27 5d | 20 49 66 20 24 28 70 2c |eorem'']| If $(p,|
|00003880| 71 29 3d 31 24 2c 20 74 | 68 65 6e 0a 66 6f 72 20 |q)=1$, t|hen.for |
|00003890| 61 6e 79 20 24 61 2c 62 | 24 2c 20 74 68 65 72 65 |any $a,b|$, there|
|000038a0| 20 69 73 20 61 6e 20 24 | 6e 24 20 77 69 74 68 20 | is an $|n$ with |
|000038b0| 24 24 5c 63 6f 20 6e 61 | 70 5c 71 75 61 64 0a 5c |$$\co na|p\quad.\|
|000038c0| 68 62 6f 78 7b 61 6e 64 | 7d 5c 71 75 61 64 5c 63 |hbox{and|}\quad\c|
|000038d0| 6f 20 6e 62 71 24 24 5c | 65 6e 64 7b 43 6f 7d 0a |o nbq$$\|end{Co}.|
|000038e0| 50 72 6f 6f 66 3a 20 54 | 68 65 6f 72 65 6d 7e 5c |Proof: T|heorem~\|
|000038f0| 72 65 66 7b 54 31 7d 20 | 69 6d 70 6c 69 65 73 20 |ref{T1} |implies |
|00003900| 74 68 65 72 65 20 61 72 | 65 20 69 6e 74 65 67 65 |there ar|e intege|
|00003910| 72 73 20 24 78 2c 79 24 | 20 73 75 63 68 20 74 68 |rs $x,y$| such th|
|00003920| 61 74 0a 24 24 70 78 2b | 61 3d 71 79 2b 62 5c 71 |at.$$px+|a=qy+b\q|
|00003930| 75 61 64 5c 68 62 6f 78 | 7b 73 6f 20 6c 65 74 20 |uad\hbox|{so let |
|00003940| 7d 6e 3d 70 78 2b 61 24 | 24 0a 5c 73 75 62 73 65 |}n=px+a$|$.\subse|
|00003950| 63 74 69 6f 6e 7b 50 6f | 77 65 72 73 20 6d 6f 64 |ction{Po|wers mod|
|00003960| 75 6c 6f 20 61 20 70 72 | 69 6d 65 7d 54 68 65 20 |ulo a pr|ime}The |
|00003970| 73 65 71 75 65 6e 63 65 | 20 24 24 61 5c 71 75 61 |sequence| $$a\qua|
|00003980| 64 20 61 5e 32 5c 71 75 | 61 64 0a 61 5e 33 5c 64 |d a^2\qu|ad.a^3\d|
|00003990| 6f 74 73 5c 71 75 61 64 | 5c 68 62 6f 78 7b 6d 6f |ots\quad|\hbox{mo|
|000039a0| 64 20 7d 70 24 24 20 68 | 61 73 20 6d 61 6e 79 20 |d }p$$ h|as many |
|000039b0| 61 70 70 6c 69 63 61 74 | 69 6f 6e 73 20 69 6e 20 |applicat|ions in |
|000039c0| 63 72 79 70 74 6f 67 72 | 61 70 68 79 2e 0a 42 65 |cryptogr|aphy..Be|
|000039d0| 66 6f 72 65 20 6c 6f 6f | 6b 69 6e 67 20 61 74 20 |fore loo|king at |
|000039e0| 74 68 65 6f 72 65 74 69 | 63 61 6c 20 70 72 6f 70 |theoreti|cal prop|
|000039f0| 65 72 74 69 65 73 2c 20 | 74 68 65 20 65 78 61 6d |erties, |the exam|
|00003a00| 70 6c 65 20 62 65 6c 6f | 77 20 28 64 6f 6e 65 0a |ple belo|w (done.|
|00003a10| 75 73 69 6e 67 20 61 20 | 70 6f 63 6b 65 74 20 63 |using a |pocket c|
|00003a20| 61 6c 63 75 6c 61 74 6f | 72 29 20 73 68 6f 75 6c |alculato|r) shoul|
|00003a30| 64 20 6d 61 6b 65 20 63 | 6c 65 61 72 20 74 68 61 |d make c|lear tha|
|00003a40| 74 20 69 74 20 69 73 20 | 70 72 61 63 74 69 63 61 |t it is |practica|
|00003a50| 6c 0a 74 6f 20 63 6f 6d | 70 75 74 65 20 74 68 65 |l.to com|pute the|
|00003a60| 73 65 20 6e 75 6d 62 65 | 72 73 2c 20 65 76 65 6e |se numbe|rs, even|
|00003a70| 20 77 68 65 6e 20 6d 61 | 6e 79 20 64 69 67 69 74 | when ma|ny digit|
|00003a80| 73 20 61 72 65 20 69 6e | 76 6f 6c 76 65 64 2e 0a |s are in|volved..|
|00003a90| 5c 70 71 20 53 75 70 70 | 6f 73 65 20 77 65 20 77 |\pq Supp|ose we w|
|00003aa0| 61 6e 74 20 74 6f 20 63 | 6f 6d 70 75 74 65 20 24 |ant to c|ompute $|
|00003ab0| 34 33 32 5e 7b 36 37 38 | 7d 24 7e 6d 6f 64 7e 39 |432^{678|}$~mod~9|
|00003ac0| 38 37 2e 20 20 54 68 65 | 20 62 61 73 69 63 20 74 |87.  The| basic t|
|00003ad0| 72 69 63 6b 0a 69 73 20 | 74 6f 20 73 74 61 72 74 |rick.is |to start|
|00003ae0| 20 77 69 74 68 20 61 20 | 6e 75 6d 62 65 72 20 61 | with a |number a|
|00003af0| 6e 64 20 6b 65 65 70 20 | 73 71 75 61 72 69 6e 67 |nd keep |squaring|
|00003b00| 3a 0a 24 24 34 33 32 5e | 32 3d 31 38 36 36 32 34 |:.$$432^|2=186624|
|00003b10| 5c 65 71 75 69 76 38 31 | 5c 71 75 61 64 34 33 32 |\equiv81|\quad432|
|00003b20| 5e 34 5c 65 71 75 69 76 | 38 31 5e 32 5c 65 71 75 |^4\equiv|81^2\equ|
|00003b30| 69 76 36 33 39 5c 71 75 | 61 64 34 33 32 5e 38 5c |iv639\qu|ad432^8\|
|00003b40| 65 71 75 69 76 0a 36 33 | 39 5e 32 5c 65 71 75 69 |equiv.63|9^2\equi|
|00003b50| 76 36 39 30 5c 64 6f 74 | 73 34 33 32 5e 7b 35 31 |v690\dot|s432^{51|
|00003b60| 32 7d 5c 65 71 75 69 76 | 38 35 38 24 24 0a 53 69 |2}\equiv|858$$.Si|
|00003b70| 6e 63 65 20 24 36 37 38 | 3d 35 31 32 2b 31 32 38 |nce $678|=512+128|
|00003b80| 2b 33 32 2b 34 2b 32 24 | 2c 20 24 24 34 33 32 5e |+32+4+2$|, $$432^|
|00003b90| 7b 36 37 38 7d 5c 65 71 | 75 69 76 28 38 31 29 28 |{678}\eq|uiv(81)(|
|00003ba0| 36 33 39 29 5c 64 6f 74 | 73 28 38 35 38 29 0a 5c |639)\dot|s(858).\|
|00003bb0| 65 71 75 69 76 32 30 34 | 5c 71 71 75 61 64 5c 68 |equiv204|\qquad\h|
|00003bc0| 62 6f 78 7b 28 49 20 68 | 6f 70 65 21 29 7d 24 24 |box{(I h|ope!)}$$|
|00003bd0| 43 61 6c 63 75 6c 61 74 | 69 6f 6e 73 20 77 69 74 |Calculat|ions wit|
|00003be0| 68 20 65 78 70 6f 6e 65 | 6e 74 73 20 0a 69 6e 76 |h expone|nts .inv|
|00003bf0| 6f 6c 76 65 20 6e 6f 74 | 2d 74 6f 6f 2d 6d 61 6e |olve not|-too-man|
|00003c00| 79 20 6d 75 6c 74 69 70 | 6c 69 63 61 74 69 6f 6e |y multip|lication|
|00003c10| 73 2e 20 20 49 66 20 74 | 68 65 20 6e 75 6d 62 65 |s.  If t|he numbe|
|00003c20| 72 73 20 68 61 76 65 20 | 73 65 76 65 72 61 6c 0a |rs have |several.|
|00003c30| 68 75 6e 64 72 65 64 20 | 64 69 67 69 74 73 2c 20 |hundred |digits, |
|00003c40| 68 6f 77 65 76 65 72 2c | 20 69 74 20 69 73 20 6e |however,| it is n|
|00003c50| 65 63 65 73 73 61 72 79 | 20 74 6f 20 64 65 73 69 |ecessary| to desi|
|00003c60| 67 6e 20 73 70 65 63 69 | 61 6c 20 73 75 62 25 0a |gn speci|al sub%.|
|00003c70| 72 6f 75 74 69 6e 65 73 | 20 74 6f 20 64 6f 20 74 |routines| to do t|
|00003c80| 68 65 20 6d 75 6c 74 69 | 70 6c 69 63 61 74 69 6f |he multi|plicatio|
|00003c90| 6e 73 20 28 73 65 65 20 | 4b 6e 75 74 68 2c 20 76 |ns (see |Knuth, v|
|00003ca0| 6f 6c 75 6d 65 7e 32 29 | 2e 0a 5c 70 71 20 4c 65 |olume~2)|..\pq Le|
|00003cb0| 74 20 75 73 20 6c 6f 6f | 6b 20 61 74 20 74 68 65 |t us loo|k at the|
|00003cc0| 20 73 65 71 75 65 6e 63 | 65 20 6f 66 20 70 6f 77 | sequenc|e of pow|
|00003cd0| 65 72 73 20 6f 66 20 32 | 20 6d 6f 64 20 31 31 3a |ers of 2| mod 11:|
|00003ce0| 0a 24 24 32 5c 20 34 5c | 20 38 5c 20 35 5c 20 31 |.$$2\ 4\| 8\ 5\ 1|
|00003cf0| 30 5c 20 39 5c 20 37 5c | 20 33 5c 20 36 5c 20 31 |0\ 9\ 7\| 3\ 6\ 1|
|00003d00| 24 24 45 61 63 68 20 6e | 75 6d 62 65 72 20 66 72 |$$Each n|umber fr|
|00003d10| 6f 6d 20 31 20 74 6f 20 | 31 30 20 61 70 70 65 61 |om 1 to |10 appea|
|00003d20| 72 73 0a 69 6e 20 74 68 | 65 20 73 65 71 75 65 6e |rs.in th|e sequen|
|00003d30| 63 65 2e 5c 62 65 67 69 | 6e 7b 54 68 7d 5c 6c 61 |ce.\begi|n{Th}\la|
|00003d40| 62 65 6c 7b 70 72 6f 6f | 74 7d 4c 65 74 20 24 70 |bel{proo|t}Let $p|
|00003d50| 24 20 62 65 20 61 20 70 | 72 69 6d 65 2e 20 20 54 |$ be a p|rime.  T|
|00003d60| 68 65 72 65 0a 69 73 20 | 61 6e 20 24 61 24 20 73 |here.is |an $a$ s|
|00003d70| 75 63 68 20 74 68 61 74 | 20 66 6f 72 20 65 76 65 |uch that| for eve|
|00003d80| 72 79 20 24 31 5c 6c 65 | 20 62 5c 6c 65 20 70 2d |ry $1\le| b\le p-|
|00003d90| 31 24 2c 20 74 68 65 72 | 65 20 69 73 20 24 31 5c |1$, ther|e is $1\|
|00003da0| 6c 65 20 78 5c 6c 65 0a | 70 2d 31 24 20 73 75 63 |le x\le.|p-1$ suc|
|00003db0| 68 20 74 68 61 74 20 24 | 5c 63 6f 7b 61 5e 78 7d |h that $|\co{a^x}|
|00003dc0| 62 70 24 2e 5c 65 6e 64 | 7b 54 68 7d 0a 49 74 20 |bp$.\end|{Th}.It |
|00003dd0| 69 73 20 6e 6f 74 20 61 | 6c 77 61 79 73 20 74 68 |is not a|lways th|
|00003de0| 65 20 63 61 73 65 20 74 | 68 61 74 20 24 61 3d 32 |e case t|hat $a=2|
|00003df0| 24 2e 20 20 54 68 65 20 | 70 6f 77 65 72 73 20 6f |$.  The |powers o|
|00003e00| 66 20 32 20 6d 6f 64 7e | 37 20 61 72 65 0a 32 2c |f 2 mod~|7 are.2,|
|00003e10| 7e 34 2c 7e 31 20 61 66 | 74 65 72 20 77 68 69 63 |~4,~1 af|ter whic|
|00003e20| 68 20 74 68 65 20 73 65 | 71 75 65 6e 63 65 20 72 |h the se|quence r|
|00003e30| 65 70 65 61 74 73 20 61 | 6e 64 20 77 65 20 6e 65 |epeats a|nd we ne|
|00003e40| 76 65 72 20 67 65 74 20 | 33 2c 20 35 2c 20 6f 72 |ver get |3, 5, or|
|00003e50| 20 36 2e 0a 5c 70 71 20 | 54 68 65 20 70 72 6f 6f | 6..\pq |The proo|
|00003e60| 66 20 6f 66 20 54 68 65 | 6f 72 65 6d 7e 5c 72 65 |f of The|orem~\re|
|00003e70| 66 7b 70 72 6f 6f 74 7d | 20 72 65 71 75 69 72 65 |f{proot}| require|
|00003e80| 73 20 73 65 76 65 72 61 | 6c 20 73 74 65 70 73 2c |s severa|l steps,|
|00003e90| 20 73 6f 20 77 65 0a 77 | 69 6c 6c 20 67 69 76 65 | so we.w|ill give|
|00003ea0| 20 69 74 20 6c 61 74 65 | 72 2e 20 20 46 6f 72 20 | it late|r.  For |
|00003eb0| 6e 6f 77 2c 20 77 65 20 | 77 61 6e 74 20 74 6f 20 |now, we |want to |
|00003ec0| 6c 6f 6f 6b 20 61 74 20 | 73 6f 6d 65 20 6f 66 20 |look at |some of |
|00003ed0| 69 74 73 20 63 6f 6e 73 | 65 71 75 65 6e 63 65 73 |its cons|equences|
|00003ee0| 2e 0a 5c 62 65 67 69 6e | 7b 43 6f 7d 4c 65 74 20 |..\begin|{Co}Let |
|00003ef0| 24 61 24 20 62 65 20 61 | 73 20 69 6e 20 54 68 65 |$a$ be a|s in The|
|00003f00| 6f 72 65 6d 7e 5c 72 65 | 66 7b 70 72 6f 6f 74 7d |orem~\re|f{proot}|
|00003f10| 2e 20 54 68 65 6e 20 24 | 5c 63 6f 7b 61 5e 7b 70 |. Then $|\co{a^{p|
|00003f20| 2d 31 7d 7d 31 70 24 2e | 0a 5c 6c 61 62 65 6c 7b |-1}}1p$.|.\label{|
|00003f30| 50 46 7d 5c 65 6e 64 7b | 43 6f 7d 50 72 6f 6f 66 |PF}\end{|Co}Proof|
|00003f40| 3a 20 57 65 20 6b 6e 6f | 77 20 74 68 61 74 20 24 |: We kno|w that $|
|00003f50| 61 5e 64 5c 65 71 75 69 | 76 31 24 20 66 6f 72 20 |a^d\equi|v1$ for |
|00003f60| 73 6f 6d 65 20 24 31 5c | 6c 65 20 64 5c 6c 65 20 |some $1\|le d\le |
|00003f70| 70 2d 31 24 2e 20 20 49 | 66 0a 24 64 3c 70 2d 31 |p-1$.  I|f.$d<p-1|
|00003f80| 24 2c 20 74 68 65 20 73 | 65 71 75 65 6e 63 65 20 |$, the s|equence |
|00003f90| 6f 66 20 70 6f 77 65 72 | 73 20 6f 66 20 24 61 24 |of power|s of $a$|
|00003fa0| 20 77 6f 75 6c 64 20 73 | 74 61 72 74 20 72 65 70 | would s|tart rep|
|00003fb0| 65 61 74 69 6e 67 20 62 | 65 66 6f 72 65 0a 77 65 |eating b|efore.we|
|00003fc0| 20 67 6f 74 20 61 6c 6c | 20 74 68 65 20 6e 75 6d | got all| the num|
|00003fd0| 62 65 72 73 3a 20 24 61 | 5e 7b 64 2b 31 7d 5c 65 |bers: $a|^{d+1}\e|
|00003fe0| 71 75 69 76 20 61 24 2c | 20 24 61 5e 7b 64 2b 32 |quiv a$,| $a^{d+2|
|00003ff0| 7d 5c 65 71 75 69 76 20 | 61 5e 32 24 2c 20 65 74 |}\equiv |a^2$, et|
|00004000| 63 2e 0a 5c 62 65 67 69 | 6e 7b 43 6f 7d 5c 6c 61 |c..\begi|n{Co}\la|
|00004010| 62 65 6c 7b 46 65 7d 46 | 6f 72 20 61 6e 79 20 24 |bel{Fe}F|or any $|
|00004020| 62 5c 6e 6f 74 5c 65 71 | 75 69 76 30 24 2c 0a 20 |b\not\eq|uiv0$,. |
|00004030| 24 5c 63 6f 7b 62 5e 7b | 70 2d 31 7d 7d 31 70 24 |$\co{b^{|p-1}}1p$|
|00004040| 2e 5c 65 6e 64 7b 43 6f | 7d 50 72 6f 6f 66 3a 20 |.\end{Co|}Proof: |
|00004050| 4c 65 74 20 24 61 24 20 | 62 65 0a 61 73 20 69 6e |Let $a$ |be.as in|
|00004060| 20 54 68 65 6f 72 65 6d | 7e 5c 72 65 66 7b 70 72 | Theorem|~\ref{pr|
|00004070| 6f 6f 74 7d 2e 20 55 73 | 69 6e 67 20 43 6f 72 6f |oot}. Us|ing Coro|
|00004080| 6c 6c 61 72 79 7e 5c 72 | 65 66 7b 50 46 7d 20 24 |llary~\r|ef{PF} $|
|00004090| 24 62 5e 7b 70 2d 31 7d | 0a 5c 65 71 75 69 76 20 |$b^{p-1}|.\equiv |
|000040a0| 61 5e 7b 78 28 70 2d 31 | 29 7d 5c 65 71 75 69 76 |a^{x(p-1|)}\equiv|
|000040b0| 5c 6c 65 66 74 28 61 5e | 7b 70 2d 31 7d 5c 72 69 |\left(a^|{p-1}\ri|
|000040c0| 67 68 74 29 5e 78 5c 65 | 71 75 69 76 31 24 24 0a |ght)^x\e|quiv1$$.|
|000040d0| 5c 62 65 67 69 6e 7b 43 | 6f 7d 49 66 20 24 5c 63 |\begin{C|o}If $\c|
|000040e0| 6f 20 78 79 7b 28 70 2d | 31 29 7d 24 2c 20 74 68 |o xy{(p-|1)}$, th|
|000040f0| 65 6e 20 24 5c 63 6f 20 | 7b 62 5e 78 7d 7b 62 5e |en $\co |{b^x}{b^|
|00004100| 79 7d 70 24 5c 65 6e 64 | 7b 43 6f 7d 0a 50 72 6f |y}p$\end|{Co}.Pro|
|00004110| 6f 66 3a 20 46 6f 72 20 | 73 6f 6d 65 20 69 6e 74 |of: For |some int|
|00004120| 65 67 65 72 20 24 72 24 | 2c 20 24 79 3d 72 28 70 |eger $r$|, $y=r(p|
|00004130| 2d 31 29 2b 78 24 20 61 | 6e 64 20 62 79 20 43 6f |-1)+x$ a|nd by Co|
|00004140| 72 6f 6c 6c 61 72 79 7e | 5c 72 65 66 7b 46 65 7d |rollary~|\ref{Fe}|
|00004150| 0a 24 24 62 5e 79 5c 65 | 71 75 69 76 5c 63 6f 7b |.$$b^y\e|quiv\co{|
|00004160| 5c 6c 65 66 74 28 62 5e | 7b 70 2d 31 7d 5c 72 69 |\left(b^|{p-1}\ri|
|00004170| 67 68 74 29 5e 72 62 5e | 78 7d 7b 62 5e 78 7d 70 |ght)^rb^|x}{b^x}p|
|00004180| 24 24 0a 5c 62 65 67 69 | 6e 7b 4c 65 7d 5c 6c 61 |$$.\begi|n{Le}\la|
|00004190| 62 65 6c 7b 64 69 7d 4c | 65 74 20 24 62 5c 6e 6f |bel{di}L|et $b\no|
|000041a0| 74 5c 65 71 75 69 76 30 | 24 2c 20 24 64 24 20 74 |t\equiv0|$, $d$ t|
|000041b0| 68 65 20 73 6d 61 6c 6c | 65 73 74 0a 70 6f 73 69 |he small|est.posi|
|000041c0| 74 69 76 65 20 6e 75 6d | 62 65 72 20 73 75 63 68 |tive num|ber such|
|000041d0| 20 74 68 61 74 20 24 62 | 5e 64 5c 65 71 75 69 76 | that $b|^d\equiv|
|000041e0| 31 24 2e 20 20 54 68 65 | 6e 20 66 6f 72 20 61 6e |1$.  The|n for an|
|000041f0| 79 20 24 65 3e 30 24 20 | 77 69 74 68 20 24 62 5e |y $e>0$ |with $b^|
|00004200| 65 0a 5c 65 71 75 69 76 | 20 31 24 20 20 24 64 24 |e.\equiv| 1$  $d$|
|00004210| 20 64 69 76 69 64 65 73 | 20 24 65 24 20 65 76 65 | divides| $e$ eve|
|00004220| 6e 6c 79 2e 20 49 6e 20 | 70 61 72 74 69 63 75 6c |nly. In |particul|
|00004230| 61 72 2c 20 62 79 20 43 | 6f 72 6f 6c 6c 61 72 79 |ar, by C|orollary|
|00004240| 7e 5c 72 65 66 7b 46 65 | 7d 2c 0a 24 64 24 20 64 |~\ref{Fe|},.$d$ d|
|00004250| 69 76 69 64 65 73 20 24 | 70 2d 31 24 20 65 76 65 |ivides $|p-1$ eve|
|00004260| 6e 6c 79 2e 5c 65 6e 64 | 7b 4c 65 7d 50 72 6f 6f |nly.\end|{Le}Proo|
|00004270| 66 3a 20 49 66 20 24 64 | 24 20 64 6f 65 73 20 6e |f: If $d|$ does n|
|00004280| 6f 74 20 64 69 76 69 64 | 65 20 24 65 24 2c 20 74 |ot divid|e $e$, t|
|00004290| 68 65 6e 0a 24 65 3d 64 | 72 2b 73 24 20 66 6f 72 |hen.$e=d|r+s$ for|
|000042a0| 20 73 6f 6d 65 20 24 30 | 3c 73 3c 64 24 2c 20 62 | some $0|<s<d$, b|
|000042b0| 75 74 20 24 24 62 5e 73 | 5c 65 71 75 69 76 20 62 |ut $$b^s|\equiv b|
|000042c0| 5e 7b 65 7d 5c 6c 65 66 | 74 28 62 5e 64 5c 72 69 |^{e}\lef|t(b^d\ri|
|000042d0| 67 68 74 29 25 0a 5e 7b | 2d 72 7d 5c 65 71 75 69 |ght)%.^{|-r}\equi|
|000042e0| 76 31 24 24 20 77 6f 75 | 6c 64 20 63 6f 6e 74 72 |v1$$ wou|ld contr|
|000042f0| 61 64 69 63 74 20 74 68 | 65 20 64 65 66 69 6e 69 |adict th|e defini|
|00004300| 74 69 6f 6e 20 6f 66 20 | 24 64 24 2e 0a 5c 73 75 |tion of |$d$..\su|
|00004310| 62 73 65 63 74 69 6f 6e | 7b 50 72 69 6d 69 74 69 |bsection|{Primiti|
|00004320| 76 65 20 72 6f 6f 74 73 | 7d 0a 54 68 65 6f 72 65 |ve roots|}.Theore|
|00004330| 6d 7e 5c 72 65 66 7b 70 | 72 6f 6f 74 7d 20 73 68 |m~\ref{p|root} sh|
|00004340| 6f 77 65 64 20 74 68 61 | 74 20 69 66 20 24 70 24 |owed tha|t if $p$|
|00004350| 20 69 73 20 61 20 70 72 | 69 6d 65 2c 20 74 68 65 | is a pr|ime, the|
|00004360| 72 65 20 69 73 20 61 6e | 20 24 61 24 20 73 75 63 |re is an| $a$ suc|
|00004370| 68 20 74 68 61 74 0a 74 | 68 65 20 65 71 75 61 74 |h that.t|he equat|
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|00004390| 24 24 68 61 73 20 61 20 | 73 6f 6c 75 74 69 6f 6e |$$has a |solution|
|000043a0| 20 66 6f 72 20 61 6e 79 | 20 24 62 5c 6e 6f 74 5c | for any| $b\not\|
|000043b0| 65 71 75 69 76 30 24 2e | 0a 53 75 63 68 20 61 6e |equiv0$.|.Such an|
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|000043e0| 72 6f 6f 74 5c 2f 7d 20 | 6f 66 20 24 70 24 2c 20 |root\/} |of $p$, |
|000043f0| 61 6e 64 20 24 78 24 0a | 69 73 20 63 61 6c 6c 65 |and $x$.|is calle|
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|00004450| 6e 7e 5c 72 65 66 7b 6e | 75 6d 74 68 7d 20 74 68 |n~\ref{n|umth} th|
|00004460| 61 74 20 69 74 20 69 73 | 20 65 61 73 79 20 74 6f |at it is| easy to|
|00004470| 20 6f 62 74 61 69 6e 20 | 24 62 24 20 67 69 76 65 | obtain |$b$ give|
|00004480| 6e 0a 24 61 24 20 61 6e | 64 20 24 78 24 2e 20 20 |n.$a$ an|d $x$.  |
|00004490| 46 69 6e 64 69 6e 67 20 | 24 78 24 20 67 69 76 65 |Finding |$x$ give|
|000044a0| 6e 20 24 61 24 20 61 6e | 64 20 24 62 24 20 69 73 |n $a$ an|d $b$ is|
|000044b0| 20 6d 75 63 68 20 68 61 | 72 64 65 72 2e 20 20 4d | much ha|rder.  M|
|000044c0| 61 6e 79 20 6d 6f 64 65 | 72 6e 0a 65 6e 63 72 79 |any mode|rn.encry|
|000044d0| 70 74 69 6f 6e 20 73 79 | 73 74 65 6d 73 20 61 72 |ption sy|stems ar|
|000044e0| 65 20 62 61 73 65 64 20 | 6f 6e 20 74 68 65 20 66 |e based |on the f|
|000044f0| 61 63 74 20 74 68 61 74 | 20 6e 6f 20 65 66 66 69 |act that| no effi|
|00004500| 63 69 65 6e 74 20 77 61 | 79 20 6f 66 0a 63 6f 6d |cient wa|y of.com|
|00004510| 70 75 74 69 6e 67 20 64 | 69 73 63 72 65 74 65 20 |puting d|iscrete |
|00004520| 6c 6f 67 61 72 69 74 68 | 6d 73 20 69 73 20 6b 6e |logarith|ms is kn|
|00004530| 6f 77 6e 2e 0a 5c 70 71 | 20 4e 6f 20 65 66 66 69 |own..\pq| No effi|
|00004540| 63 69 65 6e 74 20 6d 65 | 74 68 6f 64 20 66 6f 72 |cient me|thod for|
|00004550| 20 61 6c 77 61 79 73 20 | 66 69 6e 64 69 6e 67 20 | always |finding |
|00004560| 70 72 69 6d 69 74 69 76 | 65 20 72 6f 6f 74 73 20 |primitiv|e roots |
|00004570| 69 73 20 6b 6e 6f 77 6e | 2e 0a 48 6f 77 65 76 65 |is known|..Howeve|
|00004580| 72 2c 20 69 74 20 69 73 | 20 6f 66 74 65 6e 20 70 |r, it is| often p|
|00004590| 6f 73 73 69 62 6c 65 20 | 74 6f 20 66 69 6e 64 20 |ossible |to find |
|000045a0| 6f 6e 65 20 69 6e 20 73 | 70 65 63 69 61 6c 20 63 |one in s|pecial c|
|000045b0| 61 73 65 73 2e 0a 57 65 | 20 77 69 6c 6c 20 75 73 |ases..We| will us|
|000045c0| 65 20 24 70 3d 31 32 32 | 33 24 20 61 73 20 61 6e |e $p=122|3$ as an|
|000045d0| 20 65 78 61 6d 70 6c 65 | 2e 20 20 24 70 2d 31 3d | example|.  $p-1=|
|000045e0| 32 5c 63 64 6f 74 31 33 | 5c 63 64 6f 74 34 37 24 |2\cdot13|\cdot47$|
|000045f0| 2e 20 42 79 20 4c 65 6d | 6d 61 7e 25 0a 5c 72 65 |. By Lem|ma~%.\re|
|00004600| 66 7b 64 69 7d 2c 20 69 | 66 20 24 61 24 20 69 73 |f{di}, i|f $a$ is|
|00004610| 20 6e 6f 74 20 61 20 70 | 72 69 6d 69 74 69 76 65 | not a p|rimitive|
|00004620| 20 72 6f 6f 74 2c 20 74 | 68 65 6e 20 77 65 20 77 | root, t|hen we w|
|00004630| 69 6c 6c 20 65 69 74 68 | 65 72 20 68 61 76 65 0a |ill eith|er have.|
|00004640| 24 61 5e 7b 32 36 7d 24 | 2c 20 24 61 5e 7b 39 34 |$a^{26}$|, $a^{94|
|00004650| 7d 24 2c 20 6f 72 20 24 | 5c 63 6f 7b 61 5e 7b 36 |}$, or $|\co{a^{6|
|00004660| 31 31 7d 7d 31 7b 31 32 | 32 33 7d 24 2e 20 20 24 |11}}1{12|23}$.  $|
|00004670| 61 3d 32 24 20 61 6e 64 | 20 33 20 66 61 69 6c 2c |a=2$ and| 3 fail,|
|00004680| 20 62 75 74 0a 24 61 3d | 35 24 20 73 61 74 69 73 | but.$a=|5$ satis|
|00004690| 66 69 65 73 20 61 6c 6c | 20 74 68 72 65 65 20 63 |fies all| three c|
|000046a0| 6f 6e 64 69 74 69 6f 6e | 73 2c 20 73 6f 20 69 74 |ondition|s, so it|
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|000046c0| 72 6f 6f 74 2e 20 20 28 | 77 65 0a 63 6f 75 6c 64 |root.  (|we.could|
|000046d0| 20 74 65 6c 6c 20 74 68 | 61 74 20 24 61 3d 34 24 | tell th|at $a=4$|
|000046e0| 20 77 6f 75 6c 64 20 6e | 6f 74 20 62 65 20 61 20 | would n|ot be a |
|000046f0| 70 72 69 6d 69 74 69 76 | 65 20 72 6f 6f 74 20 77 |primitiv|e root w|
|00004700| 69 74 68 6f 75 74 20 74 | 65 73 74 69 6e 67 2e 20 |ithout t|esting. |
|00004710| 20 57 68 79 3f 29 0a 5c | 70 71 20 49 74 20 69 73 | Why?).\|pq It is|
|00004720| 20 65 61 73 79 20 74 6f | 20 73 68 6f 77 20 74 68 | easy to| show th|
|00004730| 61 74 2c 20 69 66 20 24 | 61 24 20 69 73 20 61 20 |at, if $|a$ is a |
|00004740| 70 72 69 6d 69 74 69 76 | 65 20 72 6f 6f 74 2c 20 |primitiv|e root, |
|00004750| 24 61 5e 78 24 20 69 73 | 20 61 0a 70 72 69 6d 69 |$a^x$ is| a.primi|
|00004760| 74 69 76 65 20 72 6f 6f | 74 20 69 66 20 61 6e 64 |tive roo|t if and|
|00004770| 20 6f 6e 6c 79 20 69 66 | 20 24 28 78 2c 70 2d 31 | only if| $(x,p-1|
|00004780| 29 3d 31 24 2e 20 20 49 | 6e 20 74 68 69 73 20 65 |)=1$.  I|n this e|
|00004790| 78 61 6d 70 6c 65 2c 20 | 74 68 69 73 20 6d 65 61 |xample, |this mea|
|000047a0| 6e 73 0a 74 68 65 20 6e | 75 6d 62 65 72 20 6f 66 |ns.the n|umber of|
|000047b0| 20 70 72 69 6d 69 74 69 | 76 65 20 72 6f 6f 74 73 | primiti|ve roots|
|000047c0| 20 69 73 24 24 31 32 32 | 32 5c 6c 65 66 74 28 5c | is$$122|2\left(\|
|000047d0| 66 72 61 63 31 32 5c 72 | 69 67 68 74 29 5c 6c 65 |frac12\r|ight)\le|
|000047e0| 66 74 28 5c 66 72 61 63 | 0a 7b 31 32 7d 7b 31 33 |ft(\frac|.{12}{13|
|000047f0| 7d 5c 72 69 67 68 74 29 | 5c 6c 65 66 74 28 5c 66 |}\right)|\left(\f|
|00004800| 72 61 63 7b 34 36 7d 7b | 34 37 7d 5c 72 69 67 68 |rac{46}{|47}\righ|
|00004810| 74 29 3d 35 35 32 24 24 | 54 68 75 73 2c 20 69 66 |t)=552$$|Thus, if|
|00004820| 20 77 65 20 68 61 64 20 | 6a 75 73 74 20 63 68 6f | we had |just cho|
|00004830| 73 65 6e 0a 24 61 24 20 | 61 74 20 72 61 6e 64 6f |sen.$a$ |at rando|
|00004840| 6d 2c 20 74 68 65 20 70 | 72 6f 62 61 62 69 6c 69 |m, the p|robabili|
|00004850| 74 79 20 74 68 61 74 20 | 69 74 20 77 6f 75 6c 64 |ty that |it would|
|00004860| 20 62 65 20 61 20 70 72 | 69 6d 69 74 69 76 65 20 | be a pr|imitive |
|00004870| 72 6f 6f 74 20 69 73 0a | 24 5c 61 70 70 72 6f 78 |root is.|$\approx|
|00004880| 2e 34 35 24 2e 20 20 43 | 68 6f 6f 73 69 6e 67 20 |.45$.  C|hoosing |
|00004890| 24 61 24 20 61 74 20 72 | 61 6e 64 6f 6d 20 61 6e |$a$ at r|andom an|
|000048a0| 64 20 74 65 73 74 69 6e | 67 20 75 6e 74 69 6c 20 |d testin|g until |
|000048b0| 77 65 20 66 6f 75 6e 64 | 20 61 0a 70 72 69 6d 69 |we found| a.primi|
|000048c0| 74 69 76 65 20 72 6f 6f | 74 20 77 6f 75 6c 64 20 |tive roo|t would |
|000048d0| 6e 6f 74 20 62 65 20 65 | 78 70 65 63 74 65 64 20 |not be e|xpected |
|000048e0| 74 6f 20 74 61 6b 65 20 | 74 6f 6f 20 6c 6f 6e 67 |to take |too long|
|000048f0| 2e 0a 5c 70 71 20 54 68 | 69 73 20 69 73 20 61 6e |..\pq Th|is is an|
|00004900| 20 65 78 61 6d 70 6c 65 | 20 6f 66 20 61 20 7b 5c | example| of a {\|
|00004910| 69 74 20 70 72 6f 62 61 | 62 69 6c 69 73 74 69 63 |it proba|bilistic|
|00004920| 20 61 6c 67 6f 72 69 74 | 68 6d 7d 2e 20 20 49 74 | algorit|hm}.  It|
|00004930| 20 69 73 20 70 6f 73 73 | 69 62 6c 65 0a 66 6f 72 | is poss|ible.for|
|00004940| 20 69 74 20 74 6f 20 74 | 61 6b 65 20 61 20 6c 6f | it to t|ake a lo|
|00004950| 6e 67 20 74 69 6d 65 2c | 20 62 75 74 20 74 68 65 |ng time,| but the|
|00004960| 20 61 6d 6f 75 6e 74 20 | 6f 66 20 74 69 6d 65 20 | amount |of time |
|00004970| 6e 65 65 64 65 64 20 6f | 6e 20 61 76 65 72 61 67 |needed o|n averag|
|00004980| 65 20 69 73 0a 72 65 61 | 73 6f 6e 61 62 6c 79 20 |e is.rea|sonably |
|00004990| 73 6d 61 6c 6c 2e 20 20 | 57 65 20 77 69 6c 6c 20 |small.  |We will |
|000049a0| 73 65 65 20 6d 61 6e 79 | 20 6f 74 68 65 72 20 70 |see many| other p|
|000049b0| 72 6f 62 61 62 69 6c 69 | 73 74 69 63 20 61 6c 67 |robabili|stic alg|
|000049c0| 6f 72 69 74 68 6d 73 20 | 6c 61 74 65 72 2e 0a 5c |orithms |later..\|
|000049d0| 73 75 62 73 75 62 73 65 | 63 74 69 6f 6e 2a 7b 50 |subsubse|ction*{P|
|000049e0| 72 6f 6f 66 20 6f 66 20 | 54 68 65 6f 72 65 6d 20 |roof of |Theorem |
|000049f0| 5c 72 65 66 7b 70 72 6f | 6f 74 7d 7d 0a 4c 65 74 |\ref{pro|ot}}.Let|
|00004a00| 20 24 61 5c 6e 6f 74 5c | 65 71 75 69 76 30 24 2c | $a\not\|equiv0$,|
|00004a10| 20 24 64 24 20 62 65 20 | 74 68 65 20 73 6d 61 6c | $d$ be |the smal|
|00004a20| 6c 65 73 74 20 70 6f 73 | 69 74 69 76 65 20 6e 75 |lest pos|itive nu|
|00004a30| 6d 62 65 72 20 66 6f 72 | 20 77 68 69 63 68 20 24 |mber for| which $|
|00004a40| 61 5e 64 0a 5c 65 71 75 | 69 76 31 24 20 28 74 68 |a^d.\equ|iv1$ (th|
|00004a50| 65 72 65 20 6d 75 73 74 | 20 62 65 20 73 75 63 68 |ere must| be such|
|00004a60| 20 61 20 24 64 24 20 73 | 69 6e 63 65 20 24 61 5e | a $d$ s|ince $a^|
|00004a70| 4b 5c 65 71 75 69 76 20 | 61 5e 4c 24 20 69 6d 70 |K\equiv |a^L$ imp|
|00004a80| 6c 69 65 73 20 24 61 5e | 7b 4b 2d 4c 7d 0a 5c 65 |lies $a^|{K-L}.\e|
|00004a90| 71 75 69 76 31 24 29 2e | 20 20 49 66 20 24 64 3d |quiv1$).|  If $d=|
|00004aa0| 70 2d 31 24 2c 20 24 61 | 24 20 69 73 20 61 20 70 |p-1$, $a|$ is a p|
|00004ab0| 72 69 6d 69 74 69 76 65 | 20 72 6f 6f 74 2e 20 20 |rimitive| root.  |
|00004ac0| 49 66 20 24 64 3c 70 2d | 31 24 2c 20 77 65 20 77 |If $d<p-|1$, we w|
|00004ad0| 69 6c 6c 0a 66 69 6e 64 | 20 24 61 27 2c 64 27 24 |ill.find| $a',d'$|
|00004ae0| 20 77 69 74 68 20 24 64 | 27 3e 64 24 2e 20 20 49 | with $d|'>d$.  I|
|00004af0| 66 20 24 64 27 3c 70 2d | 31 24 20 74 68 65 20 70 |f $d'<p-|1$ the p|
|00004b00| 72 6f 63 65 73 73 20 69 | 73 20 72 65 70 65 61 74 |rocess i|s repeat|
|00004b10| 65 64 20 75 6e 74 69 6c | 0a 77 65 20 65 76 65 6e |ed until|.we even|
|00004b20| 74 75 61 6c 6c 79 20 6f | 62 74 61 69 6e 20 61 20 |tually o|btain a |
|00004b30| 70 72 69 6d 69 74 69 76 | 65 20 72 6f 6f 74 2e 5c |primitiv|e root.\|
|00004b40| 62 65 67 69 6e 7b 4c 65 | 7d 54 68 65 72 65 20 61 |begin{Le|}There a|
|00004b50| 72 65 20 61 74 20 6d 6f | 73 74 20 24 64 24 0a 73 |re at mo|st $d$.s|
|00004b60| 6f 6c 75 74 69 6f 6e 73 | 20 74 6f 20 61 20 63 6f |olutions| to a co|
|00004b70| 6e 67 72 75 65 6e 63 65 | 20 69 6e 76 6f 6c 76 69 |ngruence| involvi|
|00004b80| 6e 67 20 61 20 70 6f 6c | 79 6e 6f 6d 69 61 6c 20 |ng a pol|ynomial |
|00004b90| 6f 66 20 64 65 67 72 65 | 65 20 24 64 24 3a 0a 24 |of degre|e $d$:.$|
|00004ba0| 24 5c 63 6f 7b 78 5e 64 | 2b 5c 61 6c 70 68 61 5f |$\co{x^d|+\alpha_|
|00004bb0| 31 78 5e 7b 64 2d 31 7d | 2b 5c 64 6f 74 73 5c 61 |1x^{d-1}|+\dots\a|
|00004bc0| 6c 70 68 61 5f 64 7d 30 | 70 24 24 49 6e 20 70 61 |lpha_d}0|p$$In pa|
|00004bd0| 72 74 69 63 75 6c 61 72 | 2c 20 74 68 65 72 65 20 |rticular|, there |
|00004be0| 61 72 65 20 61 74 0a 6d | 6f 73 74 20 24 64 24 20 |are at.m|ost $d$ |
|00004bf0| 24 78 24 20 77 69 74 68 | 20 24 78 5e 64 5c 65 71 |$x$ with| $x^d\eq|
|00004c00| 75 69 76 31 24 2e 5c 6c | 61 62 65 6c 7b 42 7d 0a |uiv1$.\l|abel{B}.|
|00004c10| 5c 65 6e 64 7b 4c 65 7d | 7b 5c 62 66 20 50 72 6f |\end{Le}|{\bf Pro|
|00004c20| 6f 66 3a 7d 20 54 68 69 | 73 20 63 61 6e 20 62 65 |of:} Thi|s can be|
|00004c30| 20 70 72 6f 76 65 64 20 | 69 6e 0a 74 68 65 20 73 | proved |in.the s|
|00004c40| 61 6d 65 20 77 61 79 20 | 61 73 20 74 68 65 20 63 |ame way |as the c|
|00004c50| 6f 72 72 65 73 70 6f 6e | 64 69 6e 67 20 74 68 65 |orrespon|ding the|
|00004c60| 6f 72 65 6d 20 69 6e 20 | 6f 72 64 69 6e 61 72 79 |orem in |ordinary|
|00004c70| 20 61 6c 67 65 62 72 61 | 3a 20 69 66 20 24 78 3d | algebra|: if $x=|
|00004c80| 5c 62 65 74 61 24 0a 69 | 73 20 61 20 73 6f 6c 75 |\beta$.i|s a solu|
|00004c90| 74 69 6f 6e 2c 20 74 68 | 65 20 70 6f 6c 79 6e 6f |tion, th|e polyno|
|00004ca0| 6d 69 61 6c 20 63 61 6e | 20 62 65 20 77 72 69 74 |mial can| be writ|
|00004cb0| 74 65 6e 20 61 73 20 24 | 28 78 2d 5c 62 65 74 61 |ten as $|(x-\beta|
|00004cc0| 29 24 20 74 69 6d 65 73 | 20 61 20 70 6f 6c 79 25 |)$ times| a poly%|
|00004cd0| 0a 6e 6f 6d 69 61 6c 20 | 6f 66 20 64 65 67 72 65 |.nomial |of degre|
|00004ce0| 65 20 24 64 2d 31 24 2c | 20 77 68 69 63 68 20 62 |e $d-1$,| which b|
|00004cf0| 79 20 69 6e 64 75 63 74 | 69 6f 6e 20 68 61 73 20 |y induct|ion has |
|00004d00| 24 5c 6c 65 20 64 2d 31 | 24 20 73 6f 6c 75 74 69 |$\le d-1|$ soluti|
|00004d10| 6f 6e 73 2e 0a 5c 70 71 | 20 57 65 20 72 65 74 75 |ons..\pq| We retu|
|00004d20| 72 6e 20 74 6f 20 74 68 | 65 20 70 72 6f 6f 66 20 |rn to th|e proof |
|00004d30| 6f 66 20 54 68 65 6f 72 | 65 6d 7e 5c 72 65 66 7b |of Theor|em~\ref{|
|00004d40| 70 72 6f 6f 74 7d 2e 20 | 54 68 65 20 73 65 71 75 |proot}. |The sequ|
|00004d50| 65 6e 63 65 20 0a 24 24 | 61 5c 71 75 61 64 20 61 |ence .$$|a\quad a|
|00004d60| 5e 32 5c 71 75 61 64 20 | 61 5e 33 5c 64 6f 74 73 |^2\quad |a^3\dots|
|00004d70| 20 5c 63 6f 20 7b 61 5e | 64 7d 31 70 24 24 63 6f | \co {a^|d}1p$$co|
|00004d80| 6e 73 69 73 74 73 20 6f | 66 20 24 64 24 20 64 69 |nsists o|f $d$ di|
|00004d90| 66 66 65 72 65 6e 74 20 | 73 6f 6c 75 74 69 6f 6e |fferent |solution|
|00004da0| 73 0a 6f 66 20 24 78 5e | 64 5c 65 71 75 69 76 31 |s.of $x^|d\equiv1|
|00004db0| 24 2e 20 20 49 66 20 24 | 64 3c 70 2d 31 24 2c 20 |$.  If $|d<p-1$, |
|00004dc0| 6c 65 74 20 24 62 24 20 | 62 65 20 61 6e 79 20 6e |let $b$ |be any n|
|00004dd0| 6f 6e 2d 6d 65 6d 62 65 | 72 20 6f 66 20 74 68 65 |on-membe|r of the|
|00004de0| 20 73 65 71 75 65 6e 63 | 65 2c 0a 77 69 74 68 20 | sequenc|e,.with |
|00004df0| 24 65 24 20 74 68 65 20 | 73 6d 61 6c 6c 65 73 74 |$e$ the |smallest|
|00004e00| 20 70 6f 73 69 74 69 76 | 65 20 6e 75 6d 62 65 72 | positiv|e number|
|00004e10| 20 77 69 74 68 20 24 62 | 5e 65 5c 65 71 75 69 76 | with $b|^e\equiv|
|00004e20| 31 24 2e 20 0a 49 66 20 | 24 65 3e 64 24 2c 20 77 |1$. .If |$e>d$, w|
|00004e30| 65 20 6d 61 79 20 74 61 | 6b 65 20 24 61 27 3d 62 |e may ta|ke $a'=b|
|00004e40| 24 2c 20 73 6f 20 77 65 | 20 77 69 6c 6c 20 61 73 |$, so we| will as|
|00004e50| 73 75 6d 65 20 24 65 5c | 6c 65 20 64 24 20 66 72 |sume $e\|le d$ fr|
|00004e60| 6f 6d 20 6e 6f 77 20 6f | 6e 2e 20 20 0a 42 79 20 |om now o|n.  .By |
|00004e70| 4c 65 6d 6d 61 7e 5c 72 | 65 66 7b 42 7d 2c 20 24 |Lemma~\r|ef{B}, $|
|00004e80| 62 5e 64 5c 6e 6f 74 5c | 65 71 75 69 76 31 24 2c |b^d\not\|equiv1$,|
|00004e90| 20 77 68 69 63 68 20 69 | 6d 70 6c 69 65 73 20 24 | which i|mplies $|
|00004ea0| 65 24 20 64 6f 65 73 20 | 6e 6f 74 20 64 69 76 69 |e$ does |not divi|
|00004eb0| 64 65 0a 24 64 24 20 61 | 6e 64 20 24 65 2f 28 64 |de.$d$ a|nd $e/(d|
|00004ec0| 2c 65 29 3e 31 24 2e 20 | 0a 24 24 5c 68 62 6f 78 |,e)>1$. |.$$\hbox|
|00004ed0| 7b 4c 65 74 5c 71 75 61 | 64 7d 61 27 3d 61 5e 7b |{Let\qua|d}a'=a^{|
|00004ee0| 28 64 2c 65 29 7d 62 5c | 71 71 75 61 64 20 63 3d |(d,e)}b\|qquad c=|
|00004ef0| 5c 66 72 61 63 20 0a 64 | 7b 28 64 2c 65 29 7d 24 |\frac .d|{(d,e)}$|
|00004f00| 24 54 6f 20 63 6f 6d 70 | 6c 65 74 65 20 74 68 65 |$To comp|lete the|
|00004f10| 20 70 72 6f 6f 66 2c 20 | 77 65 20 77 69 6c 6c 20 | proof, |we will |
|00004f20| 73 68 6f 77 20 74 68 61 | 74 20 69 66 20 24 61 27 |show tha|t if $a'|
|00004f30| 5e 78 5c 65 71 75 69 76 | 31 24 2c 0a 74 68 65 6e |^x\equiv|1$,.then|
|00004f40| 20 24 78 24 20 69 73 20 | 64 69 76 69 73 69 62 6c | $x$ is |divisibl|
|00004f50| 65 20 62 79 20 24 63 65 | 3d 64 65 2f 28 64 2c 65 |e by $ce|=de/(d,e|
|00004f60| 29 3e 64 24 2e 5c 70 71 | 0a 53 69 6e 63 65 20 24 |)>d$.\pq|.Since $|
|00004f70| 28 63 2c 65 29 3d 31 24 | 2c 20 54 68 65 6f 72 65 |(c,e)=1$|, Theore|
|00004f80| 6d 7e 5c 72 65 66 7b 54 | 31 7d 20 69 6d 70 6c 69 |m~\ref{T|1} impli|
|00004f90| 65 73 20 74 68 65 72 65 | 20 61 72 65 20 24 4b 2c |es there| are $K,|
|00004fa0| 4c 24 0a 77 69 74 68 20 | 24 63 4b 2b 65 4c 3d 31 |L$.with |$cK+eL=1|
|00004fb0| 24 2e 20 49 66 20 24 61 | 27 5e 78 5c 65 71 75 69 |$. If $a|'^x\equi|
|00004fc0| 76 31 24 2c 20 74 68 65 | 6e 20 24 61 27 5e 7b 63 |v1$, the|n $a'^{c|
|00004fd0| 78 7d 5c 65 71 75 69 76 | 20 62 5e 7b 63 78 7d 5c |x}\equiv| b^{cx}\|
|00004fe0| 65 71 75 69 76 31 24 2e | 0a 42 79 20 4c 65 6d 6d |equiv1$.|.By Lemm|
|00004ff0| 61 7e 5c 72 65 66 7b 64 | 69 7d 2c 20 24 63 78 3d |a~\ref{d|i}, $cx=|
|00005000| 65 4d 24 20 66 6f 72 20 | 73 6f 6d 65 20 69 6e 74 |eM$ for |some int|
|00005010| 65 67 65 72 20 24 4d 24 | 20 61 6e 64 0a 24 24 78 |eger $M$| and.$$x|
|00005020| 3d 28 63 4b 2b 65 4c 29 | 78 3d 65 28 4b 4d 2b 4c |=(cK+eL)|x=e(KM+L|
|00005030| 78 29 24 24 73 6f 20 24 | 78 3d 65 78 27 24 20 66 |x)$$so $|x=ex'$ f|
|00005040| 6f 72 20 73 6f 6d 65 20 | 69 6e 74 65 67 65 72 20 |or some |integer |
|00005050| 24 78 27 24 2e 20 20 54 | 6f 67 65 74 68 65 72 20 |$x'$.  T|ogether |
|00005060| 77 69 74 68 0a 24 61 27 | 5e 78 5c 65 71 75 69 76 |with.$a'|^x\equiv|
|00005070| 31 24 20 61 6e 64 20 4c | 65 6d 6d 61 7e 5c 72 65 |1$ and L|emma~\re|
|00005080| 66 7b 64 69 7d 2c 20 74 | 68 69 73 20 69 6d 70 6c |f{di}, t|his impl|
|00005090| 69 65 73 20 66 6f 72 20 | 73 6f 6d 65 20 69 6e 74 |ies for |some int|
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|00005e30| 70 72 65 76 69 6f 75 73 | 20 73 65 63 74 69 6f 6e |previous| section|
|00005e40| 20 74 68 61 74 20 64 6f | 65 73 20 6e 6f 74 20 69 | that do|es not i|
|00005e50| 6e 76 6f 6c 76 65 20 64 | 65 74 65 72 25 0a 6d 69 |nvolve d|eter%.mi|
|00005e60| 6e 69 6e 67 20 24 70 24 | 20 61 6e 64 20 24 71 24 |ning $p$| and $q$|
|00005e70| 2e 20 49 6e 20 74 68 65 | 20 6f 72 69 67 69 6e 61 |. In the| origina|
|00005e80| 6c 20 52 53 41 20 70 61 | 70 65 72 2c 20 74 68 65 |l RSA pa|per, the|
|00005e90| 20 61 75 74 68 6f 72 73 | 20 73 61 79 0a 5c 62 65 | authors| say.\be|
|00005ea0| 67 69 6e 7b 71 75 6f 74 | 61 74 69 6f 6e 7d 49 74 |gin{quot|ation}It|
|00005eb0| 20 6d 61 79 20 62 65 20 | 70 6f 73 73 69 62 6c 65 | may be |possible|
|00005ec0| 20 74 6f 20 70 72 6f 76 | 65 20 74 68 61 74 20 61 | to prov|e that a|
|00005ed0| 6e 79 20 67 65 6e 65 72 | 61 6c 20 6d 65 74 68 6f |ny gener|al metho|
|00005ee0| 64 0a 6f 66 20 62 72 65 | 61 6b 69 6e 67 20 6f 75 |d.of bre|aking ou|
|00005ef0| 72 20 73 63 68 65 6d 65 | 20 79 69 65 6c 64 73 20 |r scheme| yields |
|00005f00| 61 6e 20 65 66 66 69 63 | 69 65 6e 74 20 66 61 63 |an effic|ient fac|
|00005f10| 74 6f 72 69 6e 67 20 61 | 6c 67 6f 72 69 74 68 6d |toring a|lgorithm|
|00005f20| 2e 20 20 54 68 69 73 0a | 77 6f 75 6c 64 20 65 73 |.  This.|would es|
|00005f30| 74 61 62 6c 69 73 68 20 | 74 68 61 74 20 61 6e 79 |tablish |that any|
|00005f40| 20 77 61 79 20 6f 66 20 | 62 72 65 61 6b 69 6e 67 | way of |breaking|
|00005f50| 20 6f 75 72 20 73 63 68 | 65 6d 65 20 6d 75 73 74 | our sch|eme must|
|00005f60| 20 62 65 20 61 73 20 64 | 69 66 66 25 0a 69 63 75 | be as d|iff%.icu|
|00005f70| 6c 74 20 61 73 20 66 61 | 63 74 6f 72 69 6e 67 2e |lt as fa|ctoring.|
|00005f80| 20 57 65 20 68 61 76 65 | 20 6e 6f 74 20 62 65 65 | We have| not bee|
|00005f90| 6e 20 61 62 6c 65 20 74 | 6f 20 70 72 6f 76 65 20 |n able t|o prove |
|00005fa0| 74 68 69 73 20 63 6f 6e | 6a 65 63 74 75 72 65 2c |this con|jecture,|
|00005fb0| 0a 68 6f 77 65 76 65 72 | 2e 5c 65 6e 64 7b 71 75 |.however|.\end{qu|
|00005fc0| 6f 74 61 74 69 6f 6e 7d | 0a 54 6f 20 73 65 65 20 |otation}|.To see |
|00005fd0| 74 68 65 20 64 69 66 66 | 69 63 75 6c 74 69 65 73 |the diff|iculties|
|00005fe0| 20 69 6e 76 6f 6c 76 65 | 64 20 69 6e 20 74 72 79 | involve|d in try|
|00005ff0| 69 6e 67 20 74 6f 20 70 | 72 6f 76 65 20 73 75 63 |ing to p|rove suc|
|00006000| 68 20 61 20 74 68 69 6e | 67 2c 0a 73 75 70 70 6f |h a thin|g,.suppo|
|00006010| 73 65 20 74 68 61 74 20 | 6f 6e 65 20 63 6f 75 6c |se that |one coul|
|00006020| 64 20 73 68 6f 77 20 74 | 68 61 74 20 6b 6e 6f 77 |d show t|hat know|
|00006030| 6c 65 64 67 65 20 6f 66 | 20 61 20 63 69 70 68 65 |ledge of| a ciphe|
|00006040| 72 74 65 78 74 20 24 66 | 28 4d 29 24 0a 61 6e 64 |rtext $f|(M)$.and|
|00006050| 20 61 20 70 6c 61 69 6e | 74 65 78 74 20 24 4d 24 | a plain|text $M$|
|00006060| 20 65 6e 61 62 6c 65 64 | 20 6f 6e 65 20 74 6f 20 | enabled| one to |
|00006070| 66 69 6e 64 20 24 70 24 | 20 61 6e 64 20 24 71 24 |find $p$| and $q$|
|00006080| 2e 20 20 54 68 65 6e 20 | 6f 6e 65 20 0a 63 6f 75 |.  Then |one .cou|
|00006090| 6c 64 20 66 61 63 74 6f | 72 20 24 6e 24 20 61 73 |ld facto|r $n$ as|
|000060a0| 20 66 6f 6c 6c 6f 77 73 | 3a 5c 62 65 67 69 6e 7b | follows|:\begin{|
|000060b0| 65 6e 75 6d 65 72 61 74 | 65 7d 0a 5c 69 74 65 6d |enumerat|e}.\item|
|000060c0| 20 43 68 6f 6f 73 65 20 | 61 6e 79 20 24 4d 24 2e | Choose |any $M$.|
|000060d0| 5c 69 74 65 6d 20 43 6f | 6d 70 75 74 65 20 24 66 |\item Co|mpute $f|
|000060e0| 28 4d 29 24 2e 20 20 5b | 52 65 6d 65 6d 62 65 72 |(M)$.  [|Remember|
|000060f0| 2c 20 77 65 20 61 72 65 | 20 61 73 73 75 6d 69 6e |, we are| assumin|
|00006100| 67 0a 24 66 24 20 69 73 | 20 70 75 62 6c 69 63 6c |g.$f$ is| publicl|
|00006110| 79 20 61 76 61 69 6c 61 | 62 6c 65 2e 20 20 46 75 |y availa|ble.  Fu|
|00006120| 72 74 68 65 72 6d 6f 72 | 65 2c 20 24 66 28 4d 29 |rthermor|e, $f(M)|
|00006130| 24 20 63 61 6e 27 74 20 | 62 65 20 74 6f 6f 20 68 |$ can't |be too h|
|00006140| 61 72 64 0a 74 6f 20 63 | 6f 6d 70 75 74 65 2c 20 |ard.to c|ompute, |
|00006150| 6f 72 20 74 68 65 20 63 | 6f 64 65 20 77 6f 75 6c |or the c|ode woul|
|00006160| 64 20 62 65 20 69 6d 70 | 72 61 63 74 69 63 61 6c |d be imp|ractical|
|00006170| 2e 5d 5c 69 74 65 6d 20 | 55 73 65 20 74 68 65 20 |.]\item |Use the |
|00006180| 61 73 73 75 6d 65 64 0a | 6d 65 74 68 6f 64 20 74 |assumed.|method t|
|00006190| 6f 20 6f 62 74 61 69 6e | 20 24 70 24 2c 20 24 71 |o obtain| $p$, $q|
|000061a0| 24 2e 5c 65 6e 64 7b 65 | 6e 75 6d 65 72 61 74 65 |$.\end{e|numerate|
|000061b0| 7d 0a 49 6e 20 77 6f 72 | 64 73 2c 20 77 65 20 61 |}.In wor|ds, we a|
|000061c0| 72 65 20 75 6e 61 62 6c | 65 20 74 6f 20 64 69 73 |re unabl|e to dis|
|000061d0| 74 69 6e 67 75 69 73 68 | 20 62 65 74 77 65 65 6e |tinguish| between|
|000061e0| 20 74 68 65 20 73 69 74 | 75 61 74 69 6f 6e 0a 69 | the sit|uation.i|
|000061f0| 6e 20 77 68 69 63 68 20 | 24 66 28 4d 29 24 20 69 |n which |$f(M)$ i|
|00006200| 73 20 6f 62 74 61 69 6e | 65 64 20 66 72 6f 6d 20 |s obtain|ed from |
|00006210| 24 4d 24 20 28 65 61 73 | 79 29 20 61 6e 64 20 74 |$M$ (eas|y) and t|
|00006220| 68 65 20 28 70 72 65 73 | 75 6d 61 62 6c 79 20 64 |he (pres|umably d|
|00006230| 69 66 66 69 63 75 6c 74 | 29 0a 73 69 74 75 61 74 |ifficult|).situat|
|00006240| 69 6f 6e 20 69 6e 20 77 | 68 69 63 68 20 24 4d 24 |ion in w|hich $M$|
|00006250| 20 69 73 20 6f 62 74 61 | 69 6e 65 64 20 66 72 6f | is obta|ined fro|
|00006260| 6d 20 24 66 28 4d 29 24 | 2e 0a 5c 70 71 20 52 61 |m $f(M)$|..\pq Ra|
|00006270| 62 69 6e 20 68 61 73 20 | 73 75 67 67 65 73 74 65 |bin has |suggeste|
|00006280| 64 20 61 6e 20 61 6c 74 | 65 72 6e 61 74 69 76 65 |d an alt|ernative|
|00006290| 20 74 6f 20 74 68 65 20 | 52 53 41 20 73 79 73 74 | to the |RSA syst|
|000062a0| 65 6d 20 69 6e 20 77 68 | 69 63 68 20 74 68 65 72 |em in wh|ich ther|
|000062b0| 65 0a 69 73 20 61 20 64 | 69 72 65 63 74 20 63 6f |e.is a d|irect co|
|000062c0| 6e 6e 65 63 74 69 6f 6e | 20 74 6f 20 66 61 63 74 |nnection| to fact|
|000062d0| 6f 72 69 6e 67 2e 20 20 | 41 73 20 69 6e 20 52 53 |oring.  |As in RS|
|000062e0| 41 2c 20 24 6e 3d 70 71 | 24 20 69 73 20 61 6e 6e |A, $n=pq|$ is ann|
|000062f0| 6f 75 6e 63 65 64 0a 70 | 75 62 6c 69 63 6c 79 2c |ounced.p|ublicly,|
|00006300| 20 77 69 74 68 20 70 72 | 69 6d 65 73 20 24 70 24 | with pr|imes $p$|
|00006310| 2c 20 24 71 24 20 6b 65 | 70 74 20 73 65 63 72 65 |, $q$ ke|pt secre|
|00006320| 74 2e 20 20 46 6f 72 20 | 74 65 63 68 6e 69 63 61 |t.  For |technica|
|00006330| 6c 20 72 65 61 73 6f 6e | 73 2c 20 77 65 0a 61 73 |l reason|s, we.as|
|00006340| 73 75 6d 65 20 24 5c 63 | 6f 20 7b 70 2c 71 7d 33 |sume $\c|o {p,q}3|
|00006350| 34 24 2e 0a 54 68 65 20 | 65 6e 63 6f 64 69 6e 67 |4$..The |encoding|
|00006360| 20 66 75 6e 63 74 69 6f | 6e 20 69 73 20 20 24 24 | functio|n is  $$|
|00006370| 5c 63 6f 20 7b 66 28 4d | 29 7d 7b 4d 5e 32 7d 6e |\co {f(M|)}{M^2}n|
|00006380| 24 24 20 20 0a 20 54 68 | 65 20 77 61 79 20 77 65 |$$  . Th|e way we|
|00006390| 20 61 76 6f 69 64 20 74 | 68 65 20 64 69 66 66 69 | avoid t|he diffi|
|000063a0| 63 75 6c 74 79 20 64 65 | 73 63 72 69 62 65 64 20 |culty de|scribed |
|000063b0| 61 62 6f 76 65 20 69 73 | 20 74 68 61 74 20 74 68 |above is| that th|
|000063c0| 65 72 65 20 61 72 65 20 | 0a 7b 5c 69 74 20 66 6f |ere are |.{\it fo|
|000063d0| 75 72 5c 2f 7d 20 6e 75 | 6d 62 65 72 73 20 24 4d |ur\/} nu|mbers $M|
|000063e0| 5f 31 2c 4d 5f 32 2c 4d | 5f 33 2c 4d 5f 34 24 20 |_1,M_2,M|_3,M_4$ |
|000063f0| 77 69 74 68 20 24 66 28 | 4d 5f 69 29 5c 65 71 75 |with $f(|M_i)\equ|
+--------+-------------------------+-------------------------+--------+--------+