Enigma Alic Bob Ev [cryptographic] ky public-ky cryptosystmalic Bob Alic Bob Ev Alic Bob Bob Alic Bob Alic Bob ( 2 ) Diffi Hllman 196 [1]public

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1 What's fun in EE 臺大電機系科普系列密碼學與模算術鄭振牟國立臺灣大學電機工程學系教授密碼學密碼學是資訊安全的基石在資訊網路技術日漸普及的現代社會中密碼學在維持社會順利運作上扮演著極為重要的角色沒有密碼學我們就無法在網路上安全地進行交易銀行金融系統也無法正常運作密碼學中一個重要的問題是如何達成祕密通信這也是古典密碼學中最重要的課題在這個問題中密碼學家喜歡用兩位同學作例子通常是 Alic 跟 Bob 他們想要利用公開的通信管道交換一些私密的信息密碼學家雖然不見得都憤世嫉俗但他們往往傾向為最壞的情況作打算比如說在這個例子裡面密碼學家就會假設壞人通常叫做 Ev 一定能夠偷看到所有 Alic 和 Bob 之間交換的信息在這樣的狀況下 Alic 和 Bob 必需使用一些特殊的編碼方式讓 Ev 就算看到中間交換的信息也無法推斷出 Alic 和 Bob 究竟想要告訴對方什麼消息用密碼學的而經過特行話來說 Alic 和 Bob 真正想要傳達給對方的消息叫做明文 plaintxt 殊編碼之後在公開通信管道上流通的那個信息就叫做密文 ciphrtxt 這個特殊編碼的過程叫做加密 ncryption 之後還原明文的過程就叫做解密 cryption 整套特殊編碼的方法則叫做加密法 ciphr 大家可以想像這樣的技術在傳遞軍事情報上有著非常大的價值而古典密碼學的發展的確至少可以追溯到古羅馬偉大的將領凱撒 Julius Casar 所使用的凱撒加密法 Casar ciphr 在二次大戰時盟軍也藉由破解了德軍所使用的 Enigma 加密機以及日軍所使用的乙式暗號機攔截到許多重要的軍事情報例如因此擊落了日本海軍聯合艦隊司令官山本五十六的座機值得一提的是當時參與 Enigma 密碼機破解工作的主要人員其中有後來被尊稱為電腦科學之父的圖靈 Alan Turing 後來人們為了紀念他對電腦科學的貢獻遂將電腦科學界最重要的獎項命名為圖靈獎 Turing awar 臺灣大學電機工程學系 1061 台北市大安區羅斯福路四段一號

交換的信息也無法推斷出 Alic 和 Bob 究竟想要告訴對方什麼消息用密碼學的而經過特行話來說 Alic 和 Bob 真正想要傳達給對方的消息叫做明文 plaintxt 殊編碼之後在公開通信管道上流通的那個信息就叫做密文 ciphrtxt 這個特殊編碼的過程叫做加密 ncryption 之後還原明文的過程就叫做解密 cryption 整套特殊編碼的方法則叫做加密法 ciphr

2 Enigma Alic Bob Ev [cryptographic] ky public-ky cryptosystmalic Bob Alic Bob Ev Alic Bob Bob Alic Bob Alic Bob ( 2 ) Diffi Hllman 196 [1]public ky privat ky Bob Alic Bob Bob Bob Ev Bob Alic Ev Bob Ev Ev Bob Alic Alic Bob PKI: public-ky infrastructur SA ivst Shamir Alman 198 SA [2] SA Alic Bob Bob Alic Alic Bob SA Ev

ky Bob Alic Bob Bob Bob Ev Bob Alic Ev Bob Ev Ev Bob Alic Alic Bob PKI:

3 Alic 1ys 0no Bob Ev Alic Golwassr Micali 1982 smantic scurity [] Alic Bob Alic Bob 1ys 0 no Ev Alic Alic Alic Bob SA Bob SA = = Alic 4 m = 4 Bob Alic (1) c c m mo 4 mo 16 m c mo 16 mo 4 Bob c m mo 4 mo 16 = (2) ( ) x 1 mo, m c mo 16 mo 4 (2) ( ) ( ) y ( yx) y x 1 mo Bob, x (mo ) yy yy yy Alic SA ( ) y ( yx) 1y x (mo ) Alic y( Y) yy 1 yy Bob p p 1 ( ) T ( T 1 p mo ) ( ab) p (mo ) SA T ( T mo ) congrunc ( ab) (mo ) rlation a b a b a = b mo quivalnc rlation rflxivity symmtrytransitivity quivalnc classs a = b mo a b a b 0 1 (1) a 1 a 2 b 1 b 2 a 1 = a 2 mo b 1 = b 2 mo a 1 + b 1 = a 2 + b 2 mo a 1 b 1 = a 2 b 2 mo moular arithmtic A a 1 B b 1 A B A + B AB x y gc (x, ) = 1 gc (y, ) = 1xy gc (xy, ) x

) T ( T 1 p mo ) ( ab) p (mo ) SA T ( T mo ) congrunc ( ab) (mo ) rlation a b a b a = b mo quivalnc rlation rflxivity symmtrytransitivity quivalnc classs a = b mo a b a b 0 1

4 xtn uclian algorithm a b ax + b = 1 ax = 1 mo a x 1 a b a x + b = 1a a x a x 1 = a mo x y f x (y) = xy mo bijction x x φ () = 1 mo, φ() Eulr s totint functionφ() 1 x p 1 = 1 mo p 1 x Y Y Y f x Y y x f x Y = φ() Y = {y 1 y <, gc(y, ) = 1} xy = f x (Y ) = {xy mo c m mo 4 mo 16 1 y <, gc(y, ) = 1} m c mo 16 mo 4 ( ) x 1 mo (), ( ) y ( yx) y x (mo ) yy yy yy 1 ( ) 1 1 p p T ( T mo ) SA ( ab) (mo ) 1 SA (1) c(2) m mo 4 mo 16 = 1 mo φ() m m = m 1+nφ() = m(m n ) φ() m c mo = 16 m (mo ) 4 ( ) x 1 mo, + kφ() = 1 = 1 mo φ() ( ) y ( yx) y x (mo ) yy yy yy (4) φ() 1 ( ) 1 (4) p p T ( T mo ) SA ( ab) (mo ) SA SA SA SA 2048-bit SA p q p q SA SA 68 bits[5] ()

yx) y x (mo ) yy yy yy 1 ( ) 1 1 p p T ( T mo ) SA ( ab) (mo ) 1 SA (1) c(2) m mo 4 mo 16 = 1 mo φ() m m = m 1+nφ() = m(m n ) φ() m c mo = 16 m (mo ) 4 ( ) x 1 mo, +

5 Alic Bob SA SA SA (1) (2) SA moular xponntiation Ptr Montgomry 1985 Montgomry ruction a b x mo x a b a + b = (a + b) (mo ) a a(a)(b)= c m mo 4 mo 16 2 (ab) (ab) = (mo ) m c mo 16 mo 4 ( ) x 1 mo, ( ) T = (a mo )(b mo ) x y ( yx) y x (mo ) yy yy T + x yy x 0 < 1 < 0 < < 1 = 1 1 = 1 mo 1 = 1 mo T + x = 0 mo x = ( ) 1 1 p T = T (mo ) p T ( T mo ) ( ab) (mo ) (5) < T < 2 < 2 (5) 0 2 = 100 Bob Bob 1. Bob c = 16 = = 2. = 1 mo =. c mo = 1600 mo = T 1 =(c mo ) 2 = x 1 = T 1 mo = (T 1 mo ) mo = 56 mo 100 = c 2 mo = 1 T 1 mo =(T 1 + x 1 )/ = ( )/100 = 25. T 2 =(c 2 mo )(c mo ) = x 2 = T 2 mo = 0

= ( ) 1 1 p T = T (mo ) p T ( T mo ) ( ab) (mo ) (5) < T < 2 < 2 (5) 0 2 = 100 Bob Bob 1. Bob c = 16 = = 2. = 1 mo =. c mo = 1600 mo = 16 4.

6 9. c mo = 1 T 2 mo =(T 2 + x 2 )/ = x = (c mo ) mo = m = c mo = 1 (c mo ) mo = ((c mo )+ x)/ = mo 2 = 0 Bob frncs 1. W. Diffi an M. E. Hllman, w irctions in cryptography, IEEE Transactions on Information Thory, Vol. IT- 22, 196, pp ivst, A. Shamir, an L. Alman, A mtho for obtaining igital signaturs an public-ky cryp-tosystms, Communications of th ACM, Vol. 21 (2), 198, pp S. Golwassr an S. Micali, Probabilistic ncryption an how to play mntal pokr kping scrt all partial information, In Annual ACM Symposium on Thory of Computing (STOC), P. L. Montgomry, Moular multiplication without trial ivision, Mathmatical Computation, Vol. 44, 1985, pp T. Klinjung, K. Aoki, J. Frank, A. Lnstra, E. Thomé, J. Bos, P. Gaury, A. Kruppa, P. L. Montgomry, D. A. Osvik, H. t il, A. Timofv, an P. Zimmrmann, Factorization of a 68-bit SA moulus, Cryptology Print Archiv: port 2010/006,

Alman, A mtho for obtaining igital signaturs an public-ky cryp-tosystms, Communications of th ACM, Vol. 21 (2), 198, pp. 120 126.. S. Golwassr an S.

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