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研究生:戴文忠
研究生(外文):Wen-Chung Tai
論文名稱:論Lucas函數及其在密碼學之應用
論文名稱(外文):On The Lucas Function and Its Applications On Cryptography
指導教授:賴溪松賴溪松引用關係
指導教授(外文):Chi-Sung Laih
學位類別:碩士
校院名稱:國立成功大學
系所名稱:電機工程研究所
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:1994
畢業學年度:82
語文別:英文
論文頁數:40
中文關鍵詞:盧卡斯函數公開金匙密碼系統公開金匙分配系統離散對數問題
外文關鍵詞:Lucas functionPKCPKDSdiscrete logarithm problem
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自從公開金匙密碼系統的觀念在1976年首次由Diffie和Hellman兩氏提出
後,許多暗門函數已被用來設計密碼系統。其中,最有名且最常被使用的暗
門函數是指數函數Revist等人及ElGamal已成功的利用指數函數來設計公
開金匙密碼系統也就是有名的RSA和ElGamal系統。Diffie和Hellman 兩氏
也利用指數函數而提出了公開金匙分配系統, 這系統的安全性是基於離散
對數問題。最近 Smith和 Lennon 兩氏提出一個新的暗門函數叫盧卡斯函
數且以此函數來設計公開金匙密碼系統分配系統。盧卡斯函數是二次線性
遞迴關係的一個特例。既然盧卡斯函數可視為指數函數的推廣式, 因此相
信任何對盧卡斯函數的系統成功的攻擊同樣也及於指數函數的系統而相反
的推論並不成立。但是, 這推論成立否並不確定。在此論文中除了提出另
一種的盧卡斯公開金匙密碼系統於質數場中, 我們也關注盧卡斯函數的安
全性。我們證明出盧卡斯函數的安全性是同等於廣義的離散對數問題。除
此之外我們也提出盧卡斯函數一些有趣的特性包括對摺性和非對稱性。這
些特性有可能是盧卡斯函數的一個缺點也可能不是。我們也對RSA和LUC兩
系統做各方面的比較。本論文也描述以軟體實現盧卡斯函數的系統之成果


Since the concept of public key cryptosystem (PKC) was first
published by Diffie and Hellman in 1976 , a lot of possible
trapdoor functions have been proposed to design PKCs. Of these,
the best-known and most widely used trapdoor function is the
exponentiation. Revist et al. and ElGamal has succesfully used
the exponentiation function to design their PKCs and are now
well-known as RSA and ElGamal schemes. Diffie and Hellman also
used the exponentiation function to propose a public key
distribution scheme (PKDS). The security of the schemes based
on the exponentiation is relied on the discrete logarithm
problems Recently, Smith and Lennon proposed a new trapdoor
function called Lucas function and used this function to design
their PKC and PKDS. The Lucas function is a special form of
second-order linear recurrence relations using a large public
integer as modulus. Since the Lucas function can be seen as a
generalization of the exponentiation function,it is believed
that any successful attack on the schemes based on the Lucas
function would automatically lead to that based on the
exponentiation and the reverse may not be true. However, it is
not really known. In this thesis,except that the alternative
Lucas public key cryptosystem over GF(P) is proposed, we are
concerned on the security of the Lucas function. It is shown
that the security of Lucas function is polynomail-time
equivalent to the generalized discrete logarithm problems.
Besides, we also present some properties of the Lucas function,
including fold property and asymmetric property. They may or
may not be a disadvantages of the LUC system. We also make the
comparisons between RSA and LUC system. The work about
implementing these Lucas-related PKCs is also described in this
thesis.

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