本文關鍵詞： SM2 數字簽名 Dickson乘法器　出處：《哈爾濱工業(yè)大學》2014年碩士論文　論文類型：學位論文

【摘要】：SM2是使用橢圓曲線加密(ECC)的一種密碼學標準,而ECC是1985年提出的一種公鑰密碼算法。與主流加密算法如RSA算法相比,ECC算法具有安全性能高、計算量小、處理速度快等特點。然而為充分保證系統(tǒng)的安全性,目前的數字簽名系統(tǒng)的公鑰和私鑰倍數一般都在256位以上,即密鑰生成和驗證過程進行都需要大數運算,因此即使采用ECC算法,無論是軟件還是硬件實現(xiàn),速度較慢仍然是數字簽名算法的一個缺陷。由于在實時性要求較高的場合,需要進行高速運算,因此,提高ECC算法的運算速度是非常重要的。在ECC算法中,需要執(zhí)行大量的加法與乘法運算。在加法運算可以通過XOR門實現(xiàn),而乘法運算則需要很多的AND和XOR門以及很長的延時。其他二位元擴域上的復雜運算如指數和點加運算等都可以通過調用乘法運算來實現(xiàn)。為了滿足數字簽名運算中數據快速處理的要求,需要設計出一種能夠快速完成二位元有限域上乘法運算的高效結構。本文的目標就是設計出一種能夠有效縮短乘法運算時間,提高數字簽名效率的乘法器。為此,本文提出了一種基于Dickson原理實現(xiàn)的新型乘法器,它利用Dickson多項式的獨特性質,將Dickson基底與傳統(tǒng)的GNB基底(Gaussian normal basis)聯(lián)系起來,并使用Dickson基底替代GNB基底。通過有限域上加減法相同的特性,使用Karatsuba分解方法對多項式進行分解,在付出增加三個加法的代價之下,減少一個乘法。接著利用Dickson多項式的性質,實現(xiàn)分解后多項式的重構,之后采用遞歸方法將一個長度為m的多項式分解成長度為2的多項式再進行基本的乘法運算,在遞歸返回之后,再利用基底轉換將Dickson基底轉換回GNB基底,最終實現(xiàn)整個乘法器的結構。本文一共提出了使用Karatsuba分解的二分法和三分法兩種乘法器結構,實驗結果表明,本文提出的新型Dickson乘法器與傳統(tǒng)乘法器以及同類改進的2型和4型GNB乘法器相比,二分法可以減少約50%的乘法運算,而三分法則可以減少約三分之二的乘法運算,并相應地減少一點加法運算。因此可知,使用本文提出的新型乘法器,可以優(yōu)化二位元有限域上的乘法結構,并提升數字簽名的效率。
[Abstract]:SM2 is a cryptographic standard using elliptic curve encryption (ECC), while ECC is a public key cryptographic algorithm proposed in 1985, which is compared with the mainstream encryption algorithms such as RSA. The ECC algorithm has the advantages of high security, small computation and fast processing speed. However, in order to fully guarantee the security of the system, the public and private key multiples of the current digital signature systems are generally more than 256-bit. That is, the key generation and verification process all need large number operation, so even if we use ECC algorithm, whether it is software or hardware implementation. The slow speed is still a defect of the digital signature algorithm. It is very important to improve the operation speed of ECC algorithm. In ECC algorithm, a large number of addition and multiplication operations need to be performed. In addition, the addition operation can be implemented by XOR gate. Multiplication requires a lot of AND and XOR gates and a long delay. Complex operations such as exponent and point addition on other binary extension fields can be implemented by calling multiplication operations. The requirement of fast data processing in signature operation. It is necessary to design an efficient structure which can quickly complete multiplication operations over binary finite fields. The goal of this paper is to design a new structure that can effectively shorten the time of multiplication operations. To improve the efficiency of digital signature, a new multiplier based on Dickson principle is proposed in this paper, which makes use of the unique properties of Dickson polynomials. The Dickson substrate is associated with the traditional GNB substrate Gaussian normal basis. The Dickson base is used to replace the GNB substrate. The polynomial is decomposed by using the Karatsuba decomposition method through the same properties of addition and subtraction over finite fields. At the cost of adding three additions, we reduce one multiplication. Then we use the properties of Dickson polynomials to reconstruct the decomposed polynomials. Then a polynomial whose length is m is decomposed into two polynomials by recursive method, and then the basic multiplication operation is carried out. After recursion returns, a polynomial with a length of m is decomposed into a polynomial with a growth degree of 2. Then the Dickson base is converted back to the GNB base using the base transformation. Finally, the structure of the whole multiplier is realized. In this paper, the dichotomy using Karatsuba decomposition and the three-point multiplier structure are proposed, and the experimental results show that. Compared with the traditional multiplier and the similar improved type 2 and 4 GNB multipliers, the new Dickson multiplier proposed in this paper can reduce the multiplication operation by about 50%. The three-point rule can reduce the multiplication operation by about 2/3, and reduce the addition operation by a little bit. Therefore, using the new multiplier proposed in this paper, we can optimize the multiplication structure on the binary finite field. And improve the efficiency of digital signatures.
【學位授予單位】：哈爾濱工業(yè)大學
【學位級別】：碩士
【學位授予年份】：2014
【分類號】：TN918.91

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1 李瑤;基于Dickson乘法器的SM2數字簽名算法研究與實現(xiàn)[D];哈爾濱工業(yè)大學;2014年

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