隨著科技的快速發展，電腦和網路漸漸取代紙本的方式協助我們傳遞和接收資訊，近年許多公司提供了低廉的雲端服務，使人們儲存檔案的方式不再侷限於實體硬碟、隨身碟等，隨時隨地都能獲取資料的關係，雲端備份的使用也逐漸取代了傳統的方式，但是當我們在享受雲端科技帶來的好處時，也必須思考這樣的方式是否也意味著我們在冒著資料被竊取的風險來換取儲存傳遞資料的便捷性，越來越多人也意識到資訊安全的重要性。密碼學的加解密演算法提供了一個方法來保護資料，藉由改變原始資料的特性使其他人無法輕易的得知真正的內容，本論文將探討加密演算法中的RSA如何透過指數運算與模除進行加密、解密，以改變傳遞給第三方之前的資料以及還原從第三方獲取的資料，透過蒙哥馬利算法的優點來簡化RSA繁雜的加密計算、餘數數字系統的加入打散了原始資料，進而增加了安全性和速度。最後將RSA加密系統用現場可程式化邏輯閘陣列合成數位電路，使用音檔作為此電路的輸入，驗證加、解密的正確性以及硬體使用時間和效率。

With the rapid development of technology, computers and the Internet have changed how we get information and communicate. Over the past decade, more companies have provided either cheap or free Cloud Storage. Since we can easily get our files from cloud service anytime and anywhere, cloud service gradually replaces the traditional method of saving files. Despite getting benefits from technology, people question if it is safe enough. Because of that, we start to pay more attention to information security and cryptology. Cryptosystems change plaintexts to ciphertexts, which means our data is converted to the unreadable one by an encryption algorithm. In this thesis, RSA, one of the cryptosystems, will be applied here. Since computation behind RSA is complicated, Montgomery modular multiplication will be used to optimize RSA. Residue Number System (RNS) will increase the safety and speed during computation. In the implementation, the designed circuit of RSA is configured on FPGA. The audio file as secret data input will be encrypted and decrypted. We will check the accuracy, timing, and space of this hardware of RSA.

[ 1 ] Biggs and Norman, “Codes: An introduction to Information Communication and Cryptography”, Springer, 2008, p. 171
[ 2 ] M. O’Keeffe, “The Paillier cryptosystem: a look into the cryptosystem and its potential application”, April 18, 2008.
[ 3 ] C. P. Schnorr and M. Jakobsson, “Security of Signed ElGamal Encryption”, Lecture Notes in Computer Science, vol. 1976, October 27, 2000.
[ 4 ] M. Calderbank, “The RSA Cryptosystem: History, Algorithm, Primes”, August 2007
[ 5 ] P. L. Montgomery, “Modular Multiplication Without Trial Division”, Mathematics of computation, vol.44, no. 170, p. 519~521, April 1985.
[ 6 ] A. Omondi and B. Premkumar, “Residue Number System Theory and Implementation”, dec. 2007
[ 7 ] Dence, Joseph B. and Dence, Thomas P., Elements of the Theory of Numbers, 1999
[ 8 ] F. E. Cerullo, “Deploying Secure Web Applications with OWASP Resources”, WebApplication Security, Madrid, Spain, December 2009.
[ 9 ]
W. Diffie and M. E. Hellman, “Multiuser cryptographic techniques”, AFIPS '76: Proceedings of the June 7-10, June 1976.
[ 10 ] N. Koblitz, "Elliptic curve cryptosystems", Mathematics of Computation 48, pp203–209, 1987
[ 11 ] Diffie, Whitfield; Hellman, Martin E., "New Directions in Cryptography," IEEE Transactions on Information Theory, pp. 644–654, November 1976
[ 12 ] Moskowitz, Martin A., “A Course in Complex Analysis in One Variable,” World Scientific Publishing Co., p. 7, 2002
[ 13 ] Z. Torabi, G. Jaberipur, A. Belghadr, "Fast division in the residue number system {2n + 1,2n, 2n-1} based on shortcut mixed radix conversion, " Computers & Electrical Engineering, Volume 83, 2020
[ 14 ] K. A. Gbolagade, R. Chaves, L. Sousa and S. D. Cotofana, "An improved RNS reverse converter for the {22n+1−1, 2n, 2n−1} moduli set," Proceedings of 2010 IEEE International Symposium on Circuits and Systems, 2010, pp. 2103-2106, doi: 10.1109/ISCAS.2010.5537062.
[ 15 ] D. Schinianakis and T. Stouraitis, "A RNS Montgomery multiplication architecture," 2011 IEEE International Symposium of Circuits and Systems (ISCAS), 2011, pp. 1167-1170, doi: 10.1109/ISCAS.2011.5937776.
[ 16 ] Microsoft Corporation, "WAVE and AVI Codec Registries - RFC 2361", IETF, June 1998
[ 17 ] Simpson, P.A., "FPGA Design, Best Practices for Team Based Reuse, 2nd edition." Switzerland: Springer International Publishing AG., p. 16., 2015