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# Class Note for CSCI 124 with Professor Vora at GW (6)

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COURSE
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TYPE
Class Notes
PAGES
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WORDS
KARMA
25 ?

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This 2 page Class Notes was uploaded by an elite notetaker on Saturday February 7, 2015. The Class Notes belongs to a course at George Washington University taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/07/15
CSCI 124224 Discrete Structures 11 Fast Modular Exponentiation Poorvi L Vora So far we have studied encryption functions that are constructed from addition and multiplication mod m These encryption functions are however not secure enough to use today Among the encryption functions in use today two of the most popular ones are based on taking the power of an element in Zm In this section we study a fast algorithm for nding a power mod m We will not pursue the encryption functions based on exponentiation any further in this course though These are taught in more detail in the cryptography courses CSci 162 undergraduate Cryptography CSci 284 graduate introductory Cryptography and CSci 381 graduate Advanced Cryptography Consider how one might compute xk mod m for large values of z k and m If one actually performed k 7 l multiplications z X z X X I that could be a large number of multiplications On the other hand one might use the square andmultiply rule which is considerably more ef cient Let s consider an example with small enough numbers to compute results by hand 513 mod 17 513 m0d17 522 X 522 X 5 which involves only 5 multiplications instead of 12 Or another example 515 m0d17 522 X 522 X 52 X 5 which involves 6 multiplications This can be formalized as follows exponentiationbc I m 11 mod m z 1 bi W bit in biliary representation of b forz39 l 7 l downto O 2 I 22 mod n ifbilzzzgtltzm0dn endfor retun1z Example 551 mod 7 CSCI 124 and CSCI 297VoIaGWU 51 110011 Example 729 mod 11 29 11101 i Zsecond 4 7 3 2 2 6 1 3 0 8

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