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# Discrete Math for Comp Sci I - Week of April 19th CS 2305

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This 4 page Class Notes was uploaded by Aaron Maynard on Monday April 25, 2016. The Class Notes belongs to CS 2305 at a university taught by Timothy Farage in Spring 2016. Since its upload, it has received 28 views.

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Date Created: 04/25/16

DiscreteMathforCompSciI SPRINGSEMESTER2016 INSTRUCTOR:DR.TimothyFarage atm150030@utdallas.edu 19 April 2016 Cryptosystem Definition: a pair of algorithms that take a key and convert plaintext to ciphertext and back. Plaintext is what you want to protect; ciphertext should appear to be random gibberish. The design and analysis of today's cryptographic algorithms is highly mathematical. Do not try to design your own algorithms. NO WAY. = PQ YCA. Ceaser EK + 2 | Cipher DK - 2 RSA Cryptosystem RSA is one of the first practical public-key cryptosystems and is widely used for secure data transmission. In such a cryptosystem, the encryption key is public and differs from the decryption key which is kept secret. 1) Public Encryption Algorithm 2) Public Decryption Algorithm 3) Public Encryption Key 4) Private Decryption Key 1 RSA Keys Steps to create an RSA key: 1) Pick two random large primes (200 - 300 digits) P 1 nd P2 2) Compute N = P 1 P2 nd Z = (P1 1) * (2 1) 3) Find a number E that is relatively prime to Z 4) Find the number D, that is the modular inverse of E; solve (E * D) MOD Z = 1. Using P1 3 and P2 11, then Z = 20, E = {1, 3, 7, 9, 11, 13, 17, 19} Say E = 7, then 7D (MOD 20) = 1, then D = 3. Our public key is N and E, N = 33 and E = 7. Our private key is D, D = 3 D and E are symmetric in a mystical way, “Almost everyone uses 65,537 for E”. Encryption Algorithm First some preliminaries; if you want to encrypt message m, then M must be less than N. M is a number. To Encrypt M (Message) : E Compute the number C = M MOD N; send C. (7) Say M = “19”, C = (19)MOD (33) C = 13 To Decrypt M (Message) : D M = C MOD N (3) M = (13)MOD 33 M = 19 2 Digital Signatures Definition: a type of electronic signature that encrypts documents with digital codes that are particularly difficult to duplicate. M = “Test 3 is on May 15th” To digitally sign M, compute C = MOD N E To take the message, M = CMOD N Math by Induction Definition: A way to establish a given statement for all natural numbers, and to prove statements about well-ordered sets. There are two ways of performing math by induction: ● Basis:prove that the statement holds for the first natural nun.er ● Inductive Step: prove that, if the statement holds for some natural numn,r then the statement holds fon+1. Basis:Show that the statement holds fon= 0. P(0) amounts to the statement: In the left-hand side of the equation, the only term is 0, and so the left-hand side is simply equal to 0. In the right-hand side of the equation, 0·(0 + 1)/2 = 0. The two sides are equal, so the statement is trun= 0. Thus it has been shown that P(0) holds. 3 Inductive st: Show thifPk) holds, then P(k+ 1) holds. This can be done as follows. Assume Pk) holds (for some unspecified k). It must then be shown(k+ 1) holds, that is: Using the induction hypothesiPk) holds, the left-hand side can be rewritten to: Algebraically: thereby showing that inP(k+ 1) holds. 4

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