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# 5-RSA - Public-Key Cryptography and RSA CSE 651...

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Public-Key Cryptography and RSA CSE 651: Introduction to Network Security

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Abstract We will discuss The concept of public-key cryptography RSA algorithm Attacks on RSA Suggested reading: Sections 4.2, 4.3, 8.1, 8.2, 8.4 Chapter 9 2
Public-Key Cryptography Also known as asymmetric-key cryptography. Each user has a pair of keys: a public key and a private key. The public key is used for encryption. The key is known to the public. The private key is used for decryption. The key is only known to the owner. 3

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4 Bob Alice
Why Public-Key Cryptography? Developed to address two main issues: key distribution digital signatures Invented by Whitfield Diffie & Martin Hellman 1976. 5

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6 1 1 trapdoor Easy: Hard: Easy: Use as the private key. Many public-key cryptosystems are based on trapdoor one-way fu trapdoor One-way function with trapdoor f f f x y x y x y - - → ← ← nctions.
Modular Arithmetic Mathematics used in RSA (Sections 4.2, 4.3, 8.1, 8.2, 8.4)

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| : divides , is a divisor of . gcd( , ): greatest common divisor of and . Coprime or relatively prime: gcd( , ) 1. Euclid's algorithm: computes gcd( , ). Extented Eucl Integers a b a b a b a b a b a b a b = id's algorithm: computes integers and such that gcd( , ). x y ax by a b + =
0 1 1 1 1 Comment: compute gcd( , ), where 1. : : for : 1, 2, until = 0 : mod return ( ) Euclidean Algorithm n i i i n a b a b r a r b i r r r r r + + - = = = = K

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1 78 2 65 1 13 6 299 221 221 78 78 65 5 5 13 0 gcd(229,221) 13 ( 2 ) 3 3 ( 299 221) 221 299 2 78 65 78 221 78 78 22 21 1 4 1 3 Extended Euclidean Algorithm:Example gcd(299,221) ? = + = + = + = + = = - = - - = - = - - = - =
id A group, denoted by ( , ), is a set with a binary operation : such that 1. ( ) ( ) (associative) 2. s.t. , ( ) 3. , s.t. entity Group G G G G G a b c a b c e G x G x x x x G y G x y y x e e × = 5 2200 ∈ = = 2200 ∈ 5 = = o o o o o o o o o o ( ) A group ( , ) is if , , . Examples: ( , abel ), ( , ), ( \{0}, ), ( , inver ), ( \{0} ian s , ). e e G x y G x y y x Z Q Q R R 2200 = + + × + × o o o

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Let 2 be an integer. Def: is congruent to modulo , written mod , if | ( ), i.e., and have the same remainder when d m ivided by . Note: d od an Integers modulo n a b n a b n n a b a b n a b n n - { } are different. Def: [ ] all integers congruent to modulo . [ ] is called a residue calss modulo , and is a representati mod ve of that class. n n a b a a n a n a n = =
[ ] [ ] if and only if mod . There are exactly residue classes modulo : [0], [1], [2], , [ 1]. If [ ], [ ], then [ ] and [ ]. Define addition and multiplication n n a b a b n n n n x a y b x y a b x y a b = - + + K for residue classes: [ ] [ ] [ ] [ ] [ ] [ ]. a b a b a b a b + = + =

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{ } { } ( 29 Define [0], [1], ..., [ 1] . Or, more conveniently, 0, 1, ..., 1 . , forms an abelian additive group. For , , ( )mod . (Or, [ ] [ ] [ ] [ mod ].) 0 is th n n n n Z n Z n Z a b Z a b a b n a b a b a b n = - = - + + = + + = + = + g g 10 e identity element. The inverse of , denoted by , is . When doing addition/substraction in , just do the regular addition/substraction and reduce the result modulo .
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