Rsa

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CRYPTOGRAPHYPUBLIC KEY CRYPTOGRAPHY: RSA

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PRIVATE-KEY CRYPTOGRAPHY

traditionalprivate/secret/single key cryptography usesone key

shared by both sender and receiver if this key is disclosed communications arecompromised

also issymmetric, parties are equal hence does not protect sender from receiverforging a message & claiming is sent by sender

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PUBLIC-KEY CRYPTOGRAPHY

probably most significant advance in the 3000

year history of cryptography

usestwo keys – a public & a private key

asymmetric since parties arenot equaluses clever application of number theoretic

concepts to function

complementsrather than replaces private key

crypto

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WHY PUBLIC-KEY CRYPTOGRAPHY?

developed to address two key issues:key distribution – how to have securecommunications in general without having totrust a KDC with your key

digital signatures – how to verify a messagecomes intact from the claimed sender

public invention due to Whitfield Diffie &Martin Hellman at Stanford Uni in 1976

known earlier in classified community

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PUBLIC-KEY CRYPTOGRAPHY

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PUBLIC-KEY CRYPTOSYSTEM:

SECRECY

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PUBLIC-KEY CRYPTOSYSTEM:

AUTHENTICATION

• Known as Digital Signature• It is impossible to alter the message without

access to A’s private key, so the message is

authenticated both in terms of source and in

terms of data integrity.

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PUBLIC-KEY CRYPTOSYSTEM: AUTHENTICATION AND SECRECY

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RSA

by Rivest, Shamir & Adleman of MIT in 1977

best known & widely used public-key scheme

based on exponentiation in a finite (Galois) field

over integers modulo a primenb. exponentiation takes O((log n)3) operations (easy)

uses large integers (eg. 1024 bits)

security due to cost of factoring large numbersnb. factorization takes O(elog n log log n) operations

(hard)

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10.2.2 Procedure

Figure 10.6 Encryption, decryption, and key generation in RSA

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Two Algebraic Structures

10.2.2 Continued

Encryption!ecryption Ring" R # $% n , &, ' (

)ey*+eneration +roup" + # $% φ n- , ' (∗

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4-1 ALGEBRAIC STRUCTURES

Cryptography requires sets of integers and specificoperations that are defined for those sets. The

combination of the set and the operations that are

applied to the elements of the set is called an

algebraic structure. In this chapter, we will define

three common algebraic structures: groups, rings,

and fields.

Topics discussed in this section:

4.1.1 Groups

4.1.2 Rings

4.1.3 Fields

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4.1 Continued

Figure 4.1 Common algebraic structure

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4.1.1 Groups

A group (G) is a set of elements with a binar

operation (•) that satisfies four properties (or a!ioms).

A "ommutati#e group satisfies an e!tra propert$

"ommutati#it%

❏ &losure%❏ Asso"iati#it%

❏ &ommutati#it%

❏ '!isten"e of identit%❏ '!isten"e of in#erse%

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4.1.1 Continued

Figure 4.2 Group

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4.1. !ing

A ring$ R *+,$ •$ -$ is an algebrai" stru"ture with

two operations.

Figure 4.4 !ing

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4.1. Continued '!ample 4.11

he set / with two operations$ addition and multipli"ation$ is a "ommutati#e ring. 0e

show it b R /$ $ -. Addition satisfies all of the fi#e properties multipli"ation

satisfies onl three properties.

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RSA ALGORITHM

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WHY RSA WORKS

because of Euler's Theorem:aø(n)mod n = 1 wheregcd(a,n)=1

in RSA have:n=p.qø(n)=(p-1)(q-1) carefully chosee &d to be inversesmod ø(n) hencee.d=1+k.ø(n) for somek

hence :

Cd = Me.d = M1+k.ø(n) = M1.(Mø(n))k = M1.(1)k = M1 = M mod n

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RSA EXAMPLE - KEY SETUP

1. Select primes: p=17 & q =11

2. Calculate n = pq =17 x 11=187

3. Calculate ø(n)=( p–1)(q-1)=16x10=160

4. Selecte:

gcd(e,160)=1; choosee=75. Determined: de= 1 mod 160 andd < 160 Value isd=! since!x7=161= 10x160+1

6. Publish public key"#=$7,187%

7. Keep secret private key"=$!,187%

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RSA EXAMPLE - EN/DECRYPTION

Public key PU = {7, 187} and private key PR = {23, 187}.

given messageM = 88 (nb.88<187)

encryption:

C = 887 mod 187 = 11

decryption:M = 11! mod 187 = 88

Exploiting the properties of modular arithmetic

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EXAMPLE

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RSA ANOTHER EXAMPLE

Rsa

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