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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
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RSA ANOTHER EXAMPLE
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RSA ANOTHER EXAMPLE
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RSA ANOTHER EXAMPLE