Lecture Data security and encryption - Chapter 32: Review 16-30

Data Security and Encryption
(CSE348)

1


Revision
Lectures 16-30

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RSA
Ø

Ø

Ø

Ø

RSA is the best known, and by far the most widely
used general public key encryption algorithm
First published by Rivest, Shamir & Adleman of
MIT in 1978 [RIVE78]
The Rivest-Shamir-Adleman (RSA) scheme has
since that time ruled supreme as the most widely
accepted
Implemented general-purpose approach
to public3

RSA
Ø

Ø

It is based on exponentiation in a finite (Galois)
field over integers modulo a prime, using large
integers (eg. 1024 bits)
Its security is due to the cost of factoring large
numbers

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RSA
Ø

By Rivest, Shamir & Adleman of MIT in 1977

Ø

Best known & widely used public-key scheme

Ø

based on exponentiation in a finite (Galois) field
over integers modulo a prime
l nb.

exponentiation takes O((log n)3) operations (easy)

Ø

Uses large integers (eg. 1024 bits)

Ø

Security due to cost of factoring large numbers
l nb.

factorization takes O(e log n log log n) operations
(hard)
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RSA En/decryption
•

•

•

•

The scheme developed by Rivest, Shamir, and
Adleman makes use of an expression with
exponentials
Plaintext is encrypted in blocks
with each block having a binary value less than

some number n
The actual RSA encryption and decryption
computations are each simply a single
exponentiation mod (n)
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RSA En/decryption
•

•

•

•

•

Both sender and receiver must know the value
of n
The sender knows the value of e, and only the
receiver knows the value of d
Thus, this is a public-key encryption algorithm
with a public key of PU = {e, n} and a private key
of PR = {d, n}
The message must be smaller than the modulus
The “magic” is in the choice of the modulus and
exponents which makes the system work
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RSA En/decryption
•

•

•

To encrypt a message M the sender:
–

obtains public key of recipient PU={e,n}

–

computes: C = Me mod n, where 0≤M
To decrypt the ciphertext C the owner:
–

uses their private key PR={d,n}

–

computes: M = Cd mod n

The message M must be smaller than the
modulus n (block if needed)
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Diffie-Hellman Key Exchange
ØPublic-key

cryptography systems (PKCSs)

ØBegins

with a description of one of the earliest
and simplest PKCS

ØDiffie-Hellman

key exchange

ØThis

first published public-key algorithm
appeared in the seminal paper by Diffie and
Hellman
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Diffie-Hellman Key Exchange
ØThat

defined public-key cryptography [DIFF76b]

ØAnd

is generally referred to as Diffie-Hellman key
exchange

ØThe

concept had been previously described in a
classified report in 1970 by Williamson (UK
CESG)

ØAnd

subsequently declassified in 1987, see
[ELLI99]
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Diffie-Hellman Key Exchange
ØThe

purpose of the algorithm is to enable two
users to securely exchange a key

ØThat

can then be used for subsequent encryption
of messages

ØThe

algorithm itself is limited to the exchange of

secret values

ØA number

of commercial products employ this
key exchange technique
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Diffie-Hellman Key Exchange
ØFirst

public-key type scheme proposed

ØBy

Diffie & Hellman in 1976 along with the
exposition of public key concepts
l
now know that Williamson (UK CESG)
secretly proposed the concept in 1970

ØPractical

method for public exchange of a secret

key
ØUsed

in a number of commercial products

12


Diffie-Hellman Key Exchange
ØThe

purpose of the algorithm is to enable two
users to securely exchange a key

ØThat

can then be used for subsequent encryption
of messages

ØThe

algorithm itself is limited to the exchange of
secret values

ØWhich

depends on the value of the public/private
keys of the participants
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Diffie-Hellman Key Exchange
ØDiffie-Hellman

algorithm uses exponentiation in a

finite (Galois) field (modulo a prime or a
polynomial)

ØAnd

depends for its effectiveness on the difficulty
of computing discrete logarithms

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Diffie-Hellman Key Exchange
ØA public-key

distribution scheme
l cannot be used to exchange an arbitrary
message
l rather it can establish a common key
l
known only to the two participants

ØValue

of key depends on the participants
Øand their private and public key information

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Diffie-Hellman Key Exchange

ØBased

on exponentiation in a finite (Galois) field
(modulo a prime or a polynomial) - easy

ØSecurity

relies on the difficulty of computing
discrete logarithms (similar to factoring) – hard

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Man-in-the-Middle Attack
Ø

Darth prepares by creating two private / public
keys

Ø

Alice transmits her public key to Bob

Ø

Darth intercepts this and transmits his first public
key to Bob

Ø

Darth also calculates a shared key with Alice

Ø

Bob receives the public key and calculates the
shared key (with Darth instead of Alice)
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Man-in-the-Middle Attack
Ø

Bob transmits his public key to Alice

Ø

Darth intercepts this and transmits his second
public key to Alice

Ø

Darth calculates a shared key with Bob

Ø

Alice receives the key and calculates the shared
key (with Darth instead of Bob)

Ø

Darth can then intercept, decrypt, re-encrypt,
forward all messages between Alice &18Bob


ElGamal Cryptography
ØIn

1984, T. Elgamal announced a public-key
scheme based on discrete logarithms

ØClosely

related to the Diffie-Hellman technique
[ELGA84, ELGA85]

ØThe

ElGamal cryptosystem is used in some form in
a number of standards

ØIncluding

the digital signature standard (DSS) and
the S/MIME email standard
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ElGamal Cryptography
ØAs

with Diffie-Hellman, the global elements of
ElGamal are a prime number q and a

ØWhich

is a primitive root of q

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ElGamal Cryptography
ØUser A generates

a private/public key pair as shown

ØThe

security of ElGamal is based on the difficulty of
computing discrete logarithms

ØTo

recover either x given y, or k given K

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ElGamal Cryptography
ØPublic-key
ØUses

cryptosystem related to D-H

exponentiation in a finite field

ØWith

security based difficulty of computing discrete
logarithms, as in D-H

ØEach

user (eg. A) generates their key
l chooses a secret key (number): 1 < xA < q-1
l
compute their public key: yA = axA mod q
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ElGamal Message Exchange
ØAny

user B that has access to A's public key can
encrypt a message as shown

ØThese

steps correspond to Figure 9.1a in that
Alice generates a public/private key pair

ØBob

encrypts using Alice's public key; and Alice
decrypts using her private key

ØSee

text for details of why these steps result in M
being recovered
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ElGamal Message Exchange

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ElGamal Message Exchange
ØK

functions as a one-time key, used to encrypt
and decrypt the message

ØIf

a message must be broken up into blocks and
sent as a sequence of encrypted blocks, a
unique value of k should be used for each
block

ØIf

k is used for more than one block, knowledge
of one block m of the message enables the
user to compute other blocks
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