Chapter 9 – Cryptography and Network
Security (William Stallings)
Public key cryptography
Basic Asymmetric Algorithms
Dr. Danish Mahmood
CS Dept., SZABIST
Islamabad.
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traditional private/secret/single key
cryptography uses one key
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shared by both sender and receiver
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if this key is disclosed communications are
compromised
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also is symmetric, parties are equal
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hence does not protect sender from receiver
forging a message & claiming is sent by
sender
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probably most significant advance in the
3000 year history of cryptography
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uses two keys – a public & a private key
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asymmetric since parties are not equal
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uses clever application of number theoretic
concepts to function
complements rather than replaces private key
crypto
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public-key/two-key/asymmetric
cryptography involves the use of two keys:
◦ a public-key, which may be known by anybody,
and can be used to encrypt messages, and verify
signatures
◦ a private-key, known only to the recipient, used
to decrypt messages, and sign (create) signatures
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is asymmetric because
◦ those who encrypt messages or verify signatures
cannot decrypt messages or create signatures
Hashin
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can classify uses into 3 categories:
◦ encryption/decryption (provide secrecy)
◦ digital signatures (provide authentication)
◦ key exchange (of session keys)
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some algorithms are suitable for all uses,
others are specific to one
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like private key schemes brute force exhaustive search attack is
always theoretically possible
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but keys used are too large (>512bits)
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security relies on a large enough difference in difficulty
between easy (en/decrypt) and hard (cryptanalyse) problems
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more generally the hard problem is known, its just made too
hard to do in practise ---- NP HARD- NP COMPLETE
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requires the use of very large numbers
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hence is slow compared to private key schemes
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Knapsack
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Diffie-Hellman
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RSA (Rivest-Shamir-Adleman)
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El Gamal
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Elliptic curve cryptosystem (ECC)
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Digital Signature Algorithm (DSA)
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Merkle-Hellman Knapsack
Thief (cash, gold, silver, household, platinum, diamond, crystal, copper)
He has a knapsack that can carry 20kg weight…
Now the problems is, what items he should be carrying / stealing.
(cash, gold, silver, household, appliances, diamond, crystal, copper)
(.5kg, .5kg, .5kg, 10kg, 7kg, .2kg, .5kh, 2kg)—one set--(1000, 3000, 1500. 5000, 2000, 10000, 5000, 3000)– 2nd set--This is a NP Complete hard problem…(nondeterministic polynomial) --- two
iterations,,, first we guess a solution, 2nd we check if our guess was correct.
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Given a set of n weights W0,W1,...,Wn-1 and a sum S,
find ai  {0,1} so that
S = a0W0+a1W1 + ... + an-1Wn-1
(technically, this is the subset sum problem)
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Example
Value of items
◦ Weights (62,93,26,52,166,48,91,141)
◦ Problem: Find a subset that sums to S = 302
◦ Answer: 62 + 26 + 166 + 48 = 302
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The (general) knapsack is NP-complete
The size of ba
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NP is the set of decision problems solvable in polynomial
time by a non-deterministic Turing machine.
This definition is the basis for the abbreviation NP;
"nondeterministic, polynomial time."
The algorithm is based on the Turing machine which
consists of two phases,
◦ the first of which consists of a guess about the solution, which is
generated in a non-deterministic way,
◦ while the second phase consists of a deterministic algorithm that
verifies if the guess is a solution to the problem.
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General knapsack (GK) is hard (NP hard) to solve
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But superincreasing knapsack (SIK) is easy
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SIK  each weight greater than the sum of all
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Example
previous weights
◦ Weights (2,3,7,14,30,57,120,251)
◦ Problem: Find subset that sums to S = 186
◦ Work from largest to smallest weight
◦ Answer: 120 + 57 + 7 + 2 = 186
1. Generate superincreasing knapsack
(SIK)
2. Convert SIK to “general” knapsack (GK)
3. Public Key: GK
4. Private Key: SIK and conversion factor
Goal…
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o
o
o
Easy to encrypt with GK
With private key, easy to decrypt (solve SIK)
Without private key, Trudy has no choice but to try to solve GK
Element in
1+2+4+10+20+40< m--110
n *value mod m
31*31^-1mod(110)=1
62*31^-1mod(110)=2
…
General set Knapsack
SIK set knapsack
6 elements in every set
(31, 62, 14, 90, 70, 30)– GSK
(1,2,4,10,20,40)- SIK
123 4 5 6
1 2 4 10 20 40
100 1 0 0
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Trapdoor: Convert SIK into “general” knapsack
using modular arithmetic
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One-way: General knapsack easy to encrypt,
hard to solve; SIK easy to solve
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This knapsack cryptosystem is insecure
◦ Broken in 1983 with Apple II computer
◦ The attack uses lattice reduction
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“General knapsack” is not general enough!
◦ This special case of knapsack is easy to break