Chapter 3 Encryption Algorithms

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Chapter 3Encryption Algorithms Systems
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Outline

• NP-completeness Encryption
• Symmetric (secret key) vs Asymmetric (public key)
  Encryptions
• Popular Encryption Algorithms
• Merkle-Hellman Knapsacks
• RSA Encryption
• El Gamal Algorithms
• DES
• Hashing Algorithms
• Key Escrow Clipper

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Review of Concepts

• Rate of growth
• Polynomial functions
• Example n, 10n, n2, 5n3,
• Exponential functions
• Example 2n, 210n,
• The order of functions matters!
• See next page.

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Order wins out!

• From Baase, Computer Algorithms (2nd ed.)
• Exercise How if a computer that runs at 2N ns.?

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Review of Concepts

• Polynomial bounded
• An algorithm is said to be polynomial bounded if
  its worst-case complexity is bounded by a
  polynomial function of the input size.
• That is, there exists a polynomial function p
  such that the algorithm terminates after at most
  p(n) steps, where n is the input size.
• A problem is said to be polynomial bounded if
  there is a polynomial bounded algorithm for it.
• Exercise Is an algorithm whose order is
  n10,000,000 polynomial-bounded?

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Nondeterminism

• A nondeterministic algorithm has two phases
• The nondeterministic phase (the guessing phase)
• A guess at a solution for the problem is
  proposed.
• The deterministic phase (the checking phase)
• A deterministic algorithm is executed, with or
  without the proposed solution from phase 1.

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Complexity Classes

• Complexity Classes
• Class P
• P is the class of problems that are
  polynomial-bounded.
• Class NP
• NP is the class of problems for which there is a
  nondeterministic polynomial-bounded algorithm.
• A theorem P ? NP
• Implication A deterministic algorithm is a
  special case of a nondeterministic algorithm
  (with phase 1 ignored).

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Characteristics of Hard Problems

• P.72
• Each problem is solvable, and a relatively simple
  approach solves it (although the approach may be
  time-consuming).
• There are 2n cases to consider if we use the
  approach of enumerating all possibilities, where
  n is the input size.
• The problems exist in various areas (logic,
  number theory, graph theory, ).
• If it were possible to guess perfectly, each
  problem could be solved in relatively little
  time. That is, the verification process is
  polynomial bounded.

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Sample NP Problems

• Satisfiability, p.71
• Knapsack
• Subset sum, a simpler version of Knapsack, p.72
• Clique, p.72
• Graph coloring
• Job scheduling with penalties
• Bin packing
• Hamilton paths (or circuits)
• Minimum tour (Traveling salesperson problem)
• . . .

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A broader Knapsack problem

• From Baase, Computer Algorithms (2nd ed.)
• Given
• a knapsack of capacity C (a positive integer
• N objects with sizes s1, , sn and profits p1,
  , pn, where s1, , sn and p1, , pn are positive
  integers.
• Problem Find the largest total profit of any
  subset of the objects that fits in the knapsack
  (and find a subset that achieves the maximum
  profit).

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The Graph Coloring Problem

• A coloring of a graph G (V, E) is a mapping C
  V?S, where S is a finite set of coloring, such
  that if vw ? E then C(v) ? C(w) that is,
  adjacent vertices are not assigned the same
  color.
• A decision problem Given G and a positive
  integer k, is there a coloring of G using at most
  k colors? (Is G k-colorable?)
• An application Scheduling the final exams at a
  university without conflicts.
• Let V be the set of courses.
• Let E be the pairs of courses whose exams must
  not be at the same time.
• Suppose there are 15 time slots (3 exams per day
  X 5 days).
• Then the exams can be scheduled in the 15 time
  slots without conflicts if and only if the graph
  G (V, E) is 15-colorable.

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NP-completeness

• P.75
• Cook 71
• A problem is called NP-complete if it can
  represent the entire class NP.
• Example The satisfiability problem (p.71) is a
  NP-complete problem. (Cooks theorem)
• Implication If a NP-complete problem can be
  solved by a deterministic, polynomial time
  algorithm, then all NP problems can be solved by
  such algorithms.
• The converse implication If for even one of the
  NP-complete problems it could be shown that there
  was no deterministic algorithm that ran in
  polynomial time, then no deterministic algorithm
  could exist for any of them.

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Does P NP or P ? NP?

• P.76
• NP-complete problems have been studied for a long
  time by many different groups of people and so
  far no polynomial, deterministic solution has
  been found for any one of the problems.
• Implication It is very likely that no
  polynomial, deterministic solution exists for
  NP-complete problems. That is, NP ? P.
• Note The question Does P NP? is still an
  open question.

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NP-completeness Cryptography

• Hard-to-solve problems are fundamental to
  cryptography, because the interceptor would need
  to work hard to break the encryption.
• But, be aware of the fallacies
• An NP-complete problem does not guarantee that
  there is no solution easier than exponential.
• Every NP-complete problem has a deterministic
  exponential time solution. That is, O(2n).
• Continuing advances in hardware make problems of
  larger size tractable.
• The interceptor does not always have to solve the
  had problem in order to crack the encryption.

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Review of Some Arithmetic Concepts

• Inverses
• A number i is an identity for an operation op if,
  for every number x, x op i x and i op x x.
• Example
• 0 is an identity for , because x 0 0 x
  x.
• 1 is an identity for , because x 1 1 x
  x.
• The inverse of a number x, x-1, under an
  operation op, is a number that makes x x-1
  x-1 x i, where i is the identity under op.
• Example The inverse of 5, under , is 5. The
  inverse of 5, under , is 1/5.

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Review of Some Arithmetic Concepts

• Prime numbers
• GCD
• Euclidean algorithm for computing GCD
• Modular arithmetic
• Two integers, a and b, are equivalent under
  modulus n, if a mod n b mod n.
• If a is equivalent to b, under modulus n,
• then (x y) k n for some k. (That is, the
  difference between x and y is divisible by n.)

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Modular arithmetic

• Table on p.79
• Reducibility
• (a b) mod n ((a mod n) (b mod n)) mod n
• (a b) mod n ((a mod n) (b mod n)) mod n
• Modular operations on negative numbers
• -13 mod 3 ?
• The inverse under modular operations
• Example The inverse of 3 mod 5, under , is 2
• because (3 2) mod 5 1. (See the
  multiplication table on p.80.)

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The inverse under modular operations

• Exercise What is the inverse of 4 mod 5 under
  ?
• Fermats Theorem
• ap mod p a (and ap-1 mod p 1),
• where p is a prime number and a lt p.
• Computing the inverse of a number
• The inverse of a number a
• ap-2 mod p (p.81)
• Example The inverse of 3 mod 5 35-2 mod 5 2.
• Verify 3 2 mod 5 1.

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Secret versus Public Key Encryptions

• Secret Key Encryption Algorithms
• Also called single key, symmetric, private key,
  or conventional algorithms.
• Both the encryptor and the decryptor use the same
  key.
• Example DES
• Public Key Encryption Algorithms
• The sender uses the receivers public key to
  encrypt the message. The receiver uses his own
  private key to decrypt the encrypted message.
• Examples Merkle-Hellman Knapsacks, RSA
  Encryption, El Gamal Algorithms

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Secret Key Encryptions

• Advantages
• The symmetry of the key provides a two-way
  channel. Both ends can both encrypt and decrypt
  information.
• Authentication of incoming messages An incoming
  ciphertext is authenticated because only
  legitimate sender can produce the ciphertext.

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Secret Key Encryptions

• Disadvantages
• A single-point failure A compromised key allows
  the interceptor not only to decrypt all the
  ciphertext encrypted by the key, but also to
  fabricate bogus ciphertext to be sent to
  legitimate receivers.
• Distribution of keys can be a problem. (See Fig.
  3-10, p.101 key distribution in pieces)
• The number of keys increases rapidly, roughly
  half of the square of the number of people. (See
  Fig. 3-11, p.101 distribution center)
• Adding a new user into an existing network of N
  users requires the addition of N new keys.
• The symmetric key encryptions can be relatively
  weak, unless based on hard problems.

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Public Key Encryptions

• Proposed by Diffie and Hellman in 1976.
• Each user has two keys a public key and a
  private key, which operate as inverses.
• P C (KPRIV, E (KPUB, P ) )
• Advantage
• No need for a shared key between every two users.

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Next

• Popular Encryption Algorithms
• Merkle-Hellman Knapsacks
• RSA Encryption
• El Gamal Algorithms
• DES
• Hashing Algorithms
• Key Escrow Clipper