Key Establishment Techniques: Key Distribution and Key Agreement

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Key Establishment TechniquesKey Distribution
and Key Agreement

• Wade Trappe

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Key Establishment The problem

• Securing communication requires that the data is
  encrypted before being transmitted.
• Associated with encryption and decryption are
  keys that must be shared by the participants.
• The problem of securing the data then becomes the
  problem of securing the establishment of keys.
• Task If the participants do not physically meet,
  then how do the participants establish a shared
  key?
• Two types of key establishment
• Key Agreement
• Key Distribution

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Key Distribution

• Key Agreement protocols the key isnt determined
  until after the protocol is performed.
• Key Distribution protocols one party generates
  the key and distributes it to Bob and/or Alice
  (Shamirs 3pass, Kerberos).
• Shamirs Three-Pass Protocol
• Alice generates and Bob generates
  .
• A key K is distributed by

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Basic TTP Key Distribution
KDC
Kb
Ka
1. A Sends Request IDA IDB N1
2. KDC Sends EKa KAB Request IDA IDB
N1EKb(KAB, IDA)
3. A Sends EKb(KAB, IDA)
4. B Sends EKAB(N2)
5. A Sends EKAB(f(N2))
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Key Agreement

• In many scenarios, it is desirable for two
  parties to exchange messages in order to
  establish a shared secret that may be used to
  generate a key.
• The Diffie-Hellman (DH) protocol is a basic tool
  used to establish shared keys in two-party
  communication.
• Two parties, A and B, establish a shared secret
  by
• The security of the DH scheme is based upon the
  intractibility of the Diffie-Hellman Problem
• The Diffie-Hellman scheme can be extended to work
  on arbitrary groups (e.g. Elliptic Curves).

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Intruder In The Middle

• The Intruder-in-the-Middle attack on
  Diffie-Hellman is based upon the following
  strategy to improve ones chess ranking
• Eve challenges two grandmasters, and uses GM1s
  moves against GM2. Eve can either win one game,
  or tie both games.
• Eve has and can perform the
  Intruder-in-the-Middle attack by

Alice
Bob
Eve
Decrypts data with KAE, uses data and encrypts
with KBE
Decrypts data with KBE
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Station-to-Station Protocol

• Digital signatures can be used to prevent this
  protocol failure (STS Protocol).
• A digital signature is a scheme that ties a
  message and its author together.
• Private sig( ) function and Public ver( )
  function.

Verifies sig
Verifies sig
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N-to-N Group Key Establishment

• Many group scenarios require contributory key
  establishment protocols.
• 1-to-1 Key Establishment Diffie-Hellman (DH)
  protocol
• Two parties, A and B, establish a shared secret
  by
• Extensions to multi-user scenarios
• Ingemarsson Requires N-1 rounds and O(N2)
  exponentiations
• Burmester-Desmedt Requires 2 rounds but full
  broadcast
• GDH (Steiner et al.) Requires N rounds and O(N)
  exp.

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Butterfly Group Diffie-Hellman
Example
u1
u2
u3
u4

• Can be extended to arbitrary radix b using
  Ingemarsson as the basic building block.
• Total Rounds
• Total Messages
• Optimal radix in both cases is 2.

u5
u6
u7
u8
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The Conference Tree

• Group key formation procedure is described by
• Communication flow diagram
• Conference Tree
• Conference tree describes the subgroups and
  subgroup keys.

u1
u2
u3
u4
u5
u6
u7
K101
K001
K011
K100
K110
K000
K010
K111
u8
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Making Primes

• Fact Let n be an odd prime and let
  , where r is odd. Let a be any integer such that
  gcd(a,n)1. Then either or
  for some .
• Definition Let n be an odd composite with
  . Let
• . If either
  or , for some
  then n is a strong pseudoprime
  base a, and a is a strong liar for n.
• Fact If n is an odd composite integer, then at
  most 1/4 of the numbers a are strong liars for
  n.
• We can use this in a Monte-Carlo algorithm to
  produce primes
• Test t different as.
• Probability of falsely identifying a prime is