Discrete Logarithm Problem

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Discrete Logarithm Problem ElGamal Cryptosystem

• Discrete logarithm problem
• Problem 6.1, page 227
• ElGamal cryptosystem
• Cryptosystem 6.1, page 227
• Example 6.1, page 228

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Authentication and digital signature

• Conventional (hand) signature
• A part of document physically
• Verification comparing it with other authentic
  signature by people, easily forgery
• Copy of signed document is easily distinguished
  from original
• Digital signature
• Not attached in the document physically,
  therefore must somehow bind signature to the
  message
• Publicly known verification algorithm, anybody
  can verify it, not easy to forgery
• Easy copy of signed digital message, thus need to
  prevent reuse of the copy such as timestamp.

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Digital signature

• Two components
• A (private) signing algorithm sigK
• A public verifying algorithm verK
• For message x, signature is ysigK(x).
• Pair (x,y) called signed message and transmitted
• Verification
• verK(x,y) true if ysigK(x), false othersiae.

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Requirements for digital signature

• Both sigK and verK are easily computed (i.e., in
  polynomial time)
• Given any message x, it is computationally
  infeasible for anyone other than Alice to compute
  a signature y such that verK(x,y) true

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RSA signature

• Cryptosystem 7.1, page 276

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Encryption along with signature

• Suppose Alice has (PA, SA) and Bob has (PB, SB)
• Alice wants to send a both signed and encrypted
  message x to Bob.

One method encrypt first and then sign the
encrypted message i.e., y xPB mod n
, z ySA mod n and transmit (y,z)
Any problem with the above method?
Secure method sign first and then encrypt the
signed message i.e., z xSA mod n , y
zPB mod n and transmit y.
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Long message and signature

• (fast) public cryptographic hash function
• h 0,1 ? Zn
• For a message x of any length, compute mdh(x)
• md is also called message digest.
• Then sign md z sigK(md)
• Send (x, z)
• Verification compute md, then verK(md,z)
• A hash function must satisfy certain properties.

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Key distribution

• Key distribution is a big problem with secret-key
  system (and group communication)
• Use public-key system to distribute key (called
  session key) and then use session key for fast
  data transmission.
• (for secure group communication), a center key
  server generates a key and distributes the key to
  group members.

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Key exchange (agreement)

• Diffie-Hellman key exchange (agreement)
• Based on DLP problem
• Suppose a prime p and generator g of Zp are
  public.
• Alice select a number a, compute yga and send y
  to Bob
• Bob select a number b, compute zgb and send z to
  Alice
• then Alice compute k za ( gab)
• And Bob compute k yb (gab).
• Therefore Alice and Bob achieve the same key
  securely without meeting together. How beautiful
  it is!!!

Could you think any problem with the protocol?
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Key management public key certificate

• Key management
• how to securely and reliably distribute the keys
  used (not only secret key, but also public key).
• not to breaking algorithms used, but to breaking
  the key distribution scheme
• have a range of possible key distribution
  techniques
• one of the most critical areas in security
  systems
• absolutely critical to get this right
• http//www.cs.adfa.edu.au/teaching/studinfo/ccs3/l
  ectures/less20.html

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Key management schemes

• Physical Delivery
• by secure courier
• registration name and password for computers
• Authentication Key Server
• have an on-line server trusted by all clients
• server has a unique secret key shared with each
  client
• server negotiates keys on behalf of clients
• use private key encryption
• e.g. Kerberos (later)

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Key management schemes

• 3. Public Notary or Certification Authority
• have an off-line server trusted by all clients
• server has a well known public key
• server signs public key certificates for each
  client
• uses public key encryption
• will consider this next

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Public Key Certificates

• public key management generally involves the use
  of public key certificates
• There is a public, well-known, trusted
  Certification Authority (CA), users know CAs
  public key.
• bind an identity (i.e. a user) to a public key
• usually with other info such as period of
  validity, rights of use etc.
• with all contents signed by the CA, called public
  key certificate (PKC)
• Any other user can use CAs public key to verify
  the certificate, thus make sure that the public
  key is an authentic public key for the user.
• CA not know the private keys of users. However it
  is possible for CA (or government) to generate
  private and public keys for users.

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X.509 - Directory Authentication Service

• Widely accepted and used international standard
• defines framework for authentication services
• directory may store public-key certificates
• also defines authentication protocols using these
  certificates
• uses public-key cryptography and digital
  signatures
• RSA is the recommended algorithm.

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X.509 Certificates

• issued by a Certification Authority (CA)
• each certificate contains
• version (1, 2, or 3)
• serial number (unique within CA) identifying
  certificate
• Signature algorithm identifier
• issuer X.500 name (CA)
• period of validity (from - to dates)
• subject X.500 name (name of owner)
• subject public-key info (algorithm, parameters,
  key)
• issuer unique identifier (v2)
• subject unique identifier (v2)
• extension fields (v3)
• signature (of hash of all fields in certificate)

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Certificate Properties

• any user with access to CA can get any
  certificate from it
• only the CA can modify a certificate
• because they cannot be forged, certificates can
  be placed in a public directory

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CA Hierarchy

• if both users share a common CA then are assumed
  to know its public key
• otherwise CA's must form a hierarchy
• use certificates linking members of hierarchy to
  validate other CA's
• each CA has certificates for clients (forward)
  and parent (backward)
• each client trusts parents certificates
• enable verification of any certificate from one
  CA by use of all other CAs in hierarchy

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CA Hierarchy --example

• A acquires B certificate following chain
• XltltWgtgtWltltVgtgtVltltYgtgtYltltZgtgtZltltBgtgt
• B acquires A certificate following chain
• ZltltYgtgtYltltVgtgtVltltWgtgtWltltXgtgtXltltAgtgt
• Notation CAltltUsergtgt means CA has signed
  certificate details for User

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Hash function and Message Digest