In The Name of GOD

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In The Name of GOD
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Introduction to Cryptography

• KNT University of Technology
• Department of Basic Science
• Mathematics group
• 16 November
• Mohsen A. Momeni
• Farnaz Farzi
• Mehdi Abbas Zadeh
• Cryptography Association of Science Faculty

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Definition

• Cryptography is the study of mathematical
  techniques related to aspects of information
  security such as confidentiality, data integrity,
  entity authentication, and data origin
  authentication.

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Cryptographic goals

• Confidentiality
• assure that data can be kept secret. Attacker
  can eavesdrop the message.
• method encryption
• Integrity
• assure that received message has not been
  altered. - Attacker can modify the message (100
  Euro to 1000 Euro).
• method digital signature
• Authentication
• about verifying identities. Attacker can send a
  different message to company B.
• method digital certificate

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How is cryptography applied ?

• - Secure Communication (SSL IPSEC)
• - Identification and Authentication (Smart Cards)
• - Secret sharing (VPN)
• - Electronic Commerce (Credit Cards)
• - Remote Access (SSH)

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CRYPTOGRAPHY TERMINOLOGY

• Encryption encoding a message so that its
  meaningis not obvious.
• Decryption the reverse
• Cipher text Encrypted form Cc1,c2,...,cm
  CE(P)
• Plain text Original form of message
  Pp1,p2,...,pn PD(C)
• PD(E(P))
• Alphabet of definition(A) a finite set which all
  text elements are
• chosen from it. i.e. 0,1 or a,b,c,,z
• Secret Key encryption and decryption depends on
  a key.

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Simple Mathematical Approach CHARACTER
REPRESENTATION

• All characters in alphabet are labled by
  mathematical objects.
• These objects can be anything such as number ,
  vector , matrix ,etc.
• i.e. A 0, B 1, ... , Z 25 (space 26)
• Then we define a 1-1 transformation from the
  object set onto itself.
• We can use modular arithmatic for this reason.

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A privacy scheme The Caesar Cipher
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Private Key Scheme
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Private Key disadvantages

• Frequent key changes (stolen, guessed, bought)
• Distribution (through different channels)
• Number of keys rapidly increases with spreading

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Public Key Scheme (Diffie Hellman, 1976)
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The mechanical analogy of Public key
A lock with two very different keys--the key E is
for LOCKING and the key D is for
UNLOCKING. Public-key cryptography is thus
sometimes called two-key cryptography or
asymmetric cryptography.
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Definitions

• One-way function
• Easy to compute f(x) but essentially for all y
  in Im(f), its computationally infeasible to find
  x s.t. f(x)y.
• Trapdoor one-way function
• One-way function with the additional property
  that given some extra information (called
  trapdoor) ,it becomes feasibe to find for any
  given y in Im(f) , an x s.t. f(x)y.
• Permutation
• A 1-1 function from a finite set onto itself.

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Trapdoor one-way permutations based on two
mathematical problems

• Integer Factoring Problem (RSA , Rabin)
• Discrete Logarithm Problem (Elgamal)

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Association Goals

• Learning mathematical background of cryptography.
• Working on Cryptanalysis of different algorithms.
• Researching about new algorithms for different
  problems posed in cryptography.

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Thanks for yourattention