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  • pg 1
									One-way and trapdoor functions

               Based on the
      Lecture Notes on Cryptography
      Shafi Goldwasser, Mihir Bellare
• easy to compute and hard to invert.
• passwords.
• Namely, in a time-shared computer system,
  instead of storing a table of login passwords, one
  can store, for each password w, the value f(w).
• Passwords can easily be checked for correctness
  at login, but even the system administrator can
  not deduce any user's password by examining the
  stored table.
   One-Way Functions: Motivation
• In secure encryption schemes, the legitimate user is
  able to decipher the messages (using some private
  information available to him), yet for an adversary (not
  having this private information) the task of decrypting
  the ciphertext (i.e., \breaking" the encryption) should
  be infeasible.
• the breaking task can be performed by a non-
  deterministic polynomial-time machine.
• the security requirement states that breaking should
  not be feasible, namely could not be performed by a
  probabilistic polynomial-time machine.
• the existence of secure encryption schemes
  implies the existence of an efficient way (i.e.
  probabilistic polynomial-time algorithm) of
  generating instances with corresponding auxiliary
  input so that
1. it is easy to solve these instances given the
   auxiliary input; and
2. it is hard on the average to solve these instances
   (when not given the auxiliary input).
  One-Way Functions: Definitions
• The most basic primitive for cryptographic
• a function which is “easy" to compute but
  “hard" to invert.
• any probabilistic polynomial time (PPT)
  algorithm attempting to invert the one-way
  function on a element in its range, will
  succeed with no more than negligible"
          One-Way Functions
• strong one-way function (or just one-way
• Weak one-way functions
• non-uniform one-way functions.
• Example: f: ZxZ  Z where f(x,y) = x.y
 Collections Of One Way Functions
• Instead of talking about a single function f
• it is often convenient to talk about collections
  of functions, each dened over some nite
  domain and nite ranges.
              Trapdoor Functions
• a trapdoor function f is a one-way function with an extra
• The trapdoor is the secret inverse function that allows its
  possessor to eciently invert f at any point in the domain of
  his choosing
• It should be easy to compute f on any point, but infeasible
  to invert f on any point without knowledge of the inverse
  function .
• it should be easy to generate matched pairs of f's and
  corresponding trapdoor.
• Once a matched pair is generated, the publication of f
  should not reveal anything about how to compute its
  inverse on any point.
             Number theory
• provides a source of candidates for one way
  and trapdoor functions.

• Finding a generator of cyclic group
• The Legendre Symbol
    The Discrete Logarithm Function
•                         has a unique inverse
    can easily be computed in polynomial time

• it is unknown whether or not its inverse DL
  can be computed in polynomial time
• Index-calculus algorithm.
• the number eld sieve algorithm
             The RSA function
• 1977 Rivest, Shamir, and Adleman

• n is the product of two large primes p and q

• The corresponding trapdoor information is d such
• RSA is easy to compute.
• the best way known to invert RSA is to factor n.
               Attacks on RSA
• Dixon's random squares algorithms
• Elliptic Curve Algorithm
  • l = |p| where p is the smallest prime divisor of n.
• Quadratic Sieve algorithm
• new number field sieve algorithm
• The recommended size for n these days is
  1024 bits.
Connection Between The Factorization
     Problem And Inverting RSA
• If some PPT algorithm A can factor n then
  there exists a PPT A’ that can invert RSA(n,e)

• The trapdoor information d can be found by
  using the extended Euclidean algorithm
    The Squaring Trapdoor Function
          Candidate by Rabin
• n = pq where p and q are distinct odd primes.

• The trapdoor information of n = pq is gcd(p,q).
• Computing square roots modulo n is as hard
  as factoring n.
     Topics for practical seminars
• Skills
   – Reading and understanding
   – Using software professionally
   – Presentation
   – Documentation
• Dates of discussing
   – The last 5 days of the course (begin in Nov 2007).
   – Slides are distributed 7 days before the
      1. GPG (GNU Privacy Guard)
•   Free cryptographic software
•   www.gnupg.org
•   www.gpg4win.org
•   www.glump.net/dokuwiki/gpg/gpg_intro
•   1 group of 4 students
 2. Open SSH (OpenBSD Secure Shell)
• a set of computer programs providing
  encrypted communication sessions over a
  computer network using the ssh protocol.
• www.openssh.org
• www.openssh.org/windows.html
• Focus on PuTTY and FileZilla
• 1 group of 2 students
                 3. Cryptool
• a free software illustrating the cryptographic
• www.cryptool.org
• 3 groups
  – Classical cryptology (4)
  – Modern Symmetric and asymmetric encryption (3)
  – Digital Signature, Hash function and random
    generator (4)

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