Agrawal Saxena Kayal Algorithm for Primality Testing in Maple

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Agrawal Saxena - Kayal Algorithm for Primality
Testing in Maple

• Nathan S. Miller
• Oregon State University
• Data Security and Cryptography - ECE 575

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Prime Numbers

• Numbers which are only divisible by 1 and itself
• Key relationship between Cryptography and Number
  Theory
• The Infinitude of Primes (Euclid)

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Primality Testing Methods

• Sieve of Eratosthenes
• Infinitude of Primes (Euclid)
• Deterministic and slow
• Other tests include
• Baillie-PSW Primality Test
• Lucas Lehmer Test
• Miller-Rabin Primality Test
• Elliptic Curve Primality Proving
• Miller-Rabin Primality Test
• False primes (small probability)
• Used on math programs (Mathematica and Maple)
• Elliptic Curve Primality Proving
• Time consuming 2000 hours on 1GHz processor for
  4769-digit prime number

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Agrawal Saxena - Kayal

• Professor and two undergraduate students
• Indian Institute of Technology in Kanpur, India
• Polynomial time algorithm (major breakthrough)
• PRIMES in P AKS 2002
• August 6, 2002
• Deterministic polynomial time algorithm that
  determines whether an input number n is prime or
  composite

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PRIMES in P

• Rooted in Complexity Theory
• Attempts to quantify the difficulty of
  computational tasks
• Measured by resources (i.e. memory needed,
  computational bandwidth, time of execution, et
  cetera)
• P is a class of decisional problems (yes or no)
  for which any instance of the problem can be
  solved by a deterministic algorithm in a time
  which can be bounded by a polynomial on the size
  of the input of the problem

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PRIMES in P continued

• PRIMES is the decisional problem of determining
  whether or not a given integer n is prime.
• What is the result of this paper?
• A proof that the problem of deciding whether or
  not an integer is prime can be solved by a
  deterministic algorithm in time bounded by a
  polynomial in the size of the input, without
  using any unproven mathematical assumptions.
• 1976 Miller comes up with such an algorithm based
  on Extended Reimann Hypothesis (not yet proven)
  which says, the first quadratic nonresidue mod p
  of a number is always less than 2(ln p)2.

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How does it work?

• Theorem Suppose a and p are relatively prime
  with pgt1. p is prime if and only if

• If p is prime, then p divides the binomial
  coefficients pCr for r1, 2, , p-1. This shows
  that (x-a)p xp-ap mod p and the equation given
  above follows given Fermats Little Theorem.
• If p is composite, then it has a prime divisor q
  . Let qk be the greatest power of q that divides
  p. Then qk does not divide pCr and would be
  relatively prime to ap-q, so the coefficients of
  the LHS are not zero, but they are on the RHS.

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The Simplification

• Too many coefficients so ASK sought to prove a
  simpler condition (x-a)p xp-a mod (xr-1, p).
  ASK proved that this holds true if p is prime and
  if rgt1 does not divide p.

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Demonstration
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References

• Agrawal, M., Saxena, N. Kayal, N. PRIMES in P
  August 6, 2002.
• Stiglic, A., PRIMES is in P little FAQ,
  http//www.cse.iitk.ac.in/news/primality.html