FACTORIZATION OF LARGE NUMBERS USING NUMBER FIELD SIEVE: SIEVING STEP

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FACTORIZATION OF LARGE NUMBERS USING NUMBER FIELD
SIEVE SIEVING STEP

• BY
• SUSHMA GUDAVALLI
• TEJA VALURUPALLI

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FACTORIZATION

• Factoring an integer n means finding
• integers x and y such that nx.y
• Prime factorization of a number, written
• as a product of prime numbers is unique.
• Factoring a large number is believed to be
• a very hard problem.

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FACTORIZATION IN CRYPTOGRAPHY

• The security of RSA relies on the difficulty of
• factorization.
• It relies on the fact that it is computationally
• difficult to factor a large integer.

Factorization
Two Prime Factors
Encrypted Message
Product
Message
Message
Encryption
Decryption
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FACTORING METHODS

• General algorithms( usable for RSA)
• Continued Fractions (CFRAC)
• Quadratic Sieve (efficient up to 110 digits)
• Number field Sieve (efficient beyond 110
  digits)

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NUMBER FIELD SIEVE

• The NFS algorithm consists of five steps which
• are mutually dependent
• Polynomial step
• Finding Factor Bases
• Sieving
• Linear Algebra (Matrix step)
• Square root

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NFS ALGORITHM

• Polynomial Step
• Chooses a good polynomial f(x) from a large
  set of
• usable polynomials.
• Factor Bases
• Chooses factor bases for the sieving step.
• RFB ( p0, p0mod m), (p1, p1mod m),
• AFB (p0, r0), (p1, r1),
• Such that f(ri) 0 mod pi

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NFS ALGORITHM (contd)

• Sieving
• The purpose of the sieving step is to find
  usable relations
• (ai, bi)
• Linear algebra
• This step finds combination of elements
  from the relation
• set which has a product that is a square.
• Square root
• The purpose of this step is to find
  rational square root and
• algebraic square root for the solutions
  obtained from the
• matrix step.

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SIEVING

• The task of Sieving step is to find usable
  relations (ai, bi)
• from many possible pairs.
• These relations should have the following three
  properties
• gcd(a,b)1
• abm is smooth over RFB
• bdeg(f) f(a/b) is smooth over AFB
• ( f(x) is the polynomial, m ? f(m) 0 mod n)

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SIEVING (contd..)

• What is meant by being smooth over factor base?
• Example
• RFB (2, 1), (3, 0), (11, 7),
• Let abm 264 264 23. 3. 11
• Thus (a,b) are smooth over RFB
• Similar is the case with AFB

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SIEVING (contd..)

• An initial pair (a, b) which is likely to be a
  relation is found on the sieving line.
• All possible relation pairs on the sieving line
  are found out by adding log2pi ( pi ? RFB) to the
  location of the initial pair (a,b)

For each b fixed
a
Pairs of elements likely to be relations
log2p
log2p
Sieving Line
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SIEVING (contd..)

• The sorted pairs from the Sieving line are
  checked with
• the above mentioned properties. Thus the
  Relations
• are obtained.
• Briefly, the Sieving step involves
• Input RFB, AFB, f(x), m, sieving interval
  (u)
• Output List of Relations (a0, b0),(at,
  bt)

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MAGMA

• Radically new system to solve computationally
  hard problems in algebra, number theory,
  geometry.
• It is both computer algebra system and a
  programming language.
• SPECIALITY
• Provision of mathematical data types such as
  groups,
• rings, fields, sets, sequences, mappings etc.
• Large collection of functions for performing
  standard
• tasks in algebra.

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QUESTIONS ???