Number Theory and
Advanced Cryptography
3. Quadratic Residues and Rabin
Public key Cryptosystem
Part I: Introduction to Number Theory
Part II: Advanced Cryptography
Chih-Hung Wang
Sept. 2012
1
Quadratic Residues
2
Quadratic Residues
3
Quadratic Residuosity Problem (1)
Proof: see page 188
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Quadratic Residuosity Problem (2)
Important!
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Quadratic Residuosity Problem (3)
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Legendre-Jacobi Symbols
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Jacobi Symbol Properties
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Algorithm for Computing Jacobi
Symbol
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Notes of Jacobi Symbol



n
   0 if m | n
m
Note that the Jacobi symbol is not defined for m  0 or
even.
Testing web-site



http://mathworld.wolfram.com/JacobiSymbol.html
http://www.math.fau.edu/Richman/jacobi.htm
http://wwwmaths.anu.edu.au/DoM/thirdyear/MATH3301/j
acobi.html
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4432  1  (443  1)(443  1)
Jacobi Examples (1)

 444  442  8(111)(221)
 111 221 is an odd number
Jacobi(384,443)=-Jacobi(192,443)
=Jacobi(96,443)
=-Jacobi(48,443)
=Jacobi(24,443)
=-Jacobi(12,443)
=Jacobi(6,443)
=-Jacobi(3,443)
 1) 4(221)

 221 (odd number) ]
=Jacobi(2,3) [ (3 1)(443
4
4
=-Jacobi(1,3) [(32-1)/8=1 odd number]
=-1
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Jacobi Examples (2)











Jacobi(1368/24571)
= -Jacobi(684/24571)
= Jacobi(342/24571)
= -Jacobi(171/24571)
= Jacobi(118/171)
= -Jacobi(59/171)
= Jacobi(53/59)
=Jacobi(6/53)
= -Jacobi (3/53)
=-Jacobi(2/3)
=Jacobi (1/3) = 1
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Square Root Modulo Integer(1)

Modulo prime Algorithm
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Square Root Modulo Integer(2)

Modulo prime in General case
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Square Root Modulo Integer(3)
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Square Root Modulo Integer(4)
Odd number
1  x
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Square Root Modulo Integer(3)

Modulo composite
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Square Root Modulo Integer(4)

Properties of modulo composite
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Example (1)
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Example (2)
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Factoring Problem

For a large n with large prime factors, factoring is a
hard problem, but not as hard as it used to be.

Example: factorize 48770428682337401 => hard problem


1977: three inventors of RSA issue “Mathematical Games”


Easy problem:
Is 223092871 a factor of 48770428682337401?
$100 reward
1994: RSA-129 (428 bits) breaking
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Progress of Factorization (1)
Number of
Decimal
Digits
Approximate
Number of
bits
Date
Achieved
MIPSYears
Algorithms
100
332
April 1991
7
Quadratic sieve
110
365
April 1992
75
Quadratic sieve
120
398
June 1993
830
Quadratic sieve
129
428
April 1994
5000
Quadratic sieve
130
431
April 1996
500
Generalized
number field sieve
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Progress of Factorization (2)
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Progress of Factorization (3)
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Blum Integers
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Properties of Blum Integers (1)
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Properties of Blum Integers (2)
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Rabin Encryption Scheme (1)
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Rabin Encryption Scheme (1)
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Example of Rabin
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Insecurity of Rabin
CPA (Chosen-plaintext attack)
CCA (Chosen-ciphertext attack)
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