CIS 5371 Cryptography

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CIS 5371 Cryptography

• 6. An Introduction to Number Theory

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Congruence and Residue classes

• Arithmetic modulo n, Zn
• Solving linear equations
• The Chinese Remainder Theorem
• Eulers phi function
• The theorems of Fermat and Euler
• Quadratic residues
• Legendre Jacobi symbols

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Arithmetic modulo n

• Examples
• Zn 0,1,2, , n-1,
• Zp 1,2, , p-1, for prime p,
• Zn all integers k, 0 lt k lt n, with gcd(k,n)
  1.

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Solving linear equations

• Theorem
• For any integer n gt 1,
• ax ? b (mod n)
• is solvable, if and only if, gcd (a,n) b .
• Examples
• 6x ? 18 (mod 36) has 6 solutions 3, 9, 15, 21,
  27, 33.
• 2x ? 5 (mod 10), has no solutions.

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The Chinese Remainder Theorem
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The Chinese Remainder Theorem
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Example
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Example
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Eulers phi function
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The theorems of Fermat and Euler

• Fermats little theorem
• If p is prime then,
• ap-1 ? 1 (mod p),
• for all integers a 0 lt a lt p.
• Eulers theorem
• If gcd (a,n)1 then a?(n) ? 1 (mod n),

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Legendre Jacobi symbols
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Legendre Jacobi symbols
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Legendre Jacobi symbols
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Example