Number Theory and Advanced Cryptography 10' Modular Exponentiation

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Number Theory and Advanced Cryptography 10.
Modular Exponentiation
Part I Introduction to Number Theory Part II
Advanced Cryptography

• Chih-Hung Wang
• Feb. 2008

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Reference

• High-Speed RSA Implementation
• Cetin Kaya Koc
• RSA Lab. RSA Data Security, Inc.
• Nov. 1994

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Modular Exponentiation
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Binary Method (1)
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Binary Method (2)

• Example

e 250 (11111010)
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m-ary Method (1)
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m-ary Method (2)
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m-ary Method (3)

• Quaternary Example

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Adaptive m-ary Method (1)

• Reducing preprocessing multiplications

6 multiplications ? m-array needs 14
multiplications
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Adaptive m-ary Method (2)

• Sliding window techniques

dmax(L(Fi))
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Constant Length Nonzero Windows (1)
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Constant Length Nonzero Windows (2)
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Variable Length Nonzero Windows (1)
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Variable Length Nonzero Windows (2)
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Addition Chains (1)
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Addition Chains (2)
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Recoding Methods (1)
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Recoding Methods (2)
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Recoding Methods (3)
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Booth Algorithm (1)
Worst case
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Booth Algorithm (2)

• Modified schemes

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The Canonical Recoding Algorithm (1)
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The Canonical Recoding Algorithm (2)
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Recoding and Cryptography
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Modular Multiplication

• Multiply and then Reduce
• Blakleys method
• Montgomerys method

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Standard Multiplication Algorithm (1)
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Standard Multiplication Algorithm (2)
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Standard Multiplication Algorithm (3)
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Standard Multiplication Algorithm (4)
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Computation of the Remainder (1)

• Restoring Division Algorithm
• Division is the most complex of the four basic
  arithmetic operations.

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Computation of the Remainder (2)

• Example 3019 mod 53
• 3019(101111001011)2 53(110101)2

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Computation of the Remainder (3)

• Nonrestoring Division Algorithm

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Computation of the Remainder (4)

• Example 513019 mod 53

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Blakleys Method (1)

• G. R. Blakley. A computer algorithm for the
  product AB modulo M. IEEE Transactions on
  Computers, 32(5)497500, May 1983.

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Blakleys Method (2)
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Blakleys Method (3)
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Montgomerys Method (1)
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Montgomerys Method (2)
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Montgomerys Method (3)
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Montgomerys Method (4)
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Montgomerys Method (5)
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Montgomery Exponentiation (1)

• The exponentiation algorithm uses the binary
  method

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Montgomery Exponentiation (2)

• Example x710 mod 13

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Montgomery Exponentiation (3)
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Montgomery Exponentiation (4)