Master in Security of Integrated Systems and Application : Secure your future

Program

Algorithms for Cryptography (28h)

Jean-Claude Bajard and Sylvain Duquesne

Cryptography fundamentals

  1. Introduction to cryptography (7h)
  2. Mathematics for cryptography (21h)
  3. Algorithms for cryptography (28h)
  4. Secret-key cryptography techniques (28h)
  5. Public-key cryptography techniques (28h)

Synopsis

Recent cryptographic protocols are based on modular and finite field arithmetics. It is particularly the case of public key cryptography. Hence, to get efficient implementations, exploiting well adapted algorithms is a critical factor of success. As multiplication is one of the most frequent but also one of the most costly operations, we present different algorithms which are more or less efficient with respect to the operand sizes. Cryptographic sizes can reach huge numbers of thousands binary digits. Those approaches are also available for extended finite fields in polynomial representations. Another aspect we deal with, is the modular reduction. Westudy different cases, from the most general using a division to the NIST recommendations where this operation can be reduced to few additions. To end this overview, we will approach the exponentiation for example in RSA and the inversion needed in some protocols like ECC.

Timetable

(A session amounts to 13/4 hours.)
Session 1
Multiplication: Classical double and add, Karatsuba-Ofman algorithm
Session 2
Practical implementations
Session 3
Multiplication: Toom-Cook Approach, Schonhage-Strassen Algorithm
Session 4
Practical implementations
Sessions 5
Reduction: Integer division, Montgomery reduction
Session 6
Practical implementations
Session 7
Reduction: Barrett reduction, Pseudo Mersenne numbers
Session 8
Practical implementations
Session 9
Finite Field Multiplication: Mastrovito Approach
Session 10
Practical implementations
Sessions 11
Finite Fields Multiplication: Normal Bases, Dual Bases
Session 12
Practical implementations
Session 13
Exponentiation: Square and multiply, sliding windows
Session 14
Practical implementations
Session 15
Finite Fields Inversion: Euclidean algorithm, Fermat approach
Session 16
Practical implementations

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