CS8792 – Cryptography and Network Security Unit #1

1. CS8792 – CRYPTOGRAPHY AND
NETWORK SECURITY
UNIT 2 – EUCLID ALGORITHM AND
CONGRUENT MATRIX
~ S. Janani, AP/CSE
KCET

2. Contents
• Euclid algorithm
• Matrix - Definition
• Operations and relations
• Determinants
• Residue Matrix
• Congruent Matrix

3. Prerequisites
• GCD & Relative prime
• Why Matrix
• Congruent
• Congruent function – applications
• Inverse of a matrix

4. GCD & Relative prime

5. Why Matrix
• Matrices are very useful in engineering calculations.
• For example, matrices are used to:
– Efficiently store a large number of values (as we
have done with arrays in MATLAB)
– Solve systems of linear simultaneous equations
– Transform quantities from one coordinate system
to another

6. Congruent

7. Congruent Application
• To find whether the given number is odd or even
• To find the day of the particular date

8. Inverse of a matrix
• https://www.khanacademy.org/math/algebra-home/alg-
matrices/alg-determinants-and-inverses-of-large-
matrices/v/inverting-3x3-part-1-calculating-matrix-of-
minors-and-cofactor-matrix

9. Euclid Algorithm
• To find gcd of two integers
• gcd (a,b) = gcd(b,a mod b)

10. Extended Euclidean Algorithm
 calculates not only GCD but x & y:
ax + by = d = gcd(a, b)
 useful for later crypto computations
 follow sequence of divisions for GCD but
assume at each step i, can find x &y:
r = ax + by
 at end find GCD value and also x & y
 if GCD(a,b)=1 these values are inverses

11. Finding Inverses
EXTENDED EUCLID(m, b)
1. (A1, A2, A3)=(1, 0, m);
(B1, B2, B3)=(0, 1, b)
2. if B3 = 0
return A3 = gcd(m, b); no inverse
3. if B3 = 1
return B3 = gcd(m, b); B2 = b–1 mod m
4. Q = A3 div B3
5. (T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3)
6. (A1, A2, A3)=(B1, B2, B3)
7. (B1, B2, B3)=(T1, T2, T3)
8. goto 2

12. Inverse of 550 in GF(1759)
Q A1 A2 A3 B1 B2 B3
— 1 0 1759 0 1 550
3 0 1 550 1 –3 109
5 1 –3 109 –5 16 5
21 –5 16 5 106 –339 4
1 106 –339 4 –111 355 1
355 is inverse of 550

13. Matrix - Definition

14. Operations and Relations

15. Matrix Multiplication

16. Scalar Multiplication

17. Determinant of a Matrix

18. Residue Matrix

19. Congruent Matrix
• In mathematics, two square
matrices A and B over a field are
called congruent if there exists
an invertible matrix (Inverse Matrix) P over
the same field such that
• PTAP = B

20. Linear Congruence

21. Single variable linear equation

24. Set of Linear Equations