Introduction to Cryptography with Coding Theory, 3rd Edition

B.4 Examples for Chapter 3

Example 5

Find $gcd(23456\text{,}987654)\text{.}$

> gcd(23456, 987654)

                         2

Example 6

Solve $23456x+987654y=2$ in integers $x\text{,}y\text{.}$

igcdex(23456, 987654,’x’,’y’)

                              2

> x;y

                            -3158
                              75

This means that 2 is the gcd and $23456\cdot (-3158)+987654\cdot 75=2\text{.}$ (The command igcdex is for integer gcd extended. Maple also calculates gcd’s for polynomials.) Variable names other than ’x’ and ’y’ can be used if these letters are going to be used elsewhere, for example, in a polynomial. We can also clear the value of x as follows:

> x:=’x’

                   x:=x

Example 7

Compute $234\cdot 456(\text{mod}789)\text{.}$

> 234*456 mod 789

                                   189

Example 8

Compute ${234567}^{876543}(\text{mod}565656565)\text{.}$

> 234567&$\stackrel{\mathbf{ˆ}}{\phantom{\mathbf{a}}}$876543 mod 565656565

                            473011223

You might need a $\mathrm{\setminus }$ before the $\hat{\phantom{a}}\text{.}$ Use the right ...