2

For some problems, even longer ones, I've been able to see the pattern and properly do back substitution to bring a series of equations I've derived using the Euclidean algorithm to the form of Bezout's theorem:

$sa+tm$

Where $s$ and $t$ are parameters.

But, on some problems I get stuck and have no idea how to move forward.

For example, starting from finding the $gcd(3454,4666)$:

Using the Euclidean Algorithm I find:

$4666 = 3454 * 1 + 1212$ ------------- $1212 = 4666 - 3454 * 1$

$3454 = 1212 * 2 + 1030$ ---------------- $1030 = 3454 - 1212 * 2$

$1212 = 1030 * 1 + 182$ ----------------- $182 = 1212 - 1030 *1$

$1030 = 182 * 5 + 120$ ------------------ $120 = 1030 - 182 * 5$

$182 = 120 * 1 + 62$ --------------------- $62 = 182 - 120 * 1$

$120 = 62 * 1 + 58$ ---------------------- $58 = 120 - 62*1$

$62 = 58 * 1 + 4$ ------------------------ $4 = 62 - 58 * 1$

$58 = 4 * 14 + 2$ ------------------------ $2 = 58 - 4 * 14$

For my first step I substitute for the $4$ :

$2 = 58 - (62 - 58) * 14$

Where do I go from here? What are some general strategies to solve problems of this form? I'm having an inordinately hard time with some of these problems, but find others trivial--what is going on? What should I look out for when approaching problems of this type?

If you would like me to clarify content, please ask me such and I will edit accordingly. Thank you for taking the time to read this!

cafeTechne
  • 91
  • 1
    It's best to aboid back-substitution: see https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Example – Angina Seng Apr 09 '19 at 08:05
  • This is the way we are being forced to structure the problem by the Professor because we are going to use this for basic RSA (and Chinese remainder problems?) I'm obviously really struggling with this chapter in Rosen, Discrete Mathematics and it's Applications. – cafeTechne Apr 09 '19 at 08:55
  • 3
    I strongly recommend that you use this version of the Extended Euclidean algorithm, which is simpler and far less error-prone than the common back-substitution version. For an example see here. – Bill Dubuque Apr 09 '19 at 15:09
  • 2
    And if you need to do this frequently by hand you may wish to learn the quicker fractional form. – Bill Dubuque Apr 09 '19 at 15:20
  • Just clarifying, is Bezout's process just another way of saying Euclid's algorithm? – GSmith May 27 '24 at 00:09
  • 1
    @GSmith the coefficients are found using the extended Euclidean algorithm. Here's more: https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity – cafeTechne May 28 '24 at 03:21
  • @cafeTechne My professor calls Euclid's algorithm "Bezout's Process". I do understand the technique, but I simply want to clarify the name. I'm supposing he calls it that because it is a way of finding the coefficients in Bezout's identity? – GSmith May 28 '24 at 07:48

2 Answers2

3

It is usually simpler and far less error prone to compute the Bezout identity in the forward direction by using this version of the Extended Euclidean algorithm, which keeps track of each remainder's expression as a linear combination of the gcd arguments. Below is the computation in your example - so simple that it can be done purely mentally in a few minutes. Here we use least magnitude remainders to speed it up, e.g. $\bmod 1212\!:\,\ 3454\equiv 1030\equiv -182$.

$$\rm\begin{eqnarray} [\![0]\!]\quad \color{}{4666}\ &=&\,\ \ \ 1&\cdot& 4666\, +\ 0&\cdot& 3454 \\ [\![1]\!]\quad \color{}{3454}\ &=&\,\ \ \ 0&\cdot& 4666\, +\ 1&\cdot& 3454 \\ \color{}{[\![0]\!]\ -\,\ [\![1]\!]}\, \rightarrow\, [\![2]\!]\quad \color{}{1212}\ &=&\,\ \ \ 1&\cdot& 4666\, -\ 1&\cdot& 3454 \\ \color{}{[\![1]\!]-3\,[\![2]\!]}\,\rightarrow\,[\![3]\!]\ \ \ \color{}{{-}182}\ &=&\, {-}3&\cdot& 4666\, +\, 4&\cdot& 3454 \\ \color{}{[\![2]\!]+7\,[\![3]\!]}\,\rightarrow\,[\![4]\!]\ \ \ \ \ \color{}{{-}62}\ &=& {-}20&\cdot& 4666\, +\color{}{27}&\cdot& \color{}{3454}\\ \color{}{[\![3]\!]-3\,[\![4]\!]}\,\rightarrow\,[\![5]\!]\qquad\ \ \color{}{4}\ &=&\, \ \ 57&\cdot& 4666\, -77&\cdot& 3454 \\ \color{}{[\![4]\!]\!+\!15[\![5]\!]}\,\rightarrow\,[\![6]\!]\quad\ \ \, \color{}{{-}2}\ &=&\ \ 835&\cdot& 4666\, {-}1128&\cdot& 3454 \\ \end{eqnarray}\qquad$$

Negating the final equation yields the Bezout equation for the gcd $= 2$.

As an optimization we can omit one of the RHS columns, it being computable from the other, e.g. $1128 = ((835\cdot 4666)+2)/3454$. Then the equations may be viewed in fractional form. But it is best to master the above explicit equational form before proceeding to the optimizations.

Bill Dubuque
  • 282,220
1

1) Look ahead the at the next two terms ($r$) that you will be substituting into to try and plan ahead.

2) Distribute the scalar.

3) Then, put everything in terms of addition and express the parameters as negatives--remember to make use of the associate property.

4) Sanity check from time to time to make sure you didn't make a sign error. To do this, simply use a calculator to make sure the right hand side of your equation equals the left hand side.

cafeTechne
  • 91
  • Substitute for the remainders. \begin{align} 2 & = 58 - 14 \cdot 4\ & = 58 - 14(62 - 58)\ & = 15 \cdot 58 - 14 \cdot 62\ & = 15(120 - 62) - 14 \cdot 62\ & =~...\end{align} If you are not comfortable with steps such as $58 - 14(62 - 58) = 15 \cdot 58 - 14 \cdot 62$, add the step $58 - 14 \cdot 62 - 14 \cdot 58$, then simplify. – N. F. Taussig Apr 09 '19 at 10:52
  • 1
    Thanks for this! I figured it out before I saw this unfortunately, but I hope our combined efforts will help someone else who has a similar issue save time on the front end! In the end, it really just came down to spending hours working through it until seeing these patterns became second nature. – cafeTechne Apr 10 '19 at 00:43