Exploring the newer GUI front-end, for use with SageMath.

One of the subjects which I had written about only yesterday, is that the Computer Algebra / Numerical Tool System called ‘SageMath‘ was available in the repositories, for Debian / Stretch – which is in itself news – and that additionally, the default way to use it under Debian is through a Web-interface called ‘SageNB’. Well what I’ve now learned is that the SageMath developers no longer support SageNB, and are continuing their work with the graphical front-end called ‘Jupyter‘.

But, installing Jupyter under Debian is a bit of a chore, because unlike how it is with custom-compiles, Debian package maintainers tend to break major software down into little bits and pieces. At one point, I had Jupyter running, but with no awareness of the existence of SageMath. What finally did the trick for me today, was to install the following packages:

  • python-notebook
  • jupyter-nbextension-jupyter-js-widgets
  • sage-math-jupyter

Needless to say, that last package out of the three is the most important, and may even pull in enough of the other packages, to be selected by itself. It’s just that I did not know immediately, to install that last package.

So this is what SageMath 7.4 looks like, through Jupyter:


(Corrected 09/18/2018, 3h50 … )

(Updated 09/18/2018, 5h40 … )

(As of 09/16/2018, 20h10 : )

Frankly, I was a bit disappointed at first. My main disappointment seemed to be with the fact, that this GUI did not offer to typeset the Math. It does allow us to ‘download’ our Notebooks as PDF-Files, but when we do, we simply get the same, highlighted text, and graphics, only as a PDF – in code – or with whatever appearance the browser-view is already showing us. Also, the support for 3D plots is lackluster, as the plot above is non-interactive. At least with SageNB, I was able to select the ‘canvas3d’ viewer, which allowed the plot to be rotated. Also, if we use SageMath from the command-line, it defaults to using ‘JMol’ as its viewer, which is full-featured.

But as it turns out, I have discovered ‘the trick’, to getting Jupyter to typeset the users’ Math…

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Integrating Sage with LyX

I have been interested in the LaTeX typesetting system, but not in actually learning the extra syntax. And so I’ve been using a WYSIWYM GUI named ‘LyX‘. The .TEX-Files LyX exports are suitable to creating HTML Files, which in turn can contain typeset Math, by way of scripts, that provide MathJax code.

But then one compromise this has meant for me was, that I could either typeset Math which I had written myself, or that I could do Computer Algebra, the latter through the use of ‘wxMaxima’, which has its own system of exporting to HTML. But so far, I could not do both in one document.

Well, now that I have ‘SageMath’ and ‘SageTeX’ installed, I can do both within the same document. It’s possible to integrate SageTeX into LyX. The following is an article which explains how to do that, under the assumption that the user has both SageTeX and LyX installed:


There is an important way in which I needed to modify the instructions however, to get the two working. First of all, there is a list of files to download – LyX does not support SageTeX out of the box – that the article above links to. Out of those files, ‘setup.sh’ is absolutely useless on a modern Linux computer. The following files from the repository above are essential:

  1. sage.module
  2. preferences
  3. compile-pdf-sage.sh
  4. example.lyx

The file ‘preferences’ needs to be edited, in that its last line needs to be uncommented.

The file ‘compile-pdf-sage.sh’ needs to be edited, in that the first two lines need to be commented out, and the next three lines which are commented by default, need to be uncommented. Then:


(As user:)

$ cp sage.module ~/.lyx/layouts
$ cat preferences >> ~/.lyx/preferences

(As root:)

# cp compile-pdf-sage.sh /usr/local/bin
# chmod a+x /usr/local/bin/compile-pdf-sage.sh


Then, within the GUI of LyX, one gives the command ‘Tools -> Reconfigure’. One shuts down and restarts LyX. Next, one opens ‘example.lyx’ to test the setup. The following is the document which I finally obtained (successfully), which was provided by the Web-site above (not written by me):





I just installed Sage (Math) under Debian / Stretch.

One of the mundane limitations which I’ve faced in past years, when installing Computer Algebra Systems etc., under Linux, that were supposed to be open-source, was that the only game in town – almost – was either ‘Maxima’ or ‘wxMaxima’, the latter of which is a fancy GUI, as well as a document exporter, for the former.

Well one fact which the rest of the computing world has known about for some time, but which I am newly finding for myself, is that software exists called ‘SageMath‘. Under Debian / Stretch, this is ‘straightforward’ to install, just by installing the meta-package from the standard repositories, named ‘sagemath’. If the reader also wants to install this, then I recommend also installing ‘sagemath-doc-en’ as well as ‘sagetex’ and ‘sagetex-doc’. Doing this will literally pull in hundreds of actual packages, so it should only be done on a strong machine, with a fast Internet connection! But once this has been done, the result will be enjoyable:


I have just clicked around a little bit, in the SageMath Notebook viewer, which is browser-based, and which I’m sure only provides a skeletal front-end to the actual software. But there is a feature which I already like: When the user wishes to Print his or her Worksheet, doing so from the browser just opens a secondary browser-window, from which we may ‘Save Page As…’ , and when we do, we discover that the HTML which gets saved, has its own, internal ‘MathJax‘ server. What this seems to suggest at first glance, is that the equations will display typeset correctly, without depending on an external CDN. Yay!

I look forward to getting more use out of this in the near future.

(Update 09/15/2018, 21h30 : )

Continue reading I just installed Sage (Math) under Debian / Stretch.

Finding the Purpose of Multiplicative Groups, within a Modulus

There exists a WiKiPedia article, which defines what a Multiplicative Group of Integers is, Modulo n. But even though the article is about 10 pages long (in my browser), it fails to explain in a common-sense way – at least that I can find at a glance – why the existence of these groups is important on a practical level.

I would paraphrase that the Multiplicative Group, consists of Integers within the Modulus, which have Multiplicative Inverses, and which are therefore also Coprime with the Modulus. While these two properties are equivalent, they are not in fact exactly the same property. But why is this combination of properties finally important, let’s say If we wanted to implement an encryption scheme, in which a base – representing a message – is to be raised to a 2048-bit long exponent, and if to do so a second time, is to reproduce the original message?

Essentially, we can multiply one integer by another, assuming they are both in the same modulus, and arrive at some numerical result when applying the remainder function on the product, even though one of the original integers was not coprime with the modulus. This is trivial, because of the way the remainder function has been defined. In other words, we could compute this:

(2 * 5) mod 10 = 0

There is a sticking point with this operation. Once the value zero has been computed, any further multiplication with zero, yield zero again. There is no magical reason why this would not be the case with modular multiplication. But there was once a (2) and a (5), which will never be recoverable in any specific way, from ( 0 mod 10 ). Not only that, but this loss of information does not take place all at once; this happens cumulatively. We could start with (7), which is coprime with the modulus, we could multiply it once by (2), and we could then multiply it again by (5), and again, we’d get zero:

C1 = (7 * 2) mod 10 = 4

T1 = (C1 * 5) mod 10 = 0 !

We can also observe, that just to perform a multiplication in which one parameter was not coprime, results in a product which is not so – in the above case, in ( 4 mod 10 ). Similarly:

C2 = (7 * 5) mod 10 = 5  (No obvious problem yet.)

T2 = (C2 * 2) mod 10 = 0  (Problem.)

Well, when we are exponentiating a number, we are essentially performing many multiplications on it. And then the cumulative result can easily be, that we cannot reproduce the original message, or even, that the message becomes zero !

Continue reading Finding the Purpose of Multiplicative Groups, within a Modulus