There are a few problems with your sentence. The colon is only one.
Here, I would like to summarize the reasons for my extension
request : as necessity of additional time for rectifying the
algorithms, lengthy literature survey than expected and etc.
The first problem is that there is no space before the colon, only after the colon.
The second is that the list of reasons is likely to be too long, which will make the sentence difficult to read and understand. I'd recommend that you use a different kind of list. For example:
I have four [However many there are] reasons for requesting an extension. First, I need more time to rectify the algorithms. Second, reviewing the literature is taking much longer than I expected. Third, yada yada yada. Fourth, blah blah blah.
You can also choose to use a bulleted list instead:
I have four reasons for requesting an extension:
- I need more time to rectify the algorithms.
- reviewing the literature is taking much longer than I expected.
- Yada yada yada.
- Blah blah blah.