"error: gnutls_handshake() failed" when connecting to https servers

This is mentioned in one of the comments to another answer but I feel it needs to be more visible https://askubuntu.com/users/177551/paul-nelson-baker has written a shell script which, for me, resolved this issue.

The script is available in his repository https://github.com/paul-nelson-baker/git-openssl-shellscript and re-compiles git with openssl instead of gnutls. Read the README for the whole story but it worked for me so thank you Paul.

For me, it ended up being that SSL certificate was self-signed. Give this a try

git config --global http.sslVerify false

Got reason of the problem, it was gnutls package. It's working weird behind a proxy. But openssl is working fine even in weak network. So workaround is that we should compile git with openssl. To do this, run the following commands:

sudo apt-get update
sudo apt-get install build-essential fakeroot dpkg-dev
sudo apt-get build-dep git
mkdir ~/git-openssl
cd ~/git-openssl
apt-get source git
dpkg-source -x git_1.7.9.5-1.dsc
cd git-1.7.9.5

(Remember to replace 1.7.9.5 with the actual version of git in your system.)

Then, edit debian/control file (run the command: gksu gedit debian/control) and replace all instances of libcurl4-gnutls-dev with libcurl4-openssl-dev.

Then build the package (if it's failing on test, you can remove the line TEST=test from the file debian/rules):

sudo apt-get install libcurl4-openssl-dev
sudo dpkg-buildpackage -rfakeroot -b

Install new package:

i386: sudo dpkg -i ../git_1.7.9.5-1_i386.deb

x86_64: sudo dpkg -i ../git_1.7.9.5-1_amd64.deb

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Inspired from here: https://github.com/xmendez/wfuzz/wiki/PyCurlSSLBug