You tried setting up public key, passwordless authentication on your Linux box, but sshd keeps asking for a password and ignores all the keys you send. You are probably running RHEL or CentOS and SELinux.

The first thing you might try is fixing the permissions, 700 for ~/.ssh, 600 or 644 for ~/.ssh/authorized_keys.
Still doesn’t work.

You start investigating the issue and in the logs (probably /var/log/secure) you find a message like this one

type=AVC msg=audit(1392479922.440:24765601): avc: denied { read } for pid=13960 comm=\“sshd\” name=\“authorized_keys\” dev=dm-0 ino=786507 scontext=unconfined_u:system_r:sshd_t:s0-s0:c0.c1023 tcontext=unconfined_u:object_r:httpd_sys_content_t:s0 tclass=file

Bingo!
It’s SELinux blocking the system from reading the files.

If you lookup on Google everyone tells you to use restorecon, something like restorecon -R -v ~/.ssh.
You try different combinations of the command, with variuos targets restorecon -R -v ~/.ssh/authorized_keys or
restorecon -R -v ~/, but it doesn’t work, it still asks for a password.

You’ve been trying to fix this issue for hours now, you’re starting to get desperate and maybe a little angry.
I’ll tell you the solution that worked for me:
run ls -laZ ~/.ssh anche check the output, it should be something similar to this

drwx------. user group system_u:object_r:default_t:s0`

Wait, wait, wait!
What’s that -Z?
According to man it’s

-Z, --context
      print any security context of each file

You need to change the security context to system_u:object_r:usr_t:s0.
How?
Just run

chcon -R -v system_u:object_r:usr_t:s0 ~/.ssh/

Et voilat, your public key, passwordless authentication is now online.