You want to use Linux/Unix and openSSH to automate your jobs and tasks. So you need automatic logins that don’t need to type your password when you connect from A host to B host.
Step 1:Check .ssh file
Ssh file is located under the user home directory.
 

#ls  -ld  .ssh
ls: .ssh: No such file or directory
#pwd
home/testA

Step 2:Create Rsa-Dsa file.

DSA is faster in signing but slower verifying.DSA signature is smaller than RSA key which is the same size. An RSA 512 bits can be cracked also a 280 DSA. When creating an RSA  or DSA key file if you don't need to specify any option press "Enter" for all questions.
 

#ssh-keygen -t rsa
Generating public/private rsa key pair.
Enter file in which to save the key (/home/testA/.ssh/id_rsa):
Created directory ‘/home/testA/.ssh’.
Enter passphrase (empty for no passphrase):
Enter same passphrase again:
Your identification has been saved in /home/testA/.ssh/id_rsa.
Your public key has been saved in /home/testA/.ssh/id_rsa.pub.
The key fingerprint is:
f6:fb:8f:ac:6a:38:7c:aa:1d:5e:69:c6:dc:70:67:be testA@Aserver[/shell]
This command will generate two files which name id_rsa.pub and id_rsa.
[shell]#ls -ld .ssh/id_rsa*
-rw——- 1 testA testA 1675 Nov 24 05:11 .ssh/id_rsa
-rw-r–r– 1 testA testA 396 Nov 24 05:11 .ssh/id_rsa.pub

Step 3:Create  .ssh directory at  B Server

Before copy, id_rsa.pub content to destination server run the same ssh-keygen command on destination B server.
Destination B server

#ssh-keygen -t rsa
Generating public/private rsa key pair.
Enter file in which to save the key (/home/testB/.ssh/id_rsa):
Created directory ‘/home/testB/.ssh’.
Enter passphrase (empty for no passphrase):
Enter same passphrase again:
Your identification has been saved in /home/testB/.ssh/id_rsa.
Your public key has been saved in /home/testB/.ssh/id_rsa.pub.
The key fingerprint is:
aa:a1:86:9a:bb:97:63:32:88:cd:69:54:a0:b6:23:0d testB@Bserver[/shell]
[shell]#ls -ld .ssh/*
-rw——- 1 testB testB 1675 Nov 24 05:21 .ssh/id_rsa
-rw-r–r– 1 testB testB 396 Nov 24 05:21 .ssh/id_rsa.pub[shell]
#pwd
/home/testB

Step 4:Copy id_rsa.pub content to authorized_key 

Run this command on the source (A) server.
 

# cat ~/.ssh/id_rsa.pub | ssh testB@serverB 'cat >> ~/.ssh/authorized_keys'
The authenticity of host ‘localhost (127.0.0.1)’ can’t be established.
RSA key fingerprint is 54:ab:52:fd:ff:7d:37:b7:75:73:2d:2f:5a:d8:5f:8d.
Are you sure you want to continue connecting (yes/no)? yes //input yes for only one time.
Warning: Permanently added ‘localhost’ (RSA) to the list of known hosts.
testB@localhost’s password: //input your password for only one time
#ssh testB@Bserver

TroubleShooting

Step 1:Check Permissions at both servers(A, B)

#[testA@Aserver ~]$ ls -ld .ssh/*
-rw-r–r– 1 testA testA 396 Nov 24 05:30 .ssh/authorized_keys
-rw——- 1 testA testA 1675 Nov 24 05:11 .ssh/id_rsa
-rw-r–r– 1 testA testA 396 Nov 24 05:11 .ssh/id_rsa.pub
-rw-r–r– 1 testA testA 792 Nov 24 05:27 .ssh/known_hosts
[testA@Aserver ~]$ ls -ld .ssh/
drwx—— 2 testA testA 4096 Nov 24 05:30 .ssh/

Step 1:Check sshd file for authorized_keys

#cat /etc/ssh/sshd_config |grep AuthorizedKeysFile
AuthorizedKeysFile .ssh/authorized_keys
#service sshd restart

If authorized_keys configuration is not enabled please remove “#” then restart sshd service for  Linux.