To authenticate the receiver $($B$),$ the sender $($A$)$ e$-$mails a challenge message to B$.$ B will use

$($tip: A$\text{'}$s or B$\text{'}$s$)$ private key to encrypt the challenge message and send it to A$.$ If A is able to use

$($tip: A$\text{'}$s or B$\text{'}$s$)$ public key to decrypt and get the plain text of the challenge message, A has authenticated B successfully.