The following script generates a compressed encrypted backup given a "fileinput" with a big and pretty much random symmetric secret key; then, this unique key will be encrypted using a asymmetric public key given a "recipient".

asymmetric-crypto-backup.sh

#!/usr/bin/env bash
#
# ASYMMETRIC CRYPTO BACKUP
#
# The following script generates a compressed encrypted backup given a "fileinput"
# with a big and pretty much random symmetric secret key; then, this unique key
# will be encrypted using an asymmetric public key given a "recipient".
#
# To decrypt an asymmetric crypto backup do:
#
# 1. Decrypt and print the symmetric key using the correspondant key to "recipient".
#    gpg -d example_key.txt.gpg
#
# 2. Use the symmetric key printed in screen to decrypt the backup.
#    gpg -d example.tar.gz.gpg | tar xvzp
#
# 3. Profit.
#
# CONFIGURATION

publickey="public@key.com"
fileinput="database.sql"

#
# SCRIPT STARTS HERE
#

filename="${fileinput%.*}"

log_echo() { 
  echo $(date +%Y-%m-%d\ %H:%M:%S) "$@";
}

# cd ~/Backup/

log_echo "Securing backup..."
log_echo "Generating a big and random symmetric key."

puked=`env LC_CTYPE=C tr -dc "\+\.\-\_\=A-Za-z0-9" < /dev/urandom | head -c 100 ; echo`

prefixdate=$(date +%Y-%m-%d_%H%M%S)

log_echo "Compressing and encrypting using --cipher-algo aes256."

tar czpf - $fileinput | gpg --passphrase $puked --symmetric --cipher-algo aes256 -o "$prefixdate"_"$filename".tar.gz.gpg

log_echo "Completed: $prefixdate"_"$filename".tar.gz.gpg
log_echo "Encrypting the symmetric key using the given public key."

echo $puked | gpg --output "$prefixdate"_"$filename"_key.txt.gpg --encrypt --recipient $publickey

log_echo "Completed: $prefixdate"_"$filename"_key.txt.gpg
log_echo "Done."