Mina deploy script allows public-private-key system access to git repository, as explained in this article: https://rightsrestricted.com/posts/rails-deployment-with-mina-utilizing-git-slash-ssh

deploy.rb

require 'mina/bundler'
require 'mina/rails'
require 'mina/git'

# specifies path to file on the server
def file_to(file)
  File.join(deploy_to, deploy_set_loc, file)
end

# builds the ssh command for the git-ssh-bashscript
# depends on git_ssh_options being set as a hash
def build_ssh_cmd
  string = "ssh "
  git_ssh_options.each_pair do |switch,value|
    string << "#{switch} \"#{value}\" "
  end
  string << "$1 $2"
end

# require 'mina/rbenv'  # for rbenv support. (http://rbenv.org)
# require 'mina/rvm'    # for rvm support. (http://rvm.io)

# Basic settings:
#   domain         - The hostname to SSH to.
#   deploy_to      - Path to deploy into.
#   repository     - Git repo to clone from. (needed by mina/git)
#   branch         - Branch name to deploy. (needed by mina/git)
#   identity_file  - ssh -key for server to SSH to.

# git-ssh deploy settings:
#  git_ssh_file   - local ssh key file for connection to git repository
#  git_ssh_key    - string containing the actual ssh key
#  deploy_set_loc - location on the server, relative to deploy_to, in which deployment settings are stored
#  git_ssh_option - options passed to the ssh command used by git
set :domain, 'my_domain_name.com'
set :deploy_to, 'path/to/rails/app'
set :repository, 'git@git-server:the_repository.git'
set :branch, 'master'
set :identity_file, ENV['HOME']+'/.ssh/id_rsa'
set :git_ssh_file, File.join(File.dirname(__FILE__), 'deployment/ssh_key')
set :git_ssh_key, File.read(git_ssh_file)
set :deploy_set_loc, "/shared/config/deployment/"
set :git_ssh_options, {
  "-i" => "#{file_to("ssh_key")}",
  "-o" => 'StrictHostKeyChecking no'
}

# Manually create these paths in shared/ (eg: shared/config/database.yml) in your server.
# They will be linked in the 'deploy:link_shared_paths' step.
set :shared_paths, ['config/database.yml', 'log']

# Optional settings:
  set :user, 'deployer'    # Username in the server to SSH to.
#   set :port, '30000'     # SSH port number.

# This task is the environment that is loaded for most commands, such as
# `mina deploy` or `mina rake`.
task :environment do

  # If you're using rbenv, use this to load the rbenv environment.
  # Be sure to commit your .rbenv-version to your repository.
  # invoke :'rbenv:load'

  # For those using RVM, use this to load an RVM version@gemset.
  # invoke :'rvm:use[ruby-1.9.3-p125@default]'
  
  # The script file used by git to establish the ssh connection
  queue! %[export GIT_SSH="#{file_to('clone.sh')}"]
  # Adding gem binaries to PATH
  queue! %[export PATH=/var/lib/gems/1.9.1/bin/:$PATH]
end

task :provider do
  # creating the deployment directory
  queue! %[mkdir -p "#{File.join(deploy_to, deploy_set_loc)}"]
  # writing the key to a file
  queue! %[echo "#{git_ssh_key}" > #{file_to('ssh_key')}]
  # setting correct permissions of ssh key
  queue! %[chmod g-rwx,o-rwx "#{file_to('ssh_key')}"]
  # echoing debug info
  queue  %[echo "-----> Integrated git ssh key."]
  queue  %[echo "-----> \\$GIT_SSH is set to $GIT_SSH"]
  # building the ssh command and writing it to a bash script
  queue! %[echo '#{build_ssh_cmd}' > #{file_to('clone.sh')}]
  # setting the x-bit correctly
  queue! %[chmod u+x,g+x "#{file_to('clone.sh')}"]
  # echoing debug info
  queue  %[echo "-----> built the git-ssh command."]
end

# Put any custom mkdir's in here for when `mina setup` is ran.
# For Rails apps, we'll make some of the shared paths that are shared between
# all releases.
task :setup => [:environment, :provider] do
  queue! %[mkdir -p "#{deploy_to}/shared/log"]
  queue! %[chmod g+rx,u+rwx "#{deploy_to}/shared/log"]

  queue! %[mkdir -p "#{deploy_to}/shared/config"]
  queue! %[chmod g+rx,u+rwx "#{deploy_to}/shared/config"]

  queue! %[touch "#{deploy_to}/shared/config/database.yml"]
  invoke :provider
  queue  %[echo "-----> Be sure to edit 'shared/config/database.yml'."]
end

desc "Deploys the current version to the server."
task :deploy => :environment do
  deploy do
    # Put things that will set up an empty directory into a fully set-up
    # instance of your project.
    invoke :'git:clone'
    invoke :'deploy:link_shared_paths'
    invoke :'bundle:install'
    invoke :'rails:db_migrate'
    invoke :'rails:assets_precompile'

    to :launch do
      queue! %[mkdir -p tmp/]
      queue 'touch tmp/restart.txt'
    end
  end
end

# For help in making your deploy script, see the Mina documentation:
#
#  - http://nadarei.co/mina
#  - http://nadarei.co/mina/tasks
#  - http://nadarei.co/mina/settings
#  - http://nadarei.co/mina/helpers