Part of the code for article http://rocinantesoftware.blogspot.com.es/2015/11/prestashop-improved-customer-password.html

PasswordController.php

<?php

class PasswordController extends PasswordControllerCore
{
    public function postProcess()
    {
        if (Tools::isSubmit('email')) {
            $this->sendForgettenPasswordEmail();
        } elseif (($token = Tools::getValue('token')) && ($id_customer = (int)Tools::getValue('id_customer'))) {
            $this->processForgettenPasswordConfirmation($token, $id_customer);
        } elseif (($new_password = Tools::getValue('new_passwd')) && ($new_password_confirm = Tools::getValue('new_passwd_confirm'))) {
            $this->processPasswordChange($new_password, $new_password_confirm);
        } else {
            parent::postProcess();
        }
    }

    protected function sendForgettenPasswordEmail()
    {
        if (!($email = trim(Tools::getValue('email'))) || !Validate::isEmail($email)) {
            $this->errors[] = Tools::displayError('Invalid email address.');
        } else {
            $customer = new Customer();
            $customer->getByemail($email);
            if (!Validate::isLoadedObject($customer)) {
                $this->errors[] = Tools::displayError('There is no account registered for this email address.');
            } elseif (!$customer->active) {
                $this->errors[] = Tools::displayError('This account is not active. Please contact customer support.');
            } else {
                //record when we are sending the forgotten password email, and generate a corresponding token
                $token = md5(uniqid(rand(), true));
                $password_request = array (
                    'datetime_password_request' => date('Y-m-d H:i:s'),
                    'token_password_request' => $token
                );

                if (Db::getInstance()->getValue("select id_customer from customer_extra_info where id_customer = {$customer->id}")) {
                    $token_saved = Db::getInstance()->update('customer_extra_info', $password_request, "id_customer = {$customer->id}");
                } else {
                    $password_request['id_customer'] = $customer->id;
                    $token_saved = Db::getInstance()->insert('customer_extra_info', $password_request);
                }

                $mail_params = array (
                    '{email}' => $customer->email,
                    '{lastname}' => $customer->lastname,
                    '{firstname}' => $customer->firstname,
                    '{url}' => $this->context->link->getPageLink('password', true, null,
                        'token=' . $token . '&id_customer=' . (int)$customer->id)
                );
                if ($token_saved && Mail::Send($this->context->language->id,
                        'password_query',
                        Mail::l('Password query confirmation'),
                        $mail_params,
                        $customer->email,
                        $customer->firstname . ' ' . $customer->lastname)
                ) {
                    $this->context->smarty->assign(array ('confirmation' => 2, 'customer_email' => $customer->email));
                } else {
                    $this->errors[] = Tools::displayError('An error occurred while preparing the email.');
                }
            }
        }
    }

    protected function processForgettenPasswordConfirmation($token, $id_customer)
    {
        $this->logOutOtherCustomers ($id_customer);
        $datetime_password_request = Db::getInstance()->getValue("
            select
                datetime_password_request
            from
              customer_extra_info as cei
            where
                id_customer = $id_customer
                and token_password_request = '$token'");
        if ($datetime_password_request) {
            $customer = new Customer($id_customer);
            if (!Validate::isLoadedObject($customer)) {
                $this->errors[] = Tools::displayError('Customer account not found');
            } elseif (!$customer->active) {
                $this->errors[] = Tools::displayError('This account is not active. Please contact customer support.');
            } elseif ((strtotime($datetime_password_request . '+60 minutes') - time()) < 0) {
                $this->errors[] = Tools::displayError('Your password change request expired after one hour. Please try again.');
            } else {
                $this->smartyForChangePassword($id_customer);
            }
        } else {
            $this->errors[] = Tools::displayError('We cannot regenerate your password with the data you\'ve submitted.');
        }
    }

    protected function logOutOtherCustomers ($id_customer)
    {
        if (isset ($this->context) &&
            isset ($this->context->customer) &&
            isset ($this->context->customer->id) &&
            $this->context->customer->id > 0 &&
            $this->context->customer->id != $id_customer)
        {
            $customer = new Customer ($this->context->customer->id);
            $customer->logout();
        }
    }

    protected function smartyForChangePassword($id_customer)
    {
        $this->context->smarty->assign(array (
            'change_password' => 1,
            'id_customer' => $id_customer
        ));
    }

    protected function processPasswordChange($new_password, $new_password_confirm)
    {
        $id_customer = (int)Tools::getValue('id_customer');
        $this->logOutOtherCustomers ($id_customer);

        $new_password = trim($new_password);
        $new_password_confirm = trim($new_password_confirm);

        if (0 != strcmp($new_password, $new_password_confirm) /* 0 indicates they are equal */) {
            $this->errors[] = Tools::displayError('Email addresses are not equal.');
        } else {
            $password_changed = false;
            if (isset ($this->context) && isset ($this->context->cookie)) {
                $customer = new Customer((int)$id_customer);
                if (Validate::isLoadedObject($customer)) {
                    $customer->passwd = $this->context->cookie->passwd = Tools::encrypt($new_password);
                    if ($customer->update()) {
                        $this->context->smarty->assign(array (
                            'password_changed' => 1,
                            'id_customer' => $id_customer
                        ));
                        $password_changed = true;
                    }
                }
            }
            if (!$password_changed) {
                $this->errors[] = Tools::displayError('Password could not be changed.');
            }
        }
        if (count ($this->errors) > 0)
        {
            $this->smartyForChangePassword($id_customer);
        }
    }
}