"The weeks following attacks of this nature … indicate to us we need to be even more vigilant in the weeks following, just to ensure we don't have any retaliation or copycat type attacks."
Bush said it was important for the public to know the police don't have any "specific threats that we are currently following up".
After the terror attack in Christchurch on March 15, New Zealand's terror threat level was raised from low to high for the first time ever.
This means, according to the independent inter-governmental agency responsible for setting New Zealand's terror threat: "[A] terrorist attack, violent protest activity or violent criminal behaviour is assessed as very likely".
The threat level is set by a group called CTAG – Combined Threat Assessment Group – which is made up of multiple government agencies.
The police – as well as the GCSB, the SIS, Defence Force, Corrections, Ministry of Foreign Affairs and the Civil Aviation Authority – are all part of the group.
It is up to each agency how they respond to a high threat level.
The response of the police has been to arm some officers with semi-automatic rifles.
Bush said, based on the feedback the police was getting, "we do know … that people do feel secure with seeing police armed and ready to respond at a moment's notice".
However, he said, the police also know that they will reach a point where they would have to reassess the option of being heavily armed as it won't be as reassuring to the public.
"Getting that timing right is critical for us because we're here to make the public feel safe, and to keep them safe."