Justice Minister Andrew Little expects New Zealanders who shared the name of Grace Millane's killer via social media will face legal action.
At the weekend police reminded the public that it is an offence to breach a court order, such as name suppression, including naming someone on social media.
The situation has arisen because the name of the man convicted on Friday of killing UK backpacker Grace Millane remains suppressed by the courts.
Mr Little told Morning Reportthat while work is underway to set up an international mutual agreement on suppression breaches with countries like the UK, Canada and Australia, the law is very straightforward in New Zealand.
"If somebody in New Zealand goes on social media and publishes details that are suppressed, that's a matter of contempt of court in New Zealand. We've already got the power to deal with that and I understand authorities will be looking at situations where that has happened in the last couple of days."
Anyone who breaks the law in New Zealand could face a fine of up to $25,000 or six months in jail.
Mr Little said he will be meeting with law ministers from the Australian states this week to discuss his plan for international enforcement of suppression orders.
The United States and Commonwealth law ministers have also accepted that international agreements on suppression orders is worthwhile and the Commonwealth Secretariat has begun work on the proposal, although that may take up to two years, Mr Little said.
"Everywhere I've gone it's been well received - this idea about mutual recognition of suppression orders from other countries."
No plans to extend bill regarding sexual history evidence
Mr Little is not considering changes to the kinds of evidence juries can hear about a murder victim's sexual history in the wake of the Grace Millane trial.
A bill before Parliament will tighten the rules around evidence about a complainant's sexual history, to better protect against unnecessary and distressing questioning, for survivors of rape or sexual assault.
The legislation, aimed at making it easier for sexual violence victims to testify in court, has passed its first reading.
It proposes that unless a judge gives permission, a witness cannot be asked about their sexual history with the defendant, nor could they be asked about their sexual preferences.
Since a guilty verdict was returned for the man accused of killing Ms Millane, there have been calls for more protection of the sexual history of those who are killed in the course of a crime.
Mr Little told Morning Report there are no plans to broaden the bill.
"We have drafted the bill with the particular objectives in mind that we have which relates to trials related to sexual offending both in the criminal and civil jurisdiction. There is no intention to extend it beyond that."
He said he was limited in what he could say about the Grace Millane trial because the convicted man may appeal.
"...The other party potentially gets away with saying all sorts of things about the person who has died and it's never very clear where the truth lies ...
"But defence counsel have a job to put a case and to challenge assumptions and to challenge Crown evidence and they do that. I make no judgement at all about what happened in the Grace Millane trial."
He said he did not want to be drawn into defence tactics for a trial and sensitive evidence was often the subject of pre-trial decisions by the judge.
The aim of the current bill was to make the legal process less traumatising for victims when they gave evidence.
Palpable public outrage over Ms Millane's portrayal - lawyer
Wellington Women Lawyers' Association's Steph Dyhrberg told Morning Report there was palpable public outrage over the way the narrative of Ms Millane as "innocent, sweet pretty Grace" who had met an untimely death was turned around into somebody who "sort of deserved it" during the trial.
She said it was vital the conversation that had started on the tactics used in the trial continued to be debated.
Ms Dyhrberg said it was only because of Sophie Elliott's killing 10 years ago that the defence of provocation used in her killer's trial - the "look what you made me do" defence - was made illegal.
So it was healthy to be debating the kind of defence used in the trial over Ms Millane's murder.
"It doesn't matter what Grace Millane may or may not have consented to on this night or even on previous nights with other partners which is the evidence I found particularly abhorrent and irrelevant.
"You can't consent to treatment that is so rough that it might reasonably result in your death and you certainly of course never consent to your own death ...
"The way that evidence was led, the way that the witnesses, particularly the young women who had previously dated or thought about dating this man - the way they were cross-examined, the way their history was even gone into or their attitudes towards him or sex or what happened or didn't happen - that was highly questionable and to say that that is a part of a fair defence. Fair to whom?"
She said a significant workstream led by Associate Minister Jan Logie was underway and consent was part of that programme, so the Sexual Violence Legislation Bill was only part of reforms.
Judges and defence lawyers needed to look at the relevance of a victim's sexual history and whether the public, from whom jury members were chosen, found it acceptable.
The Criminal Bar Association president, Len Andersen, said a lawyer had to follow the instructions of his or her client, and in this case, it was that death occurred as a result of an accident during sex, so it was difficult to see how the current sexual practices of both people could not be raised.