From the Membership of the National Insurance Scheme page on Nav.no:

Membership of the National Insurance Scheme is the key to your entitlements from NAV. You do not have to be a Norwegian citizen, registered with the National Population Register or pay taxes in Norway. What is crucial is that your residence in Norway is lawful.

National insurance entitlements depend on present or previous membership. Some entitlements may however, be granted to the family of a member of the National Insurance Scheme.

The benefits that you can claim will be determined on what coverage your membership provides. Some national insurance benefits require that you have been a member of the National Insurance Scheme for a certain time. You can read more about this under each separate benefit.

When are you a member of the National Insurance Scheme?

Residing in Norway – compulsory member of the National Insurance Scheme
If you are residing in Norway, you will, as a general rule, be a member of the National Insurance Scheme.
To be considered as residing in Norway, you must have stayed in Norway for at least 12 months. When you move to Norway with the intention of being here at least 12 months, you will become a member from the date of entry.

If you do not reside in Norway
Even if you do not reside in Norway, you are a member of the National Insurance Scheme if you are:
– An employee in Norway
– A Norwegian citizen/EEA-citizen and employed on a Norwegian ship, except for catering staff on a tourist ship registered on the Norwegian International Ship Register NIS
– A Norwegian citizen/EEA-citizen and employed in the service of the Norwegian government.
– A Norwegian citizen/EEA-citizen and studying abroad with a loan or scholarship from the Norwegian State Educational Loan Fund.
– Working temporarily abroad for an employer who pays compulsory employer’s contribution in Norway

Voluntary membership of the National Insurance Scheme while residing in Norway
If you will be residing in Norway for three to twelve months without working here and have strong ties to Norway, you can apply to become a member of the National Insurance Scheme.

Voluntary membership of the National Insurance Scheme while residing outside Norway
If you are a member of the National Insurance Scheme and will be going to reside abroad, that stay abroad can mean that your membership of the National Insurance Scheme ceases. You can apply for voluntary membership of the National Insurance Scheme.

Social security agreements
Norway has social security agreements with several countries. The most comprehensive agreement is with the EEA-countries. If you are covered by a social security agreement, the agreement may determine in which country you will be a member of the social security scheme. In that case, what is stated above may be inaccurate. You should enquire whether a social security agreement applies in your case.

National insurance contributions
Membership of the National Insurance Scheme means that you must pay national insurance contributions.
How much you must pay in contributions will depend on whether
You are liable to taxes in Norway and national insurance contributions are paid together with taxes
Your employer pays compulsory employer’s contribution

When does membership cease?
Working or residing abroad can mean that membership of the National Insurance Scheme ceases.

Implications of not being a member of the National Insurance Scheme
If you are not a member of the National Insurance Scheme, you have no entitlements under the National Insurance Act.
This means that you do not have to pay national insurance contributions.

http://www.nav.no/English/Membership+in+The+National+Insurance+Scheme 

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