Anne-Elisabeth Falkevik Hagen, who is married to businessman Tom Hagen, disappeared on October 31 a year ago from her home in Lorenskog.
Anne-Elisabeth Hagen, 68, who is married to multimillionaire Tom Hagen, vanished on October 31 - but news of her disappearance was only just made public.
The newspaper reported that the note said Falkevik Hagen would be killed if police were involved.
"A ransom demand and serious threats have been issued", Insp Tommy Brøske told reporters.
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"Our main theory is that the victim was kidnapped by unidentified perpetrators at her home" in Lorenskog, about 20 kilometres (12 miles) from Oslo, he said.
There has been no evidence since she disappeared that Mrs Hagen is safe, police say.
Currently, the police have no suspects.
The house of Norwegian multi-millionaire Tom Hagen is cordonned off in Fjellhamar, east of Oslo.
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Launched in 2014 with a mission to protect the users from having their transactions traced, Monero is well-known as a virtually anonymous cryptocurrency. The police have also recommended the family not to pay the ransom that was requested in exchange for the promise that she would be released.
Kidnappings are rare in Norway, a Scandinavian nation of 5.3 million that prides itself on low crime rates.
Brøske said investigators had had "very limited" contacts with the purported kidnappers online, but refused to disclose the date of the most recent contact.
Her husband Tom Hagen - a real estate investor and owner of power facilities - is number 172 on a list of the country's 400 most wealthy people published by Norway's financial magazine Kapital.
The parents of three adult children, the couple lived a quiet, retiring life in an elegant but understated home, now blocked off by police tape.
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