Sutarman said that boats taking asylum seekers to Australia might be given rights of passage through Indonesia as they "do not come under our authority".
Yudhoyono said co-operation in areas such as people-smuggling would not resume until Australia signed new protocols and codes of conduct governing intelligence operations.
"For me it is a requirement and a stepping stone and the bilateral cooperation protocol that I suggested and has been agreed on by the Prime Minister of Australia," he said.
Yudhoyono said that he would personally ensure that the draft agreement answered "all the wishes of Indonesia", and that the final deal must be signed by both himself and Abbott.
But he did say that in his letter Abbott had promised that Australia would never in the future do anything that would "disadvantage or disturb" Indonesia.
Abbott said yesterday that Yudhoyono's statement was "warm and positive toward Australia" and that he agreed that senior envoys should meet soon to resolve outstanding issues.
"What I'd like to see some time in the future is some sort of security round table where we are more open with each other, where we build even stronger relationships and trust," he said. "I want Australia to be Indonesia's trusted partner, just as I want Indonesia to be our trusted partner."
But Australia faces international criticism of its policies on asylum seekers, which have been increasingly tightened.
Labor reintroduced much of former Liberal Prime Minister John Howard's draconian "Pacific solution", including the re-opening of camps on Nauru and Manus.
The new Government has introduced more harsh measures, despite earlier UN condemnation of Labor's regime. It is unlikely to be swayed by the new reports by the UN High Commissioner for Refugees.
The reports were based on visits to Nauru and Manus after the election, and said that treatment of asylum seekers was inhumane and in likely breach of international law. Australia will not accept asylum seekers arriving by boat even if they are accepted as genuine refugees, and the UNHCR attacked policies using oppressive conditions to encourage them to return to their homelands.
The reports criticised conditions in both centres, including overcrowded conditions in tents in which some people, including families with children and pregnant women, slept on mattresses on the floor.
On Nauru more than 400 people shared eight toilets and two urinals. Conditions were too hot for children to attend school, and UN officials said their mental health was deteriorating. On Manus growing numbers of detainees - from 300 to more than 1000 since June - had increased densities. Conditions were cramped and poor, with one block "putrid" from blocked shower drains and flooded floors.
The reports said that when policies and practices were based mainly on deterrence the consequences for the people affected could be harmful and punishing.
