"START III has become a truly important achievement in this field (nuclear threat reduction) but it is possible to do more, such as accelerate the fulfillment of treaty provisions (is it necessary to wait until 2018 for reaching the levels set by the treaty?) and start new bilateral negotiations on further reduction of nuclear arsenals," Ivanov said in an interview with Interfax.
Ivanov said he presented the ideas in detail in a joint article with former U.S. Secretary of State Madeleine Albright published by the New York Times on December 31, 2012.
He quoted a number of key extracts from the article to Interfax.
"Russia and the United States control 90 to 95 percent of the world's nuclear weapons," the article runs. "We can readily continue negotiations of further reductions and still safely ensure our security. If we do, we will be more persuasive when asking other nuclear-weapons states to join in the nuclear-arms reduction process and will enhance the credibility of our diplomacy in mobilizing international pressure on Iran to refrain from trying to build a nuclear weapon."
"The long-running dispute over missile defense continues to cast a shadow over possible progress on arms control, even though both NATO and Russia say they want to cooperate in that sphere. Now is the time to be creative. With some imagination on both sides, missile defense could prove a game-changer, making NATO and Russia allies in protecting Europe," it says.
Ivanov and Albright mentioned some other disagreements, as well.
"Differences of perspective can sometimes be sharp, as over Syria, human rights and democracy. Nevertheless, disagreements, no matter how complex and painful, must not block the development of ties along other lines. It is essential not to interrupt dialog even on those issues where positions differ substantially," the article says.
"Cooperation between Russia and the United states makes sense on a range of other issues. As the end of major operations by coalition forces draws near, Washington and Moscow, together with others, should support Afghan leaders in constructing a stable society, able to withstand pressure from violent extremist groups," it says.
"Another area of mutual interest is promoting an expansion of U.S.-Russian trade and investment relations. The current level of bilateral commerce falls significantly short of its potential, given the size of the countries' economies," the article says.
"An increase would benefit both. Russia's recent entry into the World Trade Organization will help, as will full implementation of the important trade and market access provisions that made that agreement possible. The U.S. Congress decision to remove Cold War restrictions and finally extend permanent normal trade relations status to Russia was long overdue," it added.
"If our countries are to derive the maximum benefit from our shared interests, Presidents Obama and Putin must make our potential for partnership a priority," Ivanov and Albright said.
"Some speculate that, with Mr. Putin's return to the Russian presidency, managing U.S.-Russian relations will become more difficult. We see no reason to assume that," they said.
"The reset improved bilateral ties, which are certainly stronger today than they were in 2008; that progress would not have happened had Mr. Putin opposed it. The challenge for our two presidents is to move now to the next stage - to embark on a historic mission to start a new chapter in bilateral relations between Russia and the United States," the article runs.