With the relationship between Egypt's new Islamist leader and Israel still in its fragile infancy, the terrorist attack on the border that the two countries share with Gaza over the weekend presented a critical opportunity — and a crucial test.

Several high-ranking officials inside Israel’s government and numerous independent experts on Israel-Egypt relations said Monday that the attack — in which masked gunmen killed 16 Egyptian soldiers on Sunday night and then barreled into Israeli territory in a stolen truck and armored vehicle — is the best evidence yet that the two countries are both threatened by lawlessness in the Sinai Peninsula. Now the question is whether Egypt’s new president, Mohamed Morsi, will make the Sinai a priority amid other challenges, and whether Israel will make concessions in modifying the 33-year-old peace treaty between the nations to allow for a more aggressive Egyptian military presence.

“Now it is obvious also to him that there is a real convergence of interests here, and this may get us closer to him,” Danny Ayalon, Israel’s deputy foreign minister, said of Mr. Morsi.

Hillel Frisch, a political science lecturer at Bar-Ilan University and senior research fellow at its Begin-Sadat Center for Strategic Studies, said the attack underlined the differences between Islamists like Mr. Morsi who support the international system of states and others who are trying to challenge it.

“The jihadists threaten any kind of order, anyone who has power, any kind of incumbency,” Mr. Frisch said. “It will strengthen Morsi’s commitment to be a status-quo actor, which is a big, big thing strategically. He runs a state, and there are greater enemies to the Egyptian state than Israel. In that sense, it’s a game-changer.”

The attack brought several early signs of cooperation and coordination. An Israeli brigadier general and his Egyptian counterpart met near the border to discuss the investigation. Israel handed over to Egypt the armored car and the bodies of those killed as they tried to enter through the Kerem Shalom crossing. The Israeli Foreign Ministry issued a statement of condolence.

But comments by both Israeli Defense Minister Ehud Barak and Prime Minister Benjamin Netanyahu on Monday included hints of concern. “I hope that this will be a wake-up call for Egypt regarding the necessity to be sharp and efficient on their side,” Mr. Barak said after visiting the area. Mr. Netanyahu expressed regret over the killing of the soldiers and said, “It is clear that Israel and Egypt have a common interest in maintaining a quiet border.” He quickly added that “when it comes to the security of the citizens of Israel, the State of Israel must and can rely only on itself.”

Mr. Morsi declared three days of mourning for the soldiers who were killed and traveled to Sinai with his defense minister, intelligence chief and interior minister. Egyptian security officials had spoken early Monday about a large-scale military operation near Egypt’s Rafah crossing into Gaza, but there was little sign of it later in the day.

New details emerged Monday about the attack, which began as the sun fell the night before. Egypt’s military said in a statement that 35 masked gunmen, packed into three Land Cruisers, stormed an Egyptian checkpoint and killed the soldiers as they were sitting down to break their Ramadan fast.

Lt. Col. Avital Leibovich, a spokeswoman for the Israeli Defense Force, said one of the men then drove a truck, taken from the military outpost and packed with a half ton of explosives, about a mile to the Israeli border fence, which he blew up along with himself and the vehicle. The armored car, also stolen, then entered Israel, where it was stopped by three Israeli airstrikes that killed six or seven men — most of them carrying explosives on their bodies — as they tried to flee. The operation took 15 minutes.

Egyptian officials have blamed militants in Sinai and said they were aided by Palestinians in Gaza. In its statement, the military called the attackers “enemies of the state” and said that “those who stand behind them must be confronted by force.”

But the Muslim Brotherhood, Egypt’s most powerful political party, posted a statement on its Web site on Monday night saying that Israel’s intelligence agency, which it said had sought to “thwart” Egypt’s revolution, could be responsible for the attack. The Brotherhood, which was not speaking for the president, said that the attack highlighted the need to “reconsider” the terms of Egypt’s treaty with Israel, which restricts the number of troops that Egypt can station in Sinai.

But several Israeli officials and analysts noted that the so-called military annex to the treaty signed in 1979 was modified two years ago to allow seven additional Egyptian battalions into Sinai and that Egypt has yet to fill that quota.

“The Egyptians will have to look in the mirror and ask themselves what they want to do” about Sinai, said Maj. Gen. Dan Harel, who retired as No. 2 in the Israeli military in 2009. “For a long time they pushed it under the rug. There’s a hill under the rug today.”

But several others said the Egyptians would need even more troops, along with more flexibility for air and intelligence operations, to make a difference in Sinai. “Israel put the Egyptians in a very difficult dilemma,” said Yoram Meital, chairman of the Herzog Center for Middle East Studies at Ben Gurion University. “On one hand, Israel is saying this is Egyptian territory, you have the responsibility to keep it secure; on the other, they are saying you should do this under the conditions of the military annex, and I think this is an impossible mission.”

If the episode offered a chance to improve the rocky relations between Israel and Egypt, it also threatened the growing ties between Mr. Morsi’s government and Hamas-controlled Gaza. In response, statements by Hamas leaders exceeded the typical condemnations and condolences, promising help in chasing down the attackers, who many believe either came from Gaza or moved freely there in the planning.

“We will not allow anybody to harm Egypt’s security,” Mohammed Awad, Hamas’s foreign minister, said in a statement. Drawing clear alliances, he accused Israel of “turning Sinai into a field of terror and crime to shake the stability of Egypt.”

Regardless of complicity, Gaza immediately suffered the consequences. Israel shut down Kerem Shalom, its only commercial crossing into Gaza; Egypt closed Rafah, through which both goods and people pass; and Hamas blocked the tunnels through which all manner of things are smuggled from Egypt. This left Gazans in long lines on Monday to load up on gas and food in fear of soaring prices once existing supplies are exhausted.

“I do not know if these products will remain here or will disappear,” said Saied Ajour, 56, pointing at boxes of Egyptian cheese on the shelves of the Zawya market in Gaza City.