In another attack blamed on militants 13 members of two families were killed in a village near the capital Algiers, the reports said.
As is common in Algeria, news of the attacks first appeared in the morning papers and without any direct official confirmation.
But the attack on the military appears to be the worst since President Abdelaziz Bouteflika came to power in 1999, pledging to end the country's decade-old civil war.
Petrol bombs
Details are scant but newspaper reports in the capital, Algiers, say the army was ambushed on Saturday near the town of Batna, about 430 kilometres (265 miles) south-east of Algiers.
The guerrillas are said to have thrown petrol bombs onto the military convoys as they passed by.
Hospital staff are reported as saying that 43 soldiers and local militiamen were killed and a further 19 are being treated in hospitals in Batna and nearby Biskra.
In the other attack, the dead came from two families, targeted in the town of Zabana.
Militants blamed
The country's two main hardline Islamist groups have been blamed for the attacks.
The Salafist Group for Preaching and Combat (GSPC), which has been linked to Osama bin Laden's al-Qaeda network, is accused of attacking the military convoy, and the Armed Islamic Group (GIA) has been blamed for the Zabana killings.
Islamist militants are usually blamed for such attacks, but as ever in the murky civil war, identities and motives remain blurred.
In recent weeks the military have been active in the Aures mountains, fighting an armed group that has been setting up roadblocks for extortion purposes.
In past years, the number of attacks have increased significantly during the fasting month of Ramadan which ended recently.
This year, that was not the case, and the month passed relatively peacefully.
But hopes that this would be sustained seem to have now been dashed, especially as the country is only a year away from elections - often another reason for an upsurge in violence.