The audio message, posted online yesterday night, comes days after militants in the Sinai Peninsula gunned down two judges, a prosecutor and their driver, in the first such attack in Egypt.
Militants who pledged allegiance to the Islamic State group have killed scores of soldiers and policemen in Sinai since the army toppled Islamist president Mohamed Morsi in 2013.
Morsi's overthrow unleashed a deadly crackdown on his followers, with hundreds killed in street clashes and thousands imprisoned.
The audio recording hailed the executed men as "brothers and brave lions" and vowed revenge.
"By God, we will seek vengeance for our brothers and others like them, from the party that sentenced them, and the party that implemented the sentence," it said.
The recording called on Egyptians to free prisoners and attack judges and soldiers.
"The least we can do is kill the tyrants who have jailed our brothers. Don't let them enjoy security while your brothers are tortured in jail."
The IS-affiliated jihadist group Sinai Province has spearheaded attacks on the military since Morsi's ouster, despite a massive army campaign in the restive peninsula bordering Israel and the Palestinian Gaza Strip.
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