The Al-Qassam Brigades said in a New Year's Eve statement that it had posted online "two video clips to mark the 23rd birthday of the Zionist soldier and prisoner Oron Shaul".
The statement went on to say the Al-Qassam Brigades had previously announced that it had killed 14 Israeli soldiers and abducted Shaul in eastern Gaza on July 20, 2014.
A 36-second video shows six people in military fatigues sitting on chairs and surrounding another person on a wheelchair who is wearing a shirt bearing the number 102.
In the middle of them on a table is a cake with three candles bearing a message written in Hebrew that reads: "Three years in Hamas jail".
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In the background a choir sings in Hebrew: "Happy Birthday Oron".
In the second clip, which lasts 54 seconds, a person whose face appears to have been digitally altered to that of Shaul's is seen sitting in a prison cell with their hands bound in ropes.
In the final scene, a man dressed as a clown and appearing to be Israeli Prime Minister Benjamin Netanyahu enters the cell.
The Israeli army believes that another soldier, Hadar Goldin, was killed along with Shaul two years ago and that the Islamist Palestinian movement Hamas is thought to hold their bodies.
In September, a senior Israeli official said that Israel had been holding since the 2014 Gaza war 18 Palestinians from the enclave as well as the bodies of 19 others and "offered to swap them for the bodies of the two Israeli soldiers".