Wednesday's attack was the deadliest act of violence at a Russian school since the Beslan terrorist attack in 2004, when 333 people, many children, died after Chechen fighters seized a school.
Russia's Investigative Committee named the shooter as 18-year-old Vladislav Roslyakov, a student at the college.
Russia's Investigative Committee, the nation's top investigative agency, said the attacker was caught on security cameras entering Kerch Polytechnic College in the Black Sea city of Kerch and firing at students.
The media reported earlier that the same Committee had confirmed earlier that "it was a terrorist attack" and that "an explosive device that was activated at the university was filled with metal fragments (for impact)".
Numerous victims from Wednesday's attack were teenage students.
Wednesday's attack was by far the worst by a student in Russian Federation, raising questions about school security in the country.
Soon after the attack, Russian officials said they were investigating the possibility that it was terrorism.
A student at the college who asked not to give his name told AFP that "I was in a class when I heard shooting on the first floor".
At least 17 teenagers have been killed by an 18-year-old gunman who apparently later killed himself in a school in Russian-annexed Crimea. More than 40 others were injured in the attack.
As of publication, the death toll stands at 20, while at least 50 others were injured. The city of Kerch is also the location of a controversial bridge that links the peninsula with the Russian mainland - a prestige project for Vladimir Putin.
Putin, speaking Wednesday after talks with his visiting Egyptian counterpart Abdel-Fattah el-Sissi in the Black Sea resort of Sochi, said investigators are looking into the circumstances of the attack and its motives.
"Everyone ran there. I saw a girl lying there".
Nine have been transferred to Moscow for further treatment, Crimean health minister Alexander Golenko said.
Many of those hospitalised were injured by an explosion that blasted metal ball bearings and other items into their bodies.
The young man's grandmother, Taisiya Roslyakova, told the Komsomolskaya Pravda daily on Thursday that her grandson was a kind boy who was always helping her with chores, unlike her son, the student's father, who reportedly was an alcoholic. His mother, a nurse, was helping to treat victims at a local hospital after the shootings, unaware yet that her son was accused of the rampage and was already dead.
After the findings, the investigators reclassified the criminal case from a "terror attack" to a case of "the murder of two or more people". RBK newspaper quoted an anonmyous fellow student as saying Roslyakov "really hated the college because of evil teachers and hinted he would take revenge on them".
Yuri Kerpek told the state-owned RIA Novosti news agency that the shooting lasted about 15 minutes.