His parents divorced not long after his birth, and his father, Arkadii Rubin, moved to England and was not part of Kaluznin's life.
His mother, Maria Pavlovna Kaluznina, moved with the young Kaluznin to Petrograd (present-day Saint Petersburg), where she brought him up.
She shared her love for Russian culture, including music and literature, with her son, and she would remain an important part of his life.
His school offered a thorough education in mathematics, and upon graduating, he entered the Humboldt University of Berlin, where he attended lectures taught by the algebraist Issai Schur.
World War II and in particular the German occupation of Paris in June 1940 made it necessary for him to halt his studies.
On 22 June 1941, Soviet citizens living in France were interned at a camp in Compiègne near Paris.
Here, Kaluznin was initially able to continue studying mathematics, carrying out research on Galois theory, which determines if certain equation solutions can be written with rational functions.
He got some work translating for the Soviet Embassy in Paris but was finally able to return to his mathematical studies.
In the following years, he published several papers, some of them with Marc Krasner, and presented his research at academic conferences.
They applied to the Soviet immigration authorities, who requested that he should work in East Germany, where there was a shortage of scientists, for some time.