The name of the system in Bushinkai Karate & Jujutsu is “Hakuda Kempo Toshu Jutsu.” This article explains the historical context of this name.
Hakuda was a method Jujutsu particular to southwest Japan (for example Nagasaki) that was influenced by Chinese Chuan Fa (Kempo).
The founder of Judo, Jigoro Kano cited Hakuda as follows:
The origin of jujutsu is related thus: There once lived in Nagasaki a physician named Akiyama (pictured below), who went to China to study medicine. There he learned an art called hakuda which consisted of kicking and striking, differing, we may note, from jujutsu, which is mainly seizing and throwing. Akiyama learned three methods of this hakuda and 28 ways of recovering a man from apparent death. When he returned to Japan, he began to teach this art, but as he had few methods, his pupils got tired of it, and left him.
The founder of Goju Ryu Karate, Chojun Miyagi spoke about the Chinese origins of this method, stating:
In China, in the old days, people called Hakuda or Baida for Chinese kungfu, Kenpo or Chuanfa.
Kobudo master Mosokatsu Inoue cited Toshu Jutsu as being the original name for Karate when he wrote:
The ancient martial arts of the Ryukyu Islands consist of Toshu-jutsu, the way of the empty hand and Emono-jutsu, the way of weapons. The former is called Karate and the latter, Ryukyu Kobujutsu.
Goju Kai master Gogen Yamaguchi referred to Toshu Jutsu as Toshukuken. He wrote:
Karate-do was born combining kakutojutsu which had been studied in Okinawa 500 years ago, and kempo, which was introduced from China. As you can see in many countries, fighting martial arts have been handed down in each country. In Okinawa, for a long time, using any kinds of weapons was prohibited because of a policy of prohibiting weapons. For that reason, they had to invent Toshukuken, the way to fight without a weapon. This was especially true in the beginning of the 17th century since it was thought that fighting martial arts, referring to Chinese kempo, was invented among Ryukyu samurai because their weapons were banned. In Okinawa, before it was called karate, it had two names, one was Naha-te and the other was Shuri-te.