Rosicrucianism

Rosenkreuz (薔薇十字(ローゼンクロイツ)) is a magic cabal which shares its name with an who is said to have created the style of  which forms the foundation of modern Western magic.

Principles
The cabal derives its name and likely more from the individual Rosenkreuz and the Rosicrucian movement that he is said to have created. The Rosicrucian movement apparently began with a false book written by a single person. Though it was originally false, magicians took it and developed it beyond the falsity, leading Rosicrucianism to grow into something much more, coming to form the foundation for modern Western magic. This development was such that when the original author revealed the truth of the lie he was completely ignored. According to Othinus, if one were to ask a random magician to name ten famous people, Rosenkreuz's name would come up among the list.

Toaru Majutsu no Index
Index refers to the magic cabal among those after her for the grimoires in her head, during her initial meeting with Kamijou Touma.