In the following initial summary of the plot of the series My Hero Academia, what is the secret that Izuku must keep and from whom?

Izuku Midoriya is a young man who dreams of becoming a Hero despite being bullied by his violent childhood friend Katsuki Bakugo for lacking a Quirk. After having a few encounters with his idol, All Might, who is Japan's greatest hero, Izuku is chosen by All Might to inherit his Quirk "One For All" (which gives him the ability to endlessly stockpile his raw power) and become his successor, as All Might was severely injured by his arch-nemesis All For One (whose Quirk allows him to steal other Quirks for his use and pass them over to others at his will) and does not have much time left to live. Izuku begins attending the U.A. High School for heroes in training alongside Bakugo and his friends from Class 1-A while keeping his connection with All Might a secret. During the course of the school year, Izuku and his friends work hard to improve themselves and have a few encounters with the League of Villains led by All For One's apprentice Tomura Shigaraki, who desires to kill All Might as part of their plan to take over the world. During one of these encounters, All Might and All For One have one last fight, which ends with All For One defeated and imprisoned, and All Might, having exhausted the last of One For All's power in himself, forced to retire.

Izuku keeps his secret that All Might had chosen Izuku to inherit All Might's Quirk "One For All" from Bakugo and Izuku's friends from Class 1-A.