on one's honor
Entrusted to behave honorably and honestly without supervision. For example, The students were on their honor not to consult notes during the exam. Originally alluding to a solemn oath, this idiom dates from the mid-1400s.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.