pepisteume,na verb participle perfect passive accusative neuter plural [UBS] pisteu,w
believe (in), have faith (in) (with God or Christ as object); believe, believe in; have
confidence (in someone or something), entrust (something to another); o[j me.n pÅ fagei/n
pa,nta one man's faith allows him to eat anything (Ro 14.2)
A more literal translation might be "Our books are only two and twenty, and contain the register of all time which was uprightly entrusted."
Possibly however the dikaius was originally meant as a reference back to Josephus' earlier use of dikaiov:
with dikaion having the following possible meaning (which is a possible translation of the Hebrew yashar):
di,kaion adjective normal accusative neuter singular no degree [UBS] di,kaioj, a, on conforming to the standard, will, or character of God; upright, righteous, good; just, right; proper; in a right relationship with God; fair, honest; innocent
This could have been an simple scribal error since later copiest could well have been ignorant of the Book of Jasher.
A literal translation of this adapted text might be "Our books are only two and twenty, and contain the register of all time which was entrusted from [the book] of the Righteous [or to use your translation: from the Authentic Annals]."
Thus we go back to Josephus C. Apion (1:6; or 1.2) "My first thought is one of intense astonishment at the current opinion that, in the study of primeval history, the Greeks alone deserve serious attention, that the truth should be sought from them, and that neither we nor any others in the world are to be trusted. In my view the very reverse of this is the case, if that is to say, we are not to take idle prejudices as our guide, but to extract these truths from the Upright (dikaion - perhaps better translated with your title, The Authentic Annals).
This research on Josephus' use of these words is fairly preliminary at this point - I have only been looking at this section of Contra Apion so far - but on the other hand I did pick what I consider to be a particularly relevant section. Contra Apion relates to a lot of historical sources that have not survived from antiquity. I believe that there are aspects of Josephus where he assumes his audience knows things that modern scholars have overlooked or which we no longer have the ancient evidence of... Could this be yet another case?