The rail service said it had notified those whose accounts had been targeted.
Other passengers will be told they have been blocked the next time they try to log in and will be asked to reset their details.
The firm declined to say whether any of the hack attacks were successful but said payment details have not been affected.
Falanx Group Limited (LON:FLX), is a global intelligence and cyber defence provider working with blue chip and government clients. It operates the MidGARD cyber monitoring platform for corporate and governmental customers which utilises a combination of proprietary and third-party processes and technologies.