These are the proceedings of WAIFI 2008, the second workshop on the Ari- metic of Finite Fields, that was held in Siena, Italy, July 6-9, 2008. The ?rst workshop, WAIFI 2007, which was held in Madrid (Spain), was received quite enthusiasticallybymathematicians, computerscientists, engineersandphysicists who are performing research on ?nite ?eld arithmetic. We believe that there is a need for a workshop series bridging the gap between the mathematical theory of ?nite ?elds and their hardware/software implementations and technical - plications. We hope that the WAIFI workshopseries, which from now on will be held on even years, will help to ?ll this gap. Therewere 34 submissionsto WAIFI 2008, of which the ProgramCommittee selected 16 for presentation. Each submission was reviewed by at least three reviewers. Our thanks go to the Program Committee members for their many contributionsandhardwork.Wearealsogratefultotheexternalreviewerslisted below for their expertise and assistance in the deliberations. In addition to the contributions appearing in these proceedings, the workshop program included an invited lecture given by Amin Shokrollahi. Special compliments go out to Enrico Martinelli, General Co-chair, and to Roberto Giorgi and Sandro Bartolini, local organizers of WAIFI 2008, who broughtthe workshopto Siena, oneofthe mostbeautiful citiesofTuscany, Italy. WAIFI 2008 was organized by the Dipartimento di Ingegneria dell'Informazione of the University of Siena, Italy. The submission and selection of papers were done using the iChair software, developed at EPFL by Thomas Baign eres and Matthieu Finiasz. We also thank Deniz Karakoyunlu for his help in this matter.