By L. Badescu, D. Popescu
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Additional resources for Algebraic Geometry, Bucharest 1982: Proceedings of the International Conference, Held in Bucharest, Romania, August 2-7, 1982
Hiding the key from the user can be achieved by employing tamper resistant hardware (so that the key-information is concealed, but still can be used in a black-box fashion), or, in the software-based setting, program obfuscation can be employed instead. Hardware tamper resistance can provide satisfactory solutions in terms of security. However the cost of constructing and distributing tamper resistant decoders to each subscriber of the system is prohibitive for many digital content providers. Especially in an Internet-based setting it would be extremely desirable that no system-speciﬁc physical device should be required and that the decoder software should be downloadable from the distributor’s web-site.
The goal of the Join protocol is to allow the user u to compute a point zu , αu , Q(zu , αu ) of the bivariate polynomial Q so that: zu is randomly selected by the system manager, and αu = αuC + αcR where αuC is a value selected and committed by the user, and the value αuR is randomly selected by the system manager. The commitment of C the user u to the value αuC will be of the form Cu = g αu , signu (Cu ) . 1. After the completion of the Join procedure, the user’s secret personal key will be set to the vector κu := Q(zu , αu ), zu , zu2 , .
Here we show that this assumption is, in fact, false: Claim 1. Any collusion of traitors of more than a single user, can generate keys that are not traceable in the scheme of . Such keys are random linear combinations of the traitors’ keys. The Break. The break depends on the following fact that we show about the  scheme: Given t user-keys zi , αi , f (zi , αi ) the vector deﬁned below, t t δ0 , . . , δk , δ0 , . . , =1 µz, =1 t µ α zk , =1 µ f (z , α ) =1 can also be used as a key, where µ1 , .
Algebraic Geometry, Bucharest 1982: Proceedings of the International Conference, Held in Bucharest, Romania, August 2-7, 1982 by L. Badescu, D. Popescu