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We present several attacks on the block cipher C2, which is used for encrypting DVD Audio discs and Secure Digital cards. C2 has a 56 bit key and a secret 8 to 8 bit S-box. We show that if the attacker is allowed to choose the key, the S-box can be recovered in 224 C2 encryptions. Attacking the 56 bit key for a known S-box can be done in complexity 248. Finally, a C2 implementation with a 8 to 8 bit secret S-box (equivalent to 2048 secret bits) and a 56 bit secret key can be attacked in 253.5 C2 encryptions on average. [ABSTRACT FROM AUTHOR]

Trivium is a stream cipher proposed for the eSTREAM project. Raddum introduced some reduced versions of Trivium, named Bivium A and Bivium B. In this article we present a numerical attack on the Biviums. The main idea is to transform the problem of solving a sparse system of quadratic equations over GF(2) into a combinatorial optimization problem. We convert the Boolean equation system into an equation system over ℝ and formulate the problem of finding a 0-1-valued solution for the system as a mixed-integer programming problem. This enables us to make use of several algorithms in the field of combinatorial optimization in order to find a solution for the problem and recover the initial state of Bivium. In particular this gives us an attack on Bivium B in estimated time complexity of 263.7 seconds. But this kind of attack is also applicable to other cryptographic algorithms. [ABSTRACT FROM AUTHOR]