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We investigate the security bounds of quantum cryptographic protocols using$d$-level systems. In particular, we focus on schemes that use two mutuallyunbiased bases, thus extending the BB84 quantum key distribution scheme tohigher dimensions. Under the assumption of general coherent attacks, we derivean analytic expression for the ultimate upper security bound of such quantumcryptography schemes. This bound is well below the predictions of optimalcloning machines. The possibility of extraction of a secret key beyondentanglement distillation is discussed. In the case of qutrits we argue thatany eavesdropping strategy is equivalent to a symmetric one. For higherdimensions such an equivalence is generally no longer valid.Comment: 12 pages, 2 figures, to appear in Phys. Rev.

Security bound of two-basis quantum-key-distribution protocols using qudits