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We describe implementations for solving the discrete logarithm problem in theclass group of an imaginary quadratic field and in the infrastructure of a realquadratic field. The algorithms used incorporate improvements overpreviously-used algorithms, and extensive numerical results are presenteddemonstrating their efficiency. This data is used as the basis forextrapolations, used to provide recommendations for parameter sizes providingapproximately the same level of security as block ciphers with $80,$ $112,$$128,$ $192,$ and $256$-bit symmetric keys.

Security Estimates for Quadratic Field Based Cryptosystems