  Algebraic Manipulation Detection (AMD) Codes detect adversarial noise that is
added to a coded message and stored in a storage that is opaque to the
adversary. We study AMD codes when the storage can leak up to \rho\log|G| bits
of information about the stored codeword, where G is the group in which the
stored codeword lives and \rho is a constant. We propose \rho-AMD codes that
provide protection in this new setting, and define weak and strong \rho-AMD
codes that provide security for a random and an arbitrary message,
respectively. We derive concrete and asymptotic bounds for the efficiency of
these codes featuring a rate upper bound of 1-\rho for the strong codes. We
also define the class of \rho^{LV}-AMD codes that provide protection when
leakage is in the form of a number of codeword components, and give
constructions featuring a strong \rho^{LV}-AMD codes that asymptotically
achieve the rate 1-\rho. We describe applications of \rho-AMD codes to, (i)
robust ramp secret sharing scheme and, (ii) wiretap II channel when the
adversary can eavesdrop a \rho fraction of codeword components and tamper with
all components of the codeword.
