Abstract. A decoding algorithm for Gabidulin codes (defined over Fqm ) is shown that directly provides the evaluation polynomial using an equivalent of the Euclidean Algorithm. To obtain low complexity, a fast symbolic product and a fast symbolic division are presented. The achieved complexity of the whole decoding algorithm for Gabidulin codes is O(m 3 log m) operations over the ground field Fq.

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Abstract. Message authentication codes (MACs) using polynomial evaluation have the advantage of requiring a very short key even for very large messages. We describe a low complexity software polynomial evaluation procedure, that for large message sizes gives a MAC that has about the same low software complexity as for bucket hashing but requires only small keys and has better security characteristics.Key words: Message authentication, universal hash functions, polynomial evaluation, software MAC generation. IntroductionThe verification of the authenticity of a text document or a datafile is one of the main applications of cryptographic techniques. A common used technique for this purpose is the application of a message authentication code (MAC). Basically we have two users called the sender S (or signer) and the verifier V. S and V share a secret random key string and a publicly known MAC. The MAC maps a message string to a shorter, so called, tag string. The sender calculates the tag corresponding to the message string and the shared secret key string and sends the message to V together with the tag. V accepts a received message if the received tag is the same as the tag for the received message and the secret key. A good MAC is designed to make it hard for an adversary to send own messages or substitute observed messages by new ones, without being detected by the receiver. Usually one distinguishes between so called unconditionally secure, computationally secure, and provable secure authentication codes, [1, page 392]. Codes belonging to the first category are codes for which the security of the MAC is independent of the computational power of the adversary. The security of these codes is expressed in the probability of success of an deception attack. A MAC is called computationally secure if the adversary is faced with the difficulty that all known

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Abstract-We consider the weight spectrum of a class of quasiperfect binary linear codes with code distance 4. For example, extended Hamming code and Panchenko code are the known members of this class. Also, it is known that in many cases Panchenko code has the minimal number of weight 4 codewords. We give exact recursive formulas for the weight spectrum of quasi-perfect codes and their dual codes. As an example of application of the weight spectrum we derive a lower estimate for the conditional probability of correction of erasure patterns of high weights (equal to or greater than code distance).