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In cryptography, XTR is an algorithm for public-key encryption. XTR stands for ‘ECSTR’, which is an abbreviation for Efficient and Compact Subgroup Trace Representation. It is a method to represent elements of a subgroup of a multiplicative group of a finite field. To do so, it uses the trace over GF(p) to represent elements of a subgroup of GF(p).
From a security point of view, XTR relies on the difficulty of solving Discrete Logarithm related problems in the full multiplicative group of a finite field. Unlike many cryptographic protocols that are based on the generator of the full multiplicative group of a finite field, XTR uses the generator g of a relatively small subgroup of some prime order q of a subgroup of GF(p). With the right choice of q, computing Discrete Logarithms in the group, generated by g, is, in general, as hard as it is in GF(p) and thus cryptographic applications of XTR use GF(p) arithmetics while achieving full GF(p) security leading to substantial savings both in communication and computational overhead without compromising security. Some other advantages of XTR are its fast key generation, small key sizes and speed.