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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

G
F
(

p

2

)

{\displaystyle GF(p^{2})}
to represent elements of a subgroup of

G
F
(

p

6

)

{\displaystyle GF(p^{6})^{*}}
.
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

{\displaystyle g}
of a relatively small subgroup of some prime order

q

{\displaystyle q}
of a subgroup of

G
F
(

p

6

)

{\displaystyle GF(p^{6})^{*}}
. With the right choice of

q

{\displaystyle q}
, computing Discrete Logarithms in the group, generated by

g

{\displaystyle g}
, is, in general, as hard as it is in

G
F
(

p

6

)

{\displaystyle GF(p^{6})^{*}}
and thus cryptographic applications of XTR use

G
F
(

p

2

)

{\displaystyle GF(p^{2})}
arithmetics while achieving full

G
F
(

p

6

)

{\displaystyle GF(p^{6})}
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.

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