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While proving compositeness of a natural number is a computational task that can be easily done in polynomial time, proving primality of an arbitrary positive i n teger is a harder task. Only two main streams of usefull algorithms are known in this direction: elliptic curve primality provers (ECPP, Mo1]) and cyclotomy ( BovdH], Mi1]). We shall give h e r e a n o verview of the properties of the primality proving algorithms together with some data illustrating the state of the art performance of these methods.doi:10.1016/s0378-4754(99)00051-8 fatcat:xlksnpewr5bylfxq6tv7dntqhu