The efficient generation of public-key parameters is a prerequisite in public-key systems. A specific example is the requirement of a prime number p to define a finite field ℤ_p for use in the Diffie-Hellman key agreement protocol and its derivatives (§12.6). In this case, an element of high order in ℤ p * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429466335/81e205fb-b60a-4276-b51a-603b35ce55a6/content/eq812.tif"/> is also required. Another example is the requirement of primes p and q for an RSA modulus n = pq (§8.2). In this case, the prime must be of sufficient size, and be “random” in the sense that the probability of any particular prime being selected must be sufficiently small to preclude an adversary from gaining advantage through optimizing a search strategy based on such probability. Prime numbers may be required to have certain additional properties, in order that they do not make the associated cryptosystems susceptible to specialized attacks. A third example is the requirement of an irreducible polynomial f(x) of degree m over the finite field ℤ_p for constructing the finite field F p m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429466335/81e205fb-b60a-4276-b51a-603b35ce55a6/content/eq813.tif"/> . In this case, an element of high order in F p m * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429466335/81e205fb-b60a-4276-b51a-603b35ce55a6/content/eq814.tif"/> is also required.