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The objective of this paper is to give a fast square root computation method. First the Frobenius mapping is adopted. Then a lot of calculations over an extension field are reduced to that over a proper subfield by the norm computation. In addition a inverse square root algorithm and an addition chain are adopted to save the computation cost. All of the above-mentioned steps have been proven to make the proposed algorithm much faster than the conventional algorithm. From the table which compares the computation between the conventional and the proposed algorithm, it is clearly shown that the proposed algorithm accelerates the square root computation 10 times and 20 times faster than the conventional algorithm in F p 11 and F p 12 respectively. At the same time, the proposed algorithm reduces the computation cost 10 times and 20 times less than the conventional algorithm.

In order to protect the equipment or some important information from evil Internet users, information security technology has played a key role. RSA cryptosystem is the most widely used public-key cryptosystem but its key for ensuring sufficient security reaches about 2000 bits long. Therefore, it is not efficient to implement RSA cryptosystem on terminal with scarce computation resources. On the other hand, elliptic curve cryptosystem (ECC) has the same security level with about 7-fold smaller length key as compared to RSA cryptosystem, it is said that ECC is more practical as compared to RSA.

This paper proposes a useful method for constructing a self-dual normal basis in an arbitrary extension field F"p"^"m such that 4p does not divide m(p-1) and m is odd. In detail, when the characteristic p and extension degree m satisfies the following conditions (1) and either (2a) or (2b); (1) 2km+1 is a prime number, (2a) the order of p in F"2"k"m"+"1 is 2km, (2b) [email protected]?km and the order of p in F"2"k"m"+"1 is km, we can consider a class of Gauss period normal bases. Using this Gauss period normal basis, this paper shows a method to construct a self-dual normal basis in the extension field F"p"^"m.

Binary extension field F2 8 is recently used for AES encryption and IoT (Internet of Things) security with PUF (Physical Unclonable Function). The calculations over tower field such as F(2 4)2 or F((2 2)2)2 become more efficient than that of over F28. This paper focuses on the efficient calculations over F(2 4)2.