By Gerasimos C. Meletiou, Arne Winterhof (auth.), Joachim von zur Gathen, José Luis Imaña, Çetin Kaya Koç (eds.)

This booklet constitutes the refereed lawsuits of the second one overseas Workshop at the mathematics of Finite Fields, WAIFI 2008, held in Siena, Italy, in July 2008.

The sixteen revised complete papers offered have been rigorously reviewed and chosen from 34 submissions. The papers are prepared in topical sections on buildings in finite fields, effective finite box mathematics, effective implementation and architectures, class and development of mappings over finite fields, and codes and cryptography.

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**Extra info for Arithmetic of Finite Fields: 2nd International Workshop, WAIFI 2008 Siena, Italy, July 6-9, 2008 Proceedings**

**Example text**

At−1 f is invariant under H. So applying corollary 1 to H gives p divides |yH |. at−1 f (y)]. Sumw 28 M. at−1 Sumw , the sum on y can be grouped by H orbits and applying corollary 1, the only remaining terms are the y with the same value n-t+1 times (bn−t+1 ). at−1 ))] mod p b=0 x∈Eqn t−1 χ w[f (x)] = q n−1 . As n − 1 = pk + t − 2 mod p when gcd(p,q) = 1 and 0 otherwise. Sharply Transitive Balanced Functions Proposition 11. M (qr, r, . . , r ) functions with r = qn−1 −1 . n Proof. First of all (f (an ))a∈Eq are a permutation of the set Eq .

1–7 (2000) 20. : On recognizing graph properties from adjacency matrices. Theoretical Computer Science 3, 371–384 (1976) 21. : Balancedness and Correlation Immunity of Symmetric Boolean Functions. In: Proc. C. Bose Centenary Symposium. Electronic Notes in Discrete Mathematics, vol. 15, pp. 178–183 (2003) 22. : Rotation symmetric Boolean Functions: Count and cryptographic properties. C. Bose Centenary Symposium on Discrete Mathematics and Applications. Indian Statistical Institute, Calcutta (December 2002) 23.

We study a tighter upperbound for symmetric functions from Eqn onto Em . 2 Size of ODD of q-Ary Symmetric Functions Proposition 13. When 1 < m < C(q − 1 + n, q − 1), 1 < n, the size s(f ) of the QRODD of a symmetric function f from Eqn onto Em is less than: n−t(n,m,q) U (n, m, q) = C(q + t(n, m, q) − 1, q) + mC(q−1+i,q−1) i=0 where t(n,m,q) is defined in lemma 5 in D. Proof. For a symmetric function f from Eqn onto Em there are also two inequalities. The ﬁrst one is more relevant on the upper part of the diagram pi (f ) ≤ C(q − 1 + i, q − 1) because the functions only depend on the partitions of (x1 , · · · , xi ) which can only take C(q − 1 + i, q − 1) diﬀerent values.