By Çetin Kaya Koç, Sihem Mesnager, Erkay Savaş

This e-book constitutes the refereed court cases of the fifth overseas Workshop at the mathematics of Finite box, WAIFI 2014, held in Gebze, Turkey, in September 2014. The nine revised complete papers and forty three invited talks offered have been conscientiously reviewed and chosen from 27 submissions. This workshop is a discussion board of mathematicians, computing device scientists, engineers and physicists acting study on finite box mathematics, drawn to speaking the advances within the idea, functions, and implementations of finite fields. The workshop may help to bridge the space among the mathematical conception of finite fields and their hardware/software implementations and technical applications.

**Read or Download Arithmetic of Finite Fields: 5th International Workshop, WAIFI 2014, Gebze, Turkey, September 27-28, 2014. Revised Selected Papers PDF**

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**Additional info for Arithmetic of Finite Fields: 5th International Workshop, WAIFI 2014, Gebze, Turkey, September 27-28, 2014. Revised Selected Papers**

**Example text**

Pn ) be a set of mutually coprime integers, which we call an RNS-basis. We deﬁne P as the product of all the pi ’s. The RNS uses the fact that any integer x within [0, P − 1] can be uniquely represented by the list (x1 , x2 , . . , xn ), where each xi is the residue of x modulo pi , which we write as xi = |x|pi . If x and y are given in their RNS representations x = (x1 , . . , xn ) and y = (y1 , . . , yn ), according to B, and such that x, y < P , RNS addition and multiplication are realized by modular addition and multiplication on each component: x+y = (|x1 + y1 |p1 , .

8 −22 . . 9 −27 . . 10 −30 . . 7 8 9 10 15 16 − 22 −24 −30 −33 . . . . . . . . . . . . . . . . . . . . . . . Table 2. Details for Table 1 entries µsym (m, l) Upper bound 2 Lower bound (5, 1) [5] (6, 1) (7, 1) (8, 1) (9, 1) (10, 1) [4] µsym (3)µsym (2) (first factor: by interpolation over P1F4 ) 4 2 [4] [5] [5] µsym (4)µsym (2) (unknown original contributor for the first) . 4 2 [6] . µsym (5)µsym (2) 4 2 (unknown original contributor for the first) . (1, 5) [7] [5] (1, 6) [7] [5] (1, 7) [7] (proved valid over a general ring, in [8]) [5] (1, 8) [8] [5] (1, 9) [8] (1, 10) (2, 2) (2, 3) [5] sym µ4 (1, 5)µsym (2), the first factor being equal to 10 2 µsym (1, 2)µsym (2) 4 2 sym µ4 (1, 3)µsym (2) 2 (3, 2) new (2, 4) µ4 sym .

For the SLCOO kernel, each warp works on a slice. Since each thread works on more than one row, it needs to have individual storage for its partial per-row results, or to be able to have exclusive access to a common resource. In [19], Schmidt et al. mentionned the two possibilities of either using the shared memory or having atomic accesses. While these needs can be fulﬁlled in [19] for the context of linear algebra over GF(2), we will observe in Sect. 6 that these constraints hamper the eﬃciency of the SLCOO in the context of large ﬁelds.