Information Theory

Algebraic Circuits by Antonio Lloris Ruiz, Encarnación Castillo Morales, Luis

By Antonio Lloris Ruiz, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos (auth.)

This ebook offers an entire and actual examine of algebraic circuits, electronic circuits whose functionality will be linked to any algebraic constitution. The authors distinguish among uncomplicated algebraic circuits, corresponding to Linear suggestions Shift Registers (LFSRs) and mobile automata and algebraic circuits, resembling finite fields or Galois fields.

The ebook contains a finished overview of illustration platforms, of mathematics circuits imposing easy and extra complicated operations and of the residue quantity platforms (RNS). It offers a examine of uncomplicated algebraic circuits similar to LFSRs and mobile automata in addition to a learn of circuits with regards to Galois fields, together with actual cryptographic functions of Galois fields.

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Extra resources for Algebraic Circuits

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The decimal value X (independently if the number is positive or negative) corresponding to the number snan-1 … a0, represented using base-1 complement (with sign bit instead sign digit), is given by: XnÀ1 a bi X ¼ Àsn ðbn À 1Þ þ i¼0 i P i In fact, if sn = 0 (positive number), it results X ¼ nÀ1 i¼0 ai b , as expected. If PnÀ1 i sn = 1 (negative number), then the value resulting for X is X ¼ 1 À bn þ i¼0 ai b . By the other hand, the value represented is: 24 1 Number Systems ( À nÀ1 X ) ðb À 1 À ai Þbi i¼0 ¼À nÀ1 X i¼0 biþ1 þ nÀ1 X i¼0 bi þ nÀ1 X i¼0 ai bi ¼ 1 À bn þ nÀ1 X ai bi i¼0 being equal to X.

2 Positional Notation Using One Base 13 Memberwise adding the equalities: À Á À Á À Á À Á N ¼ c0 bk À a þ c1 bk À a þ Á Á Á þ ci bk À a þ Á Á Á þ cpÀ1 bk À a þ r0 þ r1 þ Á Á Á þ rp And: Á À Á À À Á Nmod bk À a ¼ r0 þ r1 þ Á Á Á þ rp mod bk À a Thus, the remainder modulo bk - a of N may be computed by multiplying iteratively by a, being N the first multiplicand. Each product is decomposed in two fragments, Aibk ? Bi. Each Ai = 0 is the new multiplicand, and the calculus ends when Ai = 0. The remainder is RBimod(bk - a).

1 B) must be negative. e. ðbnþ1 ÀAÞ þ ðbnþ1 ÀBÞ, with A þ B\bn ðif A þ B ! bn , an overflow will be produced, as will be seen in Sect. 1 - (A ? B). The direct sum of both of the representations also provides the correct result. In fact: À nþ1 Á À nþ1 Á ffi É b ÀA þ b ÀB ¼ bnþ1 þ bnþ1 ÀðA þ BÞ ffi nþ1 É b ÀðA þ BÞ is a negative number. 1. Thus, again, the carry must be discarded for obtaining the final result. Resuming, the direct sum when using base complement representations, always generates the correct result (except overflow situations).

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