Boolean functions form crucial components in crypto- .... Boolean functions in polar form are represented here as ... tivates the new approach described below.
(-1)f(x)âg(xâz) is the crosscorrelation of f and g at z. The autocorrelation of f â bn at u â Zn. 2 is Cf,f (u) above, which we denote by Cf (u). Let ζ = e2Ïı/q be the ...
The procedure described in the preceding paragraph of constructing an iso- ...... 389. However, the interior sum in M is bounded by ÐQR\?ay2 f^)dt + /"(l^l ~ a)>.
facts with the required properties and seek one which is ac- tually a Boolean function. .... Boolean functions in polar form are represented here as vectors in R2n.
find out minimum (or low) degree functions g1,g2 such that f â g1 = 0 and ...... FSE 2004, number 3017 in Lecture Notes in Computer Science, pages 83â94.
To simplify the complexity of a circuit, the designer must find another circuit that computes the same function as the original but does so with fewer or simpler ...
Boolean function deciding if a given n-bit integer is square-free. These results ..... 1â17. Available from http://www.maths.uwa.edu.au/~woods/alan research.html.
2 Department of Computer Science and Technology. University of ... on the case of best quadratic approximations of a wide class of cubic functions of arbitrary.
function f that is independent of the permutation of the other input variables of f. .... pseudo Boolean function f : {0, 1}n â Z can be represented by a âBmd. Just.
lean functions that approximate the maximum clique size of a graph. ..... 3.4. Approximating the maximum clique size. Theorem 3.9 says that a monotone circuit ...
Doma-2-Dong, Seo-Gu ... cryptographic properties, namely balancedness, good local and global avalanche charac- ... and good avalanche characteristics.
MODp) AC0 with MODm gates for any integer m can't compute the MODn function when there is a prime dividing n but not m (Smolensky 1987). Indeed, ...
Feb 9, 2005 - resultant output spectrum comprises just 2n spectral elements. ... (measuring) any specific qubit configuration is as small as possible, ... Moreover, the set of all LC-equivalent graphs form an LC-orbit. ... characterise and analyse th
Computing Boolean Functions: Exact Quantum Query. Algorithms and Low Degree Polynomials. Alina Dubrovska, Taisija Mischenko-Slatenkova. Department of ...
Nov 7, 1989 - [23] H. Minc, Permanents, Cambridge University Press. (Cambridge), 1984. ... Tokyo Women's Christian College Journal of Mathematics nos.
Chartwell-Bratt Ltd., Bromley, 1993. Xiangyong Zeng was born in Hubei Prov- ince, China, on November 18, 1973. He re- ceived his B.S. degree from Hubei ...
Definition 1.1: Let / and g be functions on Πto A. We say that g is conjugate of / if, for anyx, y e A, we have f(x) y =0 if, and only iί,g(y) *x = 0. If, in particular, a ...
unique cover of co(f,xi) independent of the variable xi. We introduce now two types of expansion of a Boolean function that yield decompositions with respect to a ...
Dedekind's problem, Boolean functions, free distributive lattices. (^Supported in .... It follows from the second property above, that if one knows the values of an isotone Boolean ..... guide the reader through the technical maze ahead. In the next
Nov 9, 1990 - 103. Table 11. Statistics of the best solutions found by GA and SA . ...... ture was disproved by Bose, Shrikande and Parker [19], who showed ...... it in the design of the IBM cipher Lucifer, upon which the Data Encryp-.
Page 1 ..... follows. Plotkin's anti-uni cation algorithm 20] is used to compute the most ... 4 = fx1 7! z3g, the join is given by h (a); (g4 ^true) _g (g0. 4 ^ y1 y2)i = ha ...
While Boolean algebra is the fundamental formal system for digital circuit ...
symbols are clearly recognisable there is no need to label them AND, OR etc.,
and ...
linking propositional logic with kernel machines. In particular ... The question of how many Boolean primitives are needed to learn a logical formula is typically an ...
Department of Computer Science, University of Latvia,. Raina bulv. ... computing the value of Boolean function and its relation to the degree of algebraic ...
Draft notes or 22C: 040. 4. Simplification using Karnaugh Maps. A. B. 0. 1. 1 ...
using NAND. A. A. A.B = A+B. B. B. (Exercise) Prove that NOR is a universal gate.
Draft notes or 22C: 040
Simplification of Boolean functions Using the theorems of Boolean Algebra, the algebraic forms of functions can often be simplified, which leads to simpler (and cheaper) implementations.
Follow the class lectures to understand how to simplify Boolean functions using K-maps. Several examples will be worked out in the class.
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Draft notes or 22C: 040
Other types of gates
A
A A.B
B
A+B
B
NAND gate
NOR gate
Be familiar with the truth tables of these gates.
A B
A + B = A.B + A.B
Exclusive OR (XOR) gate
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Draft notes or 22C: 040
NAND and NOR are universal gates
Any function can be implemented using only NAND or only NOR gates. How can we prove this?
(Proof for NAND gates)
Any boolean function
can be implemented using AND, OR and NOT gates. So if AND, OR and NOT gates can be implemented using NAND gates only, then we prove our point.
1. Implement NOT using NAND
A
A
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Draft notes or 22C: 040
2. Implementation of AND using NAND
A
A.B
B
A
1. Implementation of OR using NAND
A
A
A.B = A+B B B
(Exercise) Prove that NOR is a universal gate.
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Draft notes or 22C: 040
Example (to be worked out in class) How to convert any circuit that uses AND, OR and NOT gates to a version that uses NAND (or NOR gates only)?
Additional properties of XOR XOR is also called modulo-2 addition. Why?
A 0 0 0 0 1 1 1 1
B 0 0 1 1 0 0 1 1
C 0 1 0 1 0 1 0 1
F 0 1 1 0 1 0 0 1
1
A=A
0
A=A
A B = 1 only when there are an odd number of 1’s in (A,B). The same is true for A B C also.
Why?
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Draft notes or 22C: 040
Logic Design Exercise
Half Adder
A B
Half Adder
S=A
A
B
S
C
Sum (S)
0
0
0
0
Carry (C)
0
1
1
0
1
0
1
0
1
1
0
1
B
C = A.B
A S B C
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Draft notes or 22C: 040
Full Adder
Sum (S) A B
Full Adder
Cin
S=A
Carry (Cout)
B
A
B
C
S
Cout
0
0
0
0
0
0
0
1
1
0
0
1
0
1
0
0
1
1
0
1
1
0
0
1
0
1
0
1
0
1
1
1
0
0
1
1
1
1
1
1
Cin
Cout = A.B + B.Cin + A.Cin
How can you add two 32-bit numbers? It will be discussed in the class.
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Draft notes or 22C: 040
Combinational vs. Sequential Circuits
Combinational circuits. The output depends only on the current values of the inputs and not on the past values. Examples are adders, subtractors, and all the circuits that we have studied so far
Sequential circuits. The output depends not only on the current values of the inputs, but also on their past values. These hold the secret of how to memorize information. We will study sequential circuits later.
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Draft notes or 22C: 040
Decoders
n
A typical decoder has n inputs and 2 outputs.
Enable
A
B
D3 D2 D1
D0
D0
0
0
0
0
0
1
A
D1
0
1
0
0
1
0
B
D2
1
0
0
1
0
0
D3
1
1
1
0
0
0
A 2-to-4 decoder and its truth table.
D3 = A.B
Draw the circuit of this decoder.
D2 = A.B D1 = A.B
The decoder works per specs
D0 = A.B
when (Enable = 1). When Enable = 0, all the outputs are 0.
Exercise.
Design a 3-to-8 decoder. 12
Draft notes or 22C: 040
Encoders n
A typical encoder has 2 inputs and n outputs.
D0
D0 D1
D2 D3 A
B
1
0
0
0
0
0
D1
A
0
1
0
0
0
1
D2
B
0
0
1
0
1
0
0
0
0
1
1
1
D3
A 4-to-2 encoder and its truth table.
A = D1 + D3 B = D2 + D3
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Draft notes or 22C: 040
Multiplexor
It is a many-to-one switch, also called a selector.
A
0
B
1
S = 0, F = A
F
S = 1, F = B
Specifications of the mux
Control S
A 2-to-1 mux
F = S. A + S. B
Exercise.
Design a 4-to-1 mux.
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Draft notes or 22C: 040
Another design of a decoder
A B
F
C D D0 D1 D2 D3 2-to-4 decoder
S
Exercise 1. Design a 2-to-4 decoder using 1-to-2 decoders only. Exercise 2. Design a 4-to-1 multiplexor using 2-1 multiplexors only.
To be discussed in the class.
15
Draft notes or 22C: 040
Demultiplexors
A demux is a one-to-many switch.
0
X
S = 0, X = A
1
Y
S = 1, Y = B
A
S
A 1-to-2 demux, and its specification.
So, X = S. A, and Y = S. B
Exercise.
Design a 1-4 demux.
We will discuss the design of a 1-bit ALU in class.
