CHAPTER 4 Number Theory and Cryptography
... and $ to represent 62 and 63). Corresponding to each such symbol would be a binary string of six digits, from
000000 for 0, through 001010 for a, 100011 for z , 100100 for A, 111101 for Z, 111110 for @, and 111111
for $ . To translate from binary to base 64, we group the binary digits from the right ...
Factoring RSA keys from certified smart cards
... While generating high-quality random numbers is critical to the security of cryptographic systems, it is also notoriously difficult to do. Non-deterministic behavior is considered to be a fault in almost every other component of a computer,
but it is a crucial component of generating random numbers ...
Non Calculator Arithmetic
... d) 0.0732 10 = 0.732 (decimal point moved one place right).
e) 9.546 1000 = 9546 (decimal point moved three places right and ‘dropped’).
f) If we try to move the decimal point three places to the right, we seem to ‘run out’, as multiplying
1.95 by 100 gives us 195, which is a whole number. We st ...
The integers Have two operations addition and multiplication
... Have two operations addition and multiplication
Addition is associative, commutative, has identity and each
element has an additive inverse
Multiplication is associative, commutative, has identity
Multiplication distributes over addition (both left and right).
Any algebraic object satisfying this co ...
... Convert from standard form to scientific notation.
Convert from scientific notation to standard form.
Add, Subtract, Multiply, and Divide with scientific
The grid method of multiplication
... Subtraction (2) – counting back on a number line
(3 digit - 3 digit)
Example: My journey is 352 kilometres. I have travelled 179
kilometres. How far is there to go?
So we need 352-179.
This is represented on a number line as shown below.
First 352 is placed towards the end of the empty number line. ...
Name : Teacher : Date : Score :
... 3 ) The sum of two numbers times a third number is equal to the sum of each
addend times the third number. For example a x (b + c) = a x b + a x c
Scientific Notation PP
... Negative Exponent
• The exponent will be negative if the number in standard
form is less than 1.
• Example: .000298
This is a number that is less than 1. To
write this in scientific notation move the decimal four (4) times to the
right in order to get a number less than 10 and greater than or equal ...
PA_M6_S2_T2_Comparing Real Numbers Transcript - NTER
... -4 < -3, while 4 > 3. I can write this as 3<4 or 4>3. Both of those are
Let's practice a few. Let's insert the correct symbol in these
expressions to make each statement true.
-2 and -12. -2 sits to the right of -12 on a number line, it is therefore
greater than -12. -2 > -12 ...
10 = 10 x 10 x 10 = 1000
... operations and facts about numbers. As a simple example we
write the numeral “352” and say “three hundred fifty two”.
We use the exponent form as a shortcut method for writing repeated multiplication. In exponent
form there are two important features, the base and the exponent. For natural number ex ...
Here is the algorithm example for the week 8 discussion
... Example 2: (to be worked out in the discussion). Write and analyze a pseudocode algorithm
that finds the product of the largest and smallest even integers in the list a 1, a2, …, an. The
algorithm should return -1 (or some other negative value) if there are no even numbers. If
there is just one even ...
Chapter 4—Statement Forms
... bytes. The unit that represents the most common integer size
on a particular hardware is called a word. Because machines
have different architectures, the number of bytes in a word
may vary from machine to machine.
... Rational numbers are numbers that
can be written as decimals or
Rational numbers include ALL
integers, fractions, and mixed
An integer is any number from the
set… continuing forever
Two's complement is a mathematical operation on binary numbers, as well as a binary signed number representation based on this operation. Its wide use in computing makes it the most important example of a radix complement.The two's complement of an N-bit number is defined as the complement with respect to 2N; in other words, it is the result of subtracting the number from 2N, which in binary is one followed by N zeroes. This is also equivalent to taking the ones' complement and then adding one, since the sum of a number and its ones' complement is all 1 bits. The two's complement of a number behaves like the negative of the original number in most arithmetic, and positive and negative numbers can coexist in a natural way.In two's-complement representation, positive numbers are simply represented as themselves, and negative numbers are represented by the two's complement of their absolute value; two tables on the right provide examples for N = 3 and N = 8. In general, negation (reversing the sign) is performed by taking the two's complement. This system is the most common method of representing signed integers on computers. An N-bit two's-complement numeral system can represent every integer in the range −(2N − 1) to +(2N − 1 − 1) while ones' complement can only represent integers in the range −(2N − 1 − 1) to +(2N − 1 − 1).The two's-complement system has the advantage that the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers (as long as the inputs are represented in the same number of bits and any overflow beyond those bits is discarded from the result). This property makes the system both simpler to implement and capable of easily handling higher precision arithmetic. Also, zero has only a single representation, obviating the subtleties associated with negative zero, which exists in ones'-complement systems.