Multiplication table modulo 26 - Department of Mathematics
Multiplication table modulo 26 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Created Date: 9/6/2016 10:41:45 AM
Addition Modulo 26 - Northern Kentucky University
Caesar cipher Assume that plaintext e(5) corresponds to ciphertext K (11) CT = pt + key mod 26 In[1]:= Solve key Multiplicative cipher 14
B A modulo 26 What is i t?) Record the vectors for the coded
modulo 26 For example, the matrix A used in questions 1 -4 has determinant 3, so A should have an inverse modulo 26, as of course it does To calculate the inverse of C modulo 26 , calculate the reduced echelon form of [C I] by doing row operations as usual, except use only integer multipliers and reduce each number modulo 26 Important: to
Everything You Need to Know About Modular Arithmetic
Now we write some multiples of 26 26,52,78,104,130,156,182,208,234 A number a has an inverse modulo 26 if there is a b such that a·b ≡ 1(mod 26)or a·b = 26·k +1 thus we are looking for numbers whose products are 1 more than a multiple of 26 We create the following table Table 2: inverses modulo 26 x 1 3 5 7 9 11 15 17 19 21 23 25
n Elements greatest common measure
1 7 15 4 26=× −× Finally, "go mod 26 " Because 26 0mod26≡ , when we "go mod 26," the equation 1 7 15 4 26=× −× becomes the congruence 1 7 15mod26≡× So, the inverse of 15 modulo 26 is 7 (and the inverse of 7 modulo 26 is 15) Gcd(6, 26) = 2; 6 and 26 are not relatively prime Therefore, 6 does not have a multiplicative inverse
Topic 1: Cryptography 1 Introduction to Cryptography
For example, the multiplicative inverse of 5 modulo 26 is 21, because 5 21 1 modulo 26 (because 5 21 = 105 = 4 26+1 1 modulo 26) (It is important to note that in modular arithmetic, a 1 does not mean 1=a In fact, we have not defined division at all ) Not all numbers have a multiplicative inverse modulo n
Modular Arithmetic - University of Washington
In our previous examples, 17 is the residue of 46 modulo 29, and 26 is the residue of 84 modulo 29 We can also do this with negative numbers For example, 5 is the residue of 7 modulo 6 There are a few important properties of modular arithmetic that will be helpful 1 Equivalence modulo m preserves sums 2 Equivalence modulo m preserves
22 UCR Solution: y x y x y howareyou h o qznhobxqd x y
26 Find the plaintext Solution: Given y, we need to solve y · 9x+2 mod 26)y ¡2 · 9x mod 26 Checking, we see that 3 is the inverse of 9 modulo 26, as 9*3 is 1 modulo 26 Thus, the above is solved by x · 3⁄9x · 3⁄(y ¡2) mod 26 Let us apply this to UCR We have U = 20, C = 2, R = 17 Thus, calculating: 3⁄(20¡2) · 54 · 2 mod 26 3
Modular Arithmetic and Cryptography
Since there are 26 letters in the English alphabet, let’s relate the letters a-z by numbers 0-25 as shown by the diagram below Notice going from \a" to \D" was a shift of 3 letters over Thus we can encrypt the word \pumpkin" by relating \p" with 15 on the wheel, adding 3 to get 18, and then we turn this back into a letter, which gives us \S"
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