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4 <title>Affine ciphers</title>
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52
53 # Affine ciphers
54
55 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
56 --|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--
57 b | e | h | k | n | q | t | w | z | c | f | i | l | o | r | u | x | a | d | g | j | m | p | s | v | y
58
59 An extension of Caesar ciphers
60
61 * Count the gaps in the letters.
62
63 ---
64
65 layout: true
66
67 .indexlink[[Index](index.html)]
68
69 ---
70
71 # How affine ciphers work
72
73 .ciphertext[_ciphertext_letter_] =.plaintext[_plaintext_letter_] × a + b
74
75 * Convert letters to numbers
76 * Take the total modulus 26
77
78 # Enciphering is easy
79
80 * Build the `affine_encipher()` function
81
82 ---
83
84 # Deciphering affine ciphers is harder
85
86 `$$p = \frac{c - b}{a} \mod 26$$`
87
88 But modular division is hard!
89
90 Define division as mutiplication by the inverse: `\(\frac{x}{y} = x \times \frac{1}{y} = x \times y^{-1}\)`
91
92 A number _x_, when multiplied by its inverse _x_<sup>-1</sup>, gives result of 1.
93
94 This is not always defined in modular arithmetic. For instance, 7 × 4 = 28 = 2 mod 26, but 20 × 4 = 80 = 2 mod 26. Therefore, 4 doesn't have a multiplicative inverse (and therefore makes a bad key for affine ciphers).
95
96 Result from number theory: only numbers coprime with _n_ have multiplicative inverses in arithmetic mod _n_.
97
98 Another result from number theory: for non-negative integers _m_ and _n_, and there exist unique integers _x_ and _y_ such that _mx_ + _ny_ = gcd(_m_, _n_)
99
100 Coprime numbers have gcd of 1.
101
102 _mx_ + _ny_ = 1 mod _n_. But _ny_ mod _n_ = 0, so _mx_ = 1 mod _m_, so _m_ = _x_<sup>-1</sup>.
103
104 Perhaps the algorithm for finding gcds could be useful?
105
106 ---
107
108 # Euclid's algorithm
109
110 .float-right[![right-aligned GCD](gcd.svg)]
111
112 World's oldest algorithm.
113
114 _a_ = _qb_ + _r_ ; gcd(_a_, _b_) = gcd(_qb_ + _r_, _b_) = gcd(_r_, _b_) = gcd(_b_, _r_)
115
116 Repeatedly apply these steps until _r_ = 0, when the other number = gcd(_a_, _b_). For instance, _a_ = 81, _b_ = 57
117
118 * 81 = 1 × 57 + 24
119 * 57 = 2 × 24 + 9
120 * 24 = 2 × 9 + 6
121 * 9 = 1 × 6 + 3
122 * 6 = 2 × 3 + 0
123
124 Therefore, gcd(81, 57) = 3 and 81_x_ + 57_y_ = 3
125
126 Now unfold the derivation to find _x_ and _y_
127
128 * 3 = 9 × 1 + 6 × -1
129 * 3 = 9 × 1 + (24 - 2 × 9) × -1 = 9 × 3 + 24 × -1
130 * 3 = (57 - 2 × 24) × 3 + 24 × -1 = 57 × 3 + 24 × -7
131 * 3 = 57 × 3 + (81 - 57 × 1) × -7 = 57 × 10 + 81 × -7
132
133 Can we do this in one pass?
134
135 ---
136
137 # Hands up if you're lost
138
139 ## (Be honest)
140
141 ---
142
143 # Triple constraints
144
145 .float-right[![right-aligned GCD](fast-good-cheap.gif)]
146
147 ## Fast, cheap, good: pick two
148
149 ## Programmer time, execution time, space: pick one, get some of another.
150
151 (Scripting languages like Python are popular because they reduce programmer time. Contrast with Java and Pascal.)
152
153 Extended Euclid's algorithm has lots of programmer time (and risk of bugs), but will take virtually no space (6 numbers).
154
155 Can we trade space for ease?
156
157 A standard technique is memoisation: store the results somewhere, then just look them up.
158
159 ---
160
161 # Modular multiplication table for 7
162
163 (7) | 0 | 1 | 2 | 3 | 4 | 5 | 6
164 ----|---|---|---|---|---|---|---
165 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
166 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6
167 2 | 0 | 2 | 4 | 6 | 1 | 3 | 5
168 3 | 0 | 3 | 6 | 2 | 5 | 1 | 4
169 4 | 0 | 4 | 1 | 5 | 2 | 6 | 3
170 5 | 0 | 5 | 3 | 1 | 6 | 4 | 2
171 6 | 0 | 6 | 5 | 4 | 3 | 2 | 1
172
173 Can use this to find the multiplicative inverses.
174
175 (7) | 1 | 2 | 3 | 4 | 5 | 6
176 ----|---|---|---|---|---|---
177 | 1 | 4 | 5 | 2 | 3 | 6
178
179 How much to store?
180
181 ---
182
183 # How much to store?
184
185 1. The inverses for this modular base.
186 2. The inverses for all bases (12 of them)
187 3. All the _x_ ÷ _y_ = _z_ mod _n_ triples...
188 4. ...for all _n_
189 5. The decipherment table for this key
190 6. The decipherment table for all keys
191
192 The choice is a design decision, taking into account space needed, time to create and use, expected use patterns, etc.
193
194 ## Thoughts?
195
196 ---
197
198 # How much to store?
199
200 Keeping the decipherment close to encipherment seems aesthetically better to me.
201
202 Giving the ability to do division is the most obvious (to me).
203
204 As there are only a few possible modular bases, might as well calculate the whole table at startup.
205
206 ## Now implement affine decipherment.
207
208 Check both enciphering and deciphering work. Round-trip some text.
209 ---
210
211 # Counting from 0 or 1
212
213 When converting letters to numbers, we're using the range 0-25.
214
215 Another convention is to use numbers in range 1-26.
216
217 Implement this.
218
219 * You'll need another parameter:
220 ```python
221 affine_encipher_letter(letter, multiplier, adder, one_based=False)
222 ```
223
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