# Affine ciphers
-## Explanation of extended Euclid's algorithm from [Programming with finite fields](http://jeremykun.com/2014/03/13/programming-with-finite-fields/)
+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
+--|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--
+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
-**Definition:** An element _d_ is called a greatest common divisor (gcd) of _a, b_ if it divides both _a_ and _b_, and for every other _z_ dividing both _a_ and _b_, _z_ divides _d_.
+An extension of Caesar ciphers
-**Theorem:** For any two integers _a, b_ there exist unique integers _x, y_ such that _ax_ + _by_ = gcd(_a, b_).
+* Count the gaps in the letters.
-We could beat around the bush and try to prove these things in various ways, but when it comes down to it there’s one algorithm of central importance that both computes the gcd and produces the needed linear combination _x, y_. The algorithm is called the Euclidean algorithm. Here is a simple version that just gives the gcd.
+---
+# How affine ciphers work
-```python
-def gcd(a, b):
- if abs(a) < abs(b):
- return gcd(b, a)
-
- while abs(b) > 0:
- q,r = divmod(a,b)
- a,b = b,r
-
- return a
-```
+_ciphertext_letter_ =_plaintext_letter_ × a + b
-This works by the simple observation that gcd(_a_, _aq_ + _r_) = gcd(_a_, _r_) (this is an easy exercise to prove directly). So the Euclidean algorithm just keeps applying this rule over and over again: take the remainder when dividing the bigger argument by the smaller argument until the remainder becomes zero. Then gcd(_x_, 0) = 0 because everything divides zero.
+* Convert letters to numbers
+* Take the total modulus 26
-Now the so-called ‘extended’ Euclidean algorithm just keeps track of some additional data as it goes (the partial quotients and remainders). Here’s the algorithm.
+# Enciphering is easy
-```python
-def extendedEuclideanAlgorithm(a, b):
- if abs(b) > abs(a):
- (x,y,d) = extendedEuclideanAlgorithm(b, a)
- return (y,x,d)
-
- if abs(b) == 0:
- return (1, 0, a)
-
- x1, x2, y1, y2 = 0, 1, 1, 0
- while abs(b) > 0:
- q, r = divmod(a,b)
- x = x2 - q*x1
- y = y2 - q*y1
- a, b, x2, x1, y2, y1 = b, r, x1, x, y1, y
-
- return (x2, y2, a)
-```
+* Build the `affine_encipher()` function
-Indeed, the reader who hasn’t seen this stuff before is encouraged to trace out a run for the numbers 4864, 3458. Their gcd is 38 and the two integers are 32 and -45, respectively.
+---
-How does this help us compute inverses? Well, if we want to find the inverse of _a_ modulo _p_, we know that their gcd is 1. So compute the _x, y_ such that _ax_ + _py_ = 1, and then reduce both sides mod _p_. You get _ax_ + 0 = 1 _mod p_, which means that _x mod p_ is the inverse of _a_. So once we have the extended Euclidean algorithm our inverse function is trivial to write!
+# Deciphering affine ciphers is harder
-```python
-def inverse(self):
- x,y,d = extendedEuclideanAlgorithm(self.n, self.p)
- return IntegerModP(x)
-```
+`$$p = \frac{c - b}{a} \mod 26$$`
+
+But modular division is hard!
+
+Define division as mutiplication by the inverse: `\(\frac{x}{y} = x \times \frac{1}{y} = x \times y^{-1}\)`
+
+A number _x_, when multiplied by its inverse _x_<sup>-1</sup>, gives result of 1.
+
+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).
+
+Result from number theory: only numbers coprime with _n_ have multiplicative inverses in arithmetic mod _n_.
+
+Another result from number theory: for non-negative integers _a_ and _n_, and there exist unique integers _x_ and _y_ such that _ax_ + _ny_ = gcd(_a_, _b_)
+
+Coprime numbers have gcd of 1.
+
+_ax_ + _ny_ = 1 mod _n_. But _ny_ mod _n_ = 0, so _ax_ = 1 mod _n_, so _a_ = _x_<sup>-1</sup>.
+
+Perhaps the algorithm for finding gcds could be useful?
+
+---
+
+# Euclid's algorithm
+
+.float-right[]
+
+World's oldest algorithm.
+
+_a_ = _qb_ + _r_ ; gcd(_a_, _b_) = gcd(_qb_ + _r_, _b_) = gcd(_r_, _b_) = gcd(_b_, _r_)
+
+Repeatedly apply these steps until _r_ = 0, when the other number = gcd(a,b). For instance, _a_ = 81, _b_ = 57
+
+* 81 = 1 × 57 + 24
+* 57 = 2 × 24 + 9
+* 24 = 2 × 9 + 6
+* 9 = 1 × 6 + 3
+* 6 = 2 × 3 + 0
+
+Therefore, gcd(81, 57) = 3 and 81_x_ + 57_y_ = 3
+
+Now unfold the derivation to find _x_ and _y_
+
+* 3 = 9 × 1 + 6 × -1
+* 3 = 9 × 1 + (24 - 2 × 9) × -1 = 9 × 3 + 24 × -1
+* 3 = (57 - 2 × 24) × 3 + 24 × -1 = 57 × 3 + 24 × -7
+* 3 = 57 × 3 + (81 - 57 × 1) × -7 = 57 × 10 + 81 × -7
+
+Can we do this in one pass?
+
+---
+
+# Hands up if you're lost
+
+## (Be honest)
+
+---
-And indeed it works as expected:
+# Triple constraints
+## Fast, cheap, good: pick two
+
+## Programmer time, execution time, space: pick one, get some of another.
+
+(Scripting languages like Python are popular because they reduce programmer time. Contrast with Java and Pascal.)
+
+Extended Euclid's algorithm has lots of programmer time (and risk of bugs), but will take virtually no space (6 numbers).
+
+Can we trade space for ease?
+
+A standard technique is memoisation: store the results somewhere, then just look them up.
+
+---
+
+# Modular multiplication table for 7
+
+(7) | 0 | 1 | 2 | 3 | 4 | 5 | 6
+----|---|---|---|---|---|---|---
+ 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
+ 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6
+ 2 | 0 | 2 | 4 | 6 | 1 | 3 | 5
+ 3 | 0 | 3 | 6 | 2 | 5 | 1 | 4
+ 4 | 0 | 4 | 1 | 5 | 2 | 6 | 3
+ 5 | 0 | 5 | 3 | 1 | 6 | 4 | 2
+ 6 | 0 | 6 | 5 | 4 | 3 | 2 | 1
+
+Can use this to find the multiplicative inverses.
+
+(7) | 1 | 2 | 3 | 4 | 5 | 6
+----|---|---|---|---|---|---
+ | 1 | 4 | 5 | 2 | 3 | 6
+
+How much to store?
+
+---
+
+# How much to store?
+
+1. The inverses for this modular base.
+2. The inverses for all bases (12 of them)
+3. All the _x_ ÷ _y_ = _z_ mod _n_ triples...
+4. ...for all _n_
+5. The decipherment table for this key
+6. The decipherment table for all keys
+
+The choice is a design decision, taking into account space needed, time to create and use, expected use patterns, etc.
+
+## Thoughts?
+
+---
+
+# How much to store?
+
+Keeping the decipherment close to encipherment seems aesthetically better to me.
+
+Giving the ability to do division is the most obvious (to me).
+
+As there are only a few possible modular bases, might as well calculate the whole table at startup.
+
+## Now implement affine decipherment.
+
+Check both enciphering and deciphering work. Round-trip some text.
+---
+
+# Counting from 0 or 1
+
+When converting letters to numbers, we're using the range 0-25.
+
+Another convention is to use numbers in range 1-26.
+
+Implement this.
+
+* You'll need another parameter:
```python
->>> mod23 = IntegersModP(23)
->>> mod23(7).inverse()
-10 (mod 23)
->>> mod23(7).inverse() * mod23(7)
-1 (mod 23)
+affine_encipher_letter(letter, multiplier, adder, one_based=False)
```
-
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projects / cipher-training.git / blobdiff