# Alternative plaintext scoring methods --- # Back to frequency of letter counts Letter | Count -------|------ a | 489107 b | 92647 c | 140497 d | 267381 e | 756288 . | . . | . . | . z | 3575 Another way of thinking about this is a 26-dimensional vector. Create a vector of our text, and one of idealised English. The distance between the vectors is how far from English the text is. --- # Vector distances .float-right[![right-aligned Vector subtraction](vector-subtraction.svg)] Several different distance measures (__metrics__, also called __norms__): * L
2
norm (Euclidean distance): `\(\|\mathbf{a} - \mathbf{b}\| = \sqrt{\sum_i (\mathbf{a}_i - \mathbf{b}_i)^2} \)` * L
1
norm (Manhattan distance, taxicab distance): `\(\|\mathbf{a} - \mathbf{b}\| = \sum_i |\mathbf{a}_i - \mathbf{b}_i| \)` * L
3
norm: `\(\|\mathbf{a} - \mathbf{b}\| = \sqrt[3]{\sum_i |\mathbf{a}_i - \mathbf{b}_i|^3} \)` The higher the power used, the more weight is given to the largest differences in components. (Extends out to: * L
0
norm (Hamming distance): `$$\|\mathbf{a} - \mathbf{b}\| = \sum_i \left\{ \begin{matrix} 1 &\mbox{if}\ \mathbf{a}_i \neq \mathbf{b}_i , \\ 0 &\mbox{if}\ \mathbf{a}_i = \mathbf{b}_i \end{matrix} \right. $$` * L
∞
norm: `\(\|\mathbf{a} - \mathbf{b}\| = \max_i{(\mathbf{a}_i - \mathbf{b}_i)} \)` neither of which will be that useful here, but they keep cropping up.) --- # Normalisation of vectors Frequency distributions drawn from different sources will have different lengths. For a fair comparison we need to scale them. * Eucliean scaling (vector with unit length): `$$ \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \sqrt{\mathbf{x}_1^2 + \mathbf{x}_2^2 + \mathbf{x}_3^2 + \dots } }$$` * Normalisation (components of vector sum to 1): `$$ \hat{\mathbf{x}} = \frac{\mathbf{x}}{\| \mathbf{x} \|} = \frac{\mathbf{x}}{ \mathbf{x}_1 + \mathbf{x}_2 + \mathbf{x}_3 + \dots }$$` --- # Angle, not distance Rather than looking at the distance between the vectors, look at the angle between them. .float-right[![right-aligned Vector dot product](vector-dot-product.svg)] Vector dot product shows how much of one vector lies in the direction of another: `\( \mathbf{A} \bullet \mathbf{B} = \| \mathbf{A} \| \cdot \| \mathbf{B} \| \cos{\theta} \)` But, `\( \mathbf{A} \bullet \mathbf{B} = \sum_i \mathbf{A}_i \cdot \mathbf{B}_i \)` and `\( \| \mathbf{A} \| = \sum_i \mathbf{A}_i^2 \)` A bit of rearranging give the cosine simiarity: `$$ \cos{\theta} = \frac{ \mathbf{A} \bullet \mathbf{B} }{ \| \mathbf{A} \| \cdot \| \mathbf{B} \| } = \frac{\sum_i \mathbf{A}_i \cdot \mathbf{B}_i}{\sum_i \mathbf{A}_i^2 \times \sum_i \mathbf{B}_i^2} $$` This is independent of vector lengths! Cosine similarity is 1 if in parallel, 0 if perpendicular, -1 if antiparallel. --- # Which is best? | Euclidean | Normalised ---|-----------|------------ L1 | x | x L2 | x | x L3 | x | x Cosine | x | x And the probability measure! * Nine different ways of measuring fitness. ## Computing is an empircal science Let's do some experiments to find the best solution! --- # Experimental harness ## Step 1: build some other scoring functions We need a way of passing the different functions to the keyfinding function. ## Step 2: find the best scoring function Try them all on random ciphertexts, see which one works best. --- # Functions are values! ```python >>> Pletters
``` ```python def caesar_break(message, fitness=Pletters): """Breaks a Caesar cipher using frequency analysis ... for shift in range(26): plaintext = caesar_decipher(message, shift) fit = fitness(plaintext) ``` --- # Changing the comparison function * Must be a function that takes a text and returns a score * Better fit must give higher score, opposite of the vector distance norms ```python def make_frequency_compare_function(target_frequency, frequency_scaling, metric, invert): def frequency_compare(text): ... return score return frequency_compare ``` --- # Data-driven processing ```python metrics = [{'func': norms.l1, 'invert': True, 'name': 'l1'}, {'func': norms.l2, 'invert': True, 'name': 'l2'}, {'func': norms.l3, 'invert': True, 'name': 'l3'}, {'func': norms.cosine_similarity, 'invert': False, 'name': 'cosine_similarity'}] scalings = [{'corpus_frequency': normalised_english_counts, 'scaling': norms.normalise, 'name': 'normalised'}, {'corpus_frequency': euclidean_scaled_english_counts, 'scaling': norms.euclidean_scale, 'name': 'euclidean_scaled'}] ``` Use this to make all nine scoring functions.