Password Entropy worksheet

1. Entropy: 77.79 bits

1. Each word is one random pick from the list, so the pool size N is the number of words in the list

   $N$$=\mathrm{8,000}$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(\mathrm{8,000})$$=12.965784\text{\hspace{0.17em}bits}$

3. Multiply by the number of words, because every random pick adds the same bits. This gives the entropy H

   $H$$=6\times 12.965784$$=77.794706\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{77.79}}{2\times \mathrm{10,000,000,000}}$$=\mathrm{13,107,200,000,000.03}\text{\hspace{0.17em}s}$$=\text{about 415,342 years}$

2. Entropy: 62.99 bits

1. Each word is one random pick from the list, so the pool size N is the number of words in the list

   $N$$=\mathrm{6,200}$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(\mathrm{6,200})$$=12.598053\text{\hspace{0.17em}bits}$

3. Multiply by the number of words, because every random pick adds the same bits. This gives the entropy H

   $H$$=5\times 12.598053$$=62.990263\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{62.99}}{2\times \mathrm{10,000,000,000}}$$=\mathrm{458,066,416}\text{\hspace{0.17em}s}$$=\text{about 14.5 years}$

3. Entropy: 65.5 bits

1. Add up the sizes of the chosen sets (lowercase, uppercase, digits) to get the pool size

   $N$$=26+26+10$$=62$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(62)$$=5.954196\text{\hspace{0.17em}bits}$

3. Multiply by the number of characters, because every random pick adds the same bits. This gives the entropy H

   $H$$=11\times 5.954196$$=65.496159\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{65.5}}{2\times \mathrm{10,000,000,000}}$$=\mathrm{2,601,828,034.191858}\text{\hspace{0.17em}s}$$=\text{about 82.4 years}$

4. Entropy: 77.4 bits

1. Add up the sizes of the chosen sets (lowercase, uppercase, digits) to get the pool size

   $N$$=26+26+10$$=62$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(62)$$=5.954196\text{\hspace{0.17em}bits}$

3. Multiply by the number of characters, because every random pick adds the same bits. This gives the entropy H

   $H$$=13\times 5.954196$$=77.404552\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{77.4}}{2\times \mathrm{10,000,000,000}}$$=\mathrm{10,001,426,963,433.426}\text{\hspace{0.17em}s}$$=\text{about 316,926 years}$

5. Entropy: 79.73 bits

1. Each word is one random pick from the list, so the pool size N is the number of words in the list

   $N$$=\mathrm{10,000}$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(\mathrm{10,000})$$=13.287712\text{\hspace{0.17em}bits}$

3. Multiply by the number of words, because every random pick adds the same bits. This gives the entropy H

   $H$$=6\times 13.287712$$=79.726274\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{79.73}}{2\times \mathrm{10,000,000,000}}$$=\mathrm{49,999,999,999,999.984}\text{\hspace{0.17em}s}$$=\text{about 1,584,404 years}$

6. Entropy: 64.46 bits

1. Each word is one random pick from the list, so the pool size N is the number of words in the list

   $N$$=\mathrm{7,600}$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(\mathrm{7,600})$$=12.891784\text{\hspace{0.17em}bits}$

3. Multiply by the number of words, because every random pick adds the same bits. This gives the entropy H

   $H$$=5\times 12.891784$$=64.458919\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{64.46}}{2\times \mathrm{10,000,000,000}}$$=\mathrm{1,267,762,688.000006}\text{\hspace{0.17em}s}$$=\text{about 40.2 years}$

7. Entropy: 71.45 bits

1. Add up the sizes of the chosen sets (lowercase, uppercase, digits) to get the pool size

   $N$$=26+26+10$$=62$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(62)$$=5.954196\text{\hspace{0.17em}bits}$

3. Multiply by the number of characters, because every random pick adds the same bits. This gives the entropy H

   $H$$=12\times 5.954196$$=71.450356\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{71.45}}{2\times \mathrm{10,000,000,000}}$$=\mathrm{161,313,338,119.8938}\text{\hspace{0.17em}s}$$=\text{about 5,112 years}$

8. Entropy: 80.55 bits

1. Each word is one random pick from the list, so the pool size N is the number of words in the list

   $N$$=\mathrm{11,000}$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(\mathrm{11,000})$$=13.425216\text{\hspace{0.17em}bits}$

3. Multiply by the number of words, because every random pick adds the same bits. This gives the entropy H

   $H$$=6\times 13.425216$$=80.551295\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{80.55}}{2\times \mathrm{10,000,000,000}}$$=\mathrm{88,578,049,999,999.89}\text{\hspace{0.17em}s}$$=\text{about 2,806,869 years}$

9. Entropy: 64.83 bits

1. Each word is one random pick from the list, so the pool size N is the number of words in the list

   $N$$=\mathrm{8,000}$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(\mathrm{8,000})$$=12.965784\text{\hspace{0.17em}bits}$

3. Multiply by the number of words, because every random pick adds the same bits. This gives the entropy H

   $H$$=5\times 12.965784$$=64.828921\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{64.83}}{2\times \mathrm{10,000,000,000}}$$=\mathrm{1,638,399,999.999998}\text{\hspace{0.17em}s}$$=\text{about 51.9 years}$

10. Entropy: 90.5 bits

1. Each word is one random pick from the list, so the pool size N is the number of words in the list

   $N$$=\mathrm{7,800}$

2. Find the bits each pick adds: log₂(N), the number of times you double 1 to reach the pool size N

   $\mathrm{log₂}(\mathrm{7,800})$$=12.929258\text{\hspace{0.17em}bits}$

3. Multiply by the number of words, because every random pick adds the same bits. This gives the entropy H

   $H$$=7\times 12.929258$$=90.504809\text{\hspace{0.17em}bits}$

4. Estimate the guessing time. On average an attacker finds it halfway through all the possibilities. At an assumed 10,000,000,000 guesses per second, that takes

   $\frac{2^{90.5}}{2\times \mathrm{10,000,000,000}}$$=8.782784e+16\text{\hspace{0.17em}s}$$=\text{about 2,783,096,442 years}$