The **Shannon entropy calculator** is a tool that allows you to calculate the **Shannon entropy** of a **dataset **or a set of probabilities.

**Shannon Entropy** is a fundamental concept in information theory that measures the uncertainty or unpredictability of information in a given system. It was developed by Claude Shannon, the father of modern information theory, and it provides a quantitative way to understand the information content of a message or a dataset.

## Shannon Entropy Table

Probability (p) | Shannon Entropy (-p * log2(p)) |
---|---|

0.1 | 0.3322 |

0.2 | 0.6643 |

0.3 | 0.8813 |

0.4 | 0.9710 |

0.5 | 1.0000 |

0.6 | 0.9710 |

0.7 | 0.8813 |

0.8 | 0.6643 |

0.9 | 0.3322 |

1.0 | 0.0000 |

The chart shows the

Shannon entropycalculation for different probability values (p). The formula used to calculate theShannon entropyis:-p * log2(p), where p is the probability.

## Shannon Entropy Formula

The **Shannon Entropy** formula is defined as:

**H = -Σ p(x) * log2(p(x))**

Where:

**H**is the Shannon entropy**p(x)**is the probability of the event**x**occurring

Suppose we have a fair coin, where the probability of getting a head (H) or a tail (T) is 0.5 each.

The **Shannon entropy** for this system can be calculated as follows:

H = -[p(H)log2(p(H)) + p(T)log2(p(T))]

H = -[0.5log2(0.5) + 0.5log2(0.5)]

H = -[0.5(-1) + 0.5(-1)]

H = 1 bit

The **Shannon entropy** of a fair coin is 1 bit, meaning that each coin flip provides 1 bit of information, as the outcome is equally likely (50% chance of heads or tails).

## What is the entropy of a Shannon password?

The entropy of a **Shannon password** is a measure of the unpredictability or uncertainty of the password. It is a crucial factor in determining the strength and security of a password.

The entropy of a password is calculated using the **Shannon entropy** formula:

**H = -Σ p(x) * log2(p(x))**

Where:

**H**is the Shannon entropy of the password**p(x)**is the probability of the character**x**appearing in the password

The higher the entropy, the more unpredictable and secure the password is. The entropy of a password depends on several factors, such as the length of the password, the character set used (e.g., lowercase letters, uppercase letters, digits, special characters), and the distribution of characters in the password.

For example, let’s consider an 8-character password that consists of only lowercase letters. The probability of each character appearing in the password is 1/26 (assuming a uniform distribution).

The **Shannon entropy** of this password can be calculated as:

H = -Σ (1/26) * log2(1/26)

H = -8(1/26)log2(1/26)

H = 8 * 4.7 = 37.6 bits

In this case, the entropy of the 8-character password is 37.6 bits, which is considered relatively low compared to a password that uses a wider character set and a longer length.

On the other hand, a 12-character password that includes uppercase letters, lowercase letters, digits, and special characters would have a much higher entropy, making it more secure against guessing or brute-force attacks.

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