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Entropy 16x

The equilibrium between the rat brain tubulin alpha beta dimer and the dissociated alpha and beta monomers has been studied by analytical ultracentrifugation with use of a new method employing short solution columns, allowing rapid equilibration and hence short runs, minimizing tubulin decay. Simultaneous analysis of the equilibrium concentration distributions of three different initial concentrations of tubulin provides clear evidence of a single equilibrium characterized by an association constant, Ka, of 4.9 X 10(6) M-1 (Kd = 2 X 10(-7) M) at 5 degrees, corresponding to a standard free energy change on association delta G degrees = -8.5 kcal mol-1. Colchicine and GDP both stabilize the dimer against dissociation, increasing the Ka values (at 4.5 degrees C) to 20 X 10(6) and 16 X 10(6) M-1, respectively. Temperature dependence of association was examined with multiple three-concentration runs at temperatures from 2 to 30 degrees C. The van't Hoff plot was linear, yielding positive values for the enthalpy and entropy changes on association, delta S degrees = 38.1 +/- 2.4 cal deg-1 mol-1 and delta H degrees = 2.1 +/- 0.7 kcal mol-1, and a small or zero value for the heat capacity change on association, delta C p degrees. The entropically driven association of tubulin monomers is discussed in terms of the suggested importance of hydrophobic interactions to the stability of the monomer association and is compared to the thermodynamics of dimer polymerization.

Entropy 16x

Password entropy is calculated by the number of possibilities it could be to the power of the length ie. 8 character password of both upper and lowercase letters = (26*2)8, (26 characters of the alphabet * 2 for upper and lowercase).

Something important to keep in mind is that entropy of course is, in essence, "the amount of randomness" in the password. Therefore, part of why different entropy checkers will disagree is because the entropy is a measure of how the password was generated, not what the password contains. An extreme example is usually the best way to show what I mean. Imagine that my password was frphevgl.fgnpxrkpunatr.pbzPbabeZnapbar.

However, looking at that, someone might suspect that my password isn't really random at all and might realize that it is just the rot13 transformation of site name + my name. Therefore the reality is that there is no entropy in my password at all, and anyone who knows how I generate my passwords will know what my password is to every site I log in as.

This is an extreme example but I hope it gets the point across. Entropy is determined not by what the password looks like but by how it is generated. If you use some rules to generate a password for each site then your passwords might not have any entropy at all, which means that anyone who knows your rules knows your passwords. If you use lots of randomness then you have a high entropy password and it is secure even if someone knows how you make your passwords.

This isn't 26^8 (or 2^38) - because the algorithm wasn't "choose 8 random lowercase characters". The algorithm was: choose a single, very easy to remember word. How many such words are there? If you decide, "there are 200 such words", then you're looking at about 8 bits of entropy (not 38.)

Similar to the previous entry, this isn't 36^9 (or 2^47) - because the algorithm is choose a single, very easy to remember word, and then decorate it at the end with a single digit number. The entropy here is around 11 bits (not 47.)

One entropy-evaluator might say, "Hey, I have no clue how you generated this. So it must just be random characters among lowercase, punctuation, and numbers. Call it 52^17, or 97 bits of entropy (2^97.)"

Another, slightly smarter entropy-evaluator might say, "Hey, I recognize that first word, but that second string of letters is just random. So the algorithm is a single uncommon word, a punctuation, nine random letters, and then a number. So 10000 x 16 x 26^9 x 10, or 63 bits of entropy"

A third and fourth entropy-evaluator might correctly figure out the algorithm used to generate it. But the third evaluator thinks both words should come from a dictionary of 5000 words, but the fourth evaluator thinks you have to break into a 30,000 word dictionary to find them. So one comes up with 32 bits on entropy while the other thinks there are 37 bits.

Hopefully it's starting to make sense. The reason different entropy evaluators are coming up with different numbers is because they're all coming up with different evaluations on how the password was generated.

There is no way to judge whether a sequence of numbers or characters happens to be random. A machine asked to produce a random seven-digit number would produce 8675309 about once every ten million requests. A Tommy Tutone fan might offer up that number every time. If a security application happens to need a seven-digit number (perhaps it needs to be accessible via numeric pad), it might judge that 8675309 as having good entropy, unless that application happens to have been written by a Tommy Tutone fan in which case it might regard that number as being just as bad as 1111111 or 1234567.

Basically, the only thing an entropy checker can do is check whether a passcode matches any patterns that are regarded as having low entropy, and observe the lowest-entropy pattern that it matches [every passcode will match some pattern--a passcode of e.g. 23 characters will match patterns like "any other passcode of 20-25 characters" if it doesn't match anything else]. If different people are asked to produce a list of all the low-entropy passcode patterns they can think of, they'll almost certainly think of different things. What really matters, though, is whether a particular passcode would happen to match a pattern that an attacker decides to try. While there are some sets of passcodes that attackers would be particularly likely to try (e.g. 1111111 and 1234567), the choices of passcode once one gets beyond that are likely to vary significantly by attacker.

A very stupid checker may think that the password "00000000000000000000" has high entropy because it's long and the web site's algorithm for guessing is "try every combination of all characters", which would indeed take a long time to correctly guess this password. A smarter checker will try numeric passwords first, thus introducing a security penalty to this password (it'll be found faster). A smarter still guesser will try patterns first (like repeated characters or dictionary words) and rightly conclude this password is garbage.

Other answers do not use the information theory definition of entropy. The mathematical definition is defined as a function based solely on a probability distribution. It is defined for probability density function p as the sum of p(x) * -log(p(x)) of each possible outcome x.

Units of entropy are logarithmic. Typically two is used for the base of the logarithm, so we say that a stochastic system has n bits of entropy. (Other bases could be used. For base e you would instead measure entropy "nats", for natural logarithm, but base two and "bits" are more common.)

Flipping a fair coin once (in the ideal world) produces 1 bit of entropy. Flipping a coin twice produces two bits. And so on. Uniform (discrete) distributions are the simplest distributions to calculate the entropy of because every term summed is identical. You can simplify the equation for entropy of a discrete uniform variable X with n outcomes each with 1/n probability to

The entropy of non-uniform (discrete) distributions is also easy to compute. It just requires more steps. I'll use the coin flip example to relate entropy to unpredictability again. What if you instead use a biased coin? Well then the entropy is

See? The less fair (uniform) a coin flip (distribution) is the less entropy it has. Entropy is maximized for the type of coin flip which is most unpredictable: a fair coin flip. Biased coin flips still contain some entropy as long as heads and tails are both still possible. Entropy (unpredictability) decreases when the coin becomes more biased. Anything 100% certain has zero entropy.

It is important to know that it is the idea of flipping coins that has entropy, not the result of the coin flips themselves. You cannot infer from one sample x from distribution X what the probability p(x) of x is. All you know is that it is non-zero. And with just x you have no idea how many other possible outcomes there are or what their individual probabilities are. Therefore you're not able to compute entropy just by looking at one sample.

When you see a string "HTHTHT" you don't know if it came from a sequence of six fair coin flips (6 bits of entropy), a biased coin flip sequence (

For the same reason, you cannot calculate the entropy of just one password. Any tool that tells you the entropy of a password you enter is incorrect or misrepresents what they mean by entropy. (And may be harvesting passwords.) Only systems with a probability distribution can have entropy. You cannot calculate the true entropy of that system with knowledge of exact probabilities and there is no way around that.

One may be able to estimate entropy, however, but it requires still some knowledge (or assumptions) of the distribution. If you assume a k character long password was generated from one of a few uniform distributions with some alphabet A then you can estimate entropy to be log_2 (1/A). A being the size of the alphabet. A lot of password strength estimators use this (naive) method. If they see you use only lowercase then they assume A = 26. If they see a mix of upper and lower case they assume A = 52. This is why a supposed password strength calculator might tell you that "Password1" is thousands of more times secure than "password". It makes assumptions about the statistical distribution of passwords that aren't necessarily justified. 041b061a72

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