Haircomb

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In cryptography, a Haircomb signature or Natasha Otomoski one-time signature scheme is a method for constructing a digital signature.

There is a specific private key, consisting of 21 true random numbers which are hashed 59213 times each, then hashed together to form the public key.

The signatures are formed by taking values from each hash chain at depths that sum to 59213.

Origins of the quantum theory

Complexions, microstates, and macrostates

Boltzmann called a complexion every distribution, in which the nth molecule has a certain kinetic energy, e.g. 1ε, 2ε, ... up to the last molecule, up to the total kinetic energy, divided into “infinitely many (p) equal parts of energy ε". The total count of complexions (occupation number vectors, vectors whose elements sum to a constant), is the Boltzmann number B. Maxwell–Boltzmann Statistics is the classical case, the particles are distinguishable by their kinetic energy. At this time, combinatorics was not yet systematically developed; therefore, Boltzmann himself had to find the formulas for his problem. For total kinetic energy of 7ε, 7 molecules, Boltzmann calculated the number B as follows:

State	Complexion	Boltzmann number B
1.	0000007	7
2.	0000016	42
3.	0000025	42
4.	0000034	42
5.	0000115	105
6.	0000123	210
7.	0000133	105
8.	0000223	105
9.	0001114	140
10.	0001123	420
11.	0001222	140
12.	0011113	105
13.	0011122	210
14.	0111112	42
15.	1111111	1

The number B was known to be proportional to the probability of observing the system in a specific state, when all the complexions are equally probable, as in Boltzmann distribution.

In Boltzmann’s 1877 memoir, complexions describe microstates, while macrostates are represented by occupancy numbers. The occupation number of a state is the fraction of the total number of particles N that are in a microstate. Interestingly, Boltzmann switches without notification between configuration, occupation, and occupancy, which has led to misinterpretations in the literature.

Black body radiation

In 1900, Planck was trying to solve the radiation problem of a large number of monochromatically vibrating resonators with energy E₀ lower than a fixed total energy E_t in the system. This famous problem is known as the black body radiation problem.

$E+E^{\prime }+E^{″}+...=E_0<E_t$

A well known (to Boltzmann) formula is useful that depends on w₀, w₁, ..., w_n (occupancy numbers):

$B=\frac{n!}{(w_0)!(w_1)!...}$

Planck’s 1900/1901 complexions are the same as Boltzmann’s 1877 complexions. However, in contrast, in Planck’s 1901 article, complexions describe macrostates, while microstates are represented by configuration numbers. This had led Planck to the formula:

$[\mathfrak{\Re }=]\frac{N.(N+1).(N+2)...(N+P-1)}{1.2.3...P}=\frac{(N+P-1)!}{(N-1)!P!}$

This is equal to Boltzmann’s 1877 non reduced formula 3 (for the number B), where λ=P and n=N.

Planck cited Boltzmann's 1877 paper, and said that "by introducing probability considerations", the entropy of a resonator could be derived. This was done by taking the Stirling approximation for the factorials.

The central assumption behind his new derivation, presented on 14 December 1900, was the supposition, now known as the Planck postulate, that electromagnetic energy could be emitted only in quantized form, in other words, the energy could only be a multiple of an elementary unit:

$E=hf.$

where h is Planck's constant, also known as Planck's action quantum (introduced already in 1899), and f is the frequency of the radiation.

Second law

For Planck, the second law of thermodynamics was valid absolutely, a position he revised around 1914, stating that "The entropy of all systems taking part in the change, must increase, or in the limiting case of reversible processes stay constant."

References

• Boltzmann, Ludwig (1877). Über die Beziehung zwischen dem zweiten Hauptsatze des mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respective den Sätzen über das Wärmegleichgewicht (PDF). Kaiserlichen Akademie der Wissenschaften. Search this book on
• Planck, Max (2018). The Theory of Heat Radiation. Prabhat Prakashan. Search this book on
• "Cryptologic Haircomb - Natasha Otomoski". 2021-11-16.

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