## Miscellaneous

Here is a collection of some miscellaneous – in some cases research related – content.

### Best Spelling Suggestions for Math Terminology

Over the years, I have written several math-related texts. I use Emacs for my LaTeX editing, and do the spell-checking with the internal ispell interface. Clearly, certain math terms are not provided in standard dictionaries. However, ispell is suggesting different words to be used when it encounters an unknown one. And especially for math terms, these suggestions are often hillarious. Here is a table of my favourite spelling suggestions. If you find more of these, please let me know and I will add them to this list. I will leave it up to the reader to find out the meanings of the respective words :-)
 Math Term Spelling suggestion adic addict affine effing ansatz unseat arithmetics asthmatics automorphism metamorphism bivariate bavaria cardinality carnality, cordiality combinatorial gubernatorial combinatorics dominatrix coprime caprice cryptosystem criticizm eigenring ignoring, gingering factorizations cauterizations filtrations flirtations finiteness faintness gradings gratins homomorphism Mormonism homotopy hometown injective invective invertible convertible, inevitable, infertile irreducibility irascibility monoid mooned monomials monorails nonabelian nobbling, nobleman otimes otiose polytope polyp projective projectile quantizations canonizations reducibility risibility scalability solubility, gullibility, scrabbled subvariety sobriety summand summoned surjection dejection tuples topless univariate unvaried, infuriate, inebriate verifier versifier

### Crypto Challenges

Based on the experimental implementation described in this paper (joint work with Reinhold Burger), we have created a set of challenges for people to break our crypto-system for different sizes for the private keys.

The challenges cover two of the presented protocols:
1. Diffie Hellman Key Exchange (Algorithm 1 in the paper)
2. Three Pass (Algorithm 2 in the paper)
The implementation that we used to create the challenges for different degrees can be found on GitHub.

We chose for all challenges $\mathbb{F}_{125}$ (viewed as $\mathbb{F}_5[\alpha]/\langle \alpha^3 + 3\alpha + 3\rangle$) as base-ring. The noncommutative ring is given by $\mathbb{F}_{125}[\partial_1, \partial_2; \sigma_1, \sigma_2]$, where $$\sigma_1: \mathbb{F}_5(\alpha) \to \mathbb{F}_5(\alpha), a_0 + a_1\alpha + a_2 \alpha^2 \mapsto a_0 + a_1 + a_2 + 3a_2\alpha + (3a_1 + 4a_2)\alpha^2$$ $$\sigma_2: \mathbb{F}_5(\alpha) \to \mathbb{F}_5(\alpha), a_0 + a_1\alpha + a_2 \alpha^2 \mapsto a_0 + 4a_1 + 3a_2 + (4a_1 + 2a_2)\alpha + 2a_1\alpha^2.$$ Furthermore, we are using the method depicted by Equation (3) in the paper to generate the commuting subsets $\mathcal{C}_l$ and $\mathcal{C}_r$.

Ad i: We assume that the eavesdropper (referred to as Eve) had full access to the communication between our communicating parties (referred to as Alice and Bob). This means, that the following information is known to Eve:
• The public parameters $L$, $P$ and $Q$.
• The message Alice sends to bob (i.e. $P_A\cdot L \cdot Q_A$).
• The message Bob sends to Alice (i.e. $P_B \cdot L \cdot Q_B$).
If one of the following is successfully obtained by an attacker, the respective challenge has been solved.
• At least one of the private parameters $P_A$, $P_B$, $Q_A$ or $Q_B$ has been obtained.
• The secret key $P_A \cdot P_B \cdot L \cdot Q_B \cdot Q_A$ has been correctly calculated using the provided information.
Here are the challenges, with difficulty in increasing order.
 ORE DIFFIE HELLMAN (ODH) CHALLENGES Problem file Description Status Challenge_Diffie_1 (~1.7MB) Mediocre security; we assume that this challenge may be solved before 2017. UNSOLVED Challenge_Diffie_2 (~4.8MB) Decent security; we assume that this challenge would not be solved before 2017. UNSOLVED Challenge_Diffie_3 (~16MB) Tough challenge at the current state of research. UNSOLVED Challenge_Diffie_4 (~100MB) We consider this to be almost impossible to solve in a feasible amount of time. UNSOLVED Challenge_Diffie_5 (~400MB) If this is ever solved, we will accuse you of cheating ;-). But we will be very interested in how you have done this. UNSOLVED
Format: The challenge files for ODH are formatted in the following way, where %s denotes a string representing the respective polynomial:

Base-Field: GF(125)
L: %s
P: %s
Q: %s
Message Alice to Bob:
%s
Message Bob to Alice:
%s


Ad ii: We assume that the eavesdropper (referred to as Eve) had full access to the communication between our communicating parties (referred to as Alice and Bob). This means, that the following information is known to Eve:
• The public parameters $P$ and $Q$.
• The initial message Alice sends to bob (i.e. $P_A\cdot L \cdot Q_A$).
• The message Bob sends back to Alice (i.e. $P_B \cdot P_A \cdot L \cdot Q_A\cdot Q_B$).
• The message Alice sends Bob in the end (i.e. $P_B \cdot L \cdot Q_B$).
If one of the following is successfully obtained by an attacker, the respective challenge has been solved.
• At least one of the private parameters $P_A$, $P_B$, $Q_A$ or $Q_B$ has been obtained.
• $L$ has been correctly calculated using the provided information.
Here are the challenges, with difficulty in increasing order.
 ORE THREE PASS PROTOCOL (OTPP) CHALLENGES Problem file Description Status Challenge_Threepass_1 (~5.4MB) Mediocre security; we assume that this challenge may be solved before 2017. UNSOLVED Challenge_Threepass_2 (~13MB) Decent security; we assume that this challenge would not be solved before 2017. UNSOLVED Challenge_Threepass_3 (~51MB) Tough challenge at the current state of research. UNSOLVED Challenge_Threepass_4 (~304MB) We consider this to be almost impossible to solve in a feasible amount of time. UNSOLVED Challenge_Threepass_5 (~1.2GB) If this is ever solved, we will accuse you of cheating ;-). But we will be very interested in how you have done this. UNSOLVED
Format: The challenge files for OTPP are formatted in the following way, where %s denotes a string representing the respective polynomial:

Base-Field: GF(125)
P: %s
Q: %s
Message Alice sends to Bob:
%s
Message Bob sends back to Alice:
%s
Message Alice sends back to Bob:
%s


Prizes: Everyone who solves one of the challenges will be named on this website (unless the person/team rather prefers to remain anonymous). Furthermore, we would like to have discussions with the people solving these challenges, which can be held during a meal (with drinks) that we would cook or provide.