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:
- Diffie Hellman Key Exchange (Algorithm 1 in the paper)
- 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.
