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Concept# Möbius transformation

Summary

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac{a z + b}{c z + d}
of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.
Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane.
These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.
The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2, C). Together with its subgroups, it has nume

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Post-quantum cryptography is a branch of cryptography which deals with cryptographic algorithms whose hardness assumptions are not based on problems known to be solvable by a quantum computer, such as the RSA problem, factoring or discrete logarithms.This thesis treats two such algorithms and provides theoretical and practical attacks against them.The first protocol is the generalised Legendre pseudorandom function - a random bit generator computed as the Legendre symbol of the evaluation of a secret polynomial at an element of a finite field. We introduce a new point of view on the protocol by analysing the action of the group of Möbius transformations on the set of secret keys (secret polynomials).We provide a key extraction attack by creating a table which is cubic in the number of the function queries, an improvement over the previous algorithms which only provided a quadratic yield. Furthermore we provide an ever stronger attack for a new set of particularly weak keys.The second protocol that we cover is SIKE - supersingular isogeny key encapsulation.In 2017 the American National Institute of Standards and Technology (NIST) opened a call for standardisation of post-quantum cryptographic algorithms. One of the candidates, currently listed as an alternative key encapsulation candidate in the third round of the standardisation process, is SIKE.We provide three practical side-channel attacks on the 32-bit ARM Cortex-M4 implementation of SIKE.The first attack targets the elliptic curve scalar multiplication, implemented as a three-point ladder in SIKE. The lack of coordinate randomisation is observed, and used to attack the ladder by means of a differential power analysis algorithm.This allows us to extract the full secret key of the target party with only one power trace.The second attack assumes coordinate randomisation is implemented and provides a zero-value attack - the target party is forced to compute the field element zero, which cannot be protected by randomisation. In particular we target both the three-point ladder and isogeny computation in two separate attacks by providing maliciously generated public keys made of elliptic curve points of irregular order.We show that an order-checking countermeasure is effective, but comes at a price of 10% computational overhead. Furthermore we show how to modify the implementation so that it can be protected from all zero-value attacks, i.e., a zero-value is never computed during the execution of the algorithm.Finally, the last attack targets a point swapping procedure which is a subroutine of the three-point ladder. The attack successfully extracts the full secret key with only one power trace even if the implementation is protected with coordinate randomisation or order-checking. We provide an effective countermeasure --- an improved point swapping algorithm which protects the implementation from our attack.

Cheryl Shaulya Lau, Nicolas Perdu, Gaurav Sharma, Sabine Süsstrunk

Color deficient individuals have trouble seeing color contrasts that could be very apparent to individuals with normal color vision. For example, for some color deficient individuals, red and green apples do not have the striking contrast they have for those with normal color vision, or the abundance of red cherries in a tree is not immediately clear due to a lack of perceived contrast. We present a smartphone app that enables color deficient users to visualize such problematic color contrasts in order to help them with daily tasks. The user interacts with the app through the touchscreen. As the user traces a path around the touchscreen, the colors in the image change continuously via a transform that enhances contrasts that are weak or imperceptible for the user under native viewing conditions. Specifically, we propose a transform that shears the data along lines parallel to the dimension corresponding to the affected cone sensitivity of the user. The amount and direction of shear are controlled by the user'sfinger movement over the touchscreen allowing them to visualize these contrasts. Using the GPU, this simple transformation, consisting of a linear shear and translation, is performed efficiently on each pixel and in real-time with the changing position of the user's finger. The user can use the app to aid daily tasks such as distinguishing between red and green apples or picking out ripe bananas.

2015