Quadratic formula | EPFL Graph Search

In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others. Given a general quadratic equation of the form ax^2+bx+c = 0 whose discriminant b^2 - 4ac is positive, with x representing an unknown, with a, b and c representing constants, and with a ≠ 0, the quadratic formula is: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where the plus–minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become: \begin{align} x_1 &= \frac{-b + \sqrt {b^2-4ac}}{2a}\quad\text{and} \ x_2 &= \frac{-b - \sqrt {b^2-4ac}}{2a} \end{align} Each of these two solutions is als

Attacks on some post-quantum cryptographic protocols: The case of the Legendre PRF and SIKE

Novak Kaluderovic

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.

EPFL2022

Minimum euclidien des ordres maximaux dans les algèbres centrales à division

Jérôme Chaubert

This work concerns the study of Euclidean minima of maximal orders in central simple algebras. In the first part, we define the concept of ideal lattice in the non-commutative case. Let A be a semi-simple algebra over Q. An ideal lattice over A is a triple (I, α, τ) where I is an ideal of A, α is a unit in AR = A ⊗Q R fixed by τ and τ is a positive involution on AR, in other words, trAR/R(xαxτ) > 0 for all x ∈ IR. We study ideal lattices, especially their Hermite invariant, to link the concepts of Euclidean minimum of a maximal order and ideal lattices of the form (Λ, α, τ). Namely, we show that we can bound the Euclidean minimum of Λ by the Hermite invariant of the ideal lattices of Λ. This upper bound allows us to calculate the Euclidean minima of some infinite families of maximal orders, when A is a quaternion skew field over Q. In the second part, we look for other ways to find upper and lower bounds of the euclidean minimum of a maximal order in a quaternion skew field. This research leads us to an exhaustive list of euclidean quaternion skew fields over Q, and their euclidean minima. The quadratic case has also been studied but remains partially unsolved. At the end, we give bounds for Euclidean minima of some maximal orders of quaternion skew fields over cyclotomic fields.

EPFL2007

Conformal Invariance of Spin Pattern Probabilities in the Planar Ising Model

Clément Hongler, Sung Chul Park

We study the 2-dimensional Ising model at critical temperature on a smooth simply-connected graph Ω.We rigorously prove the conformal invariance of arbitrary spin-pattern probabilities centered at a point a and derive formulas to compute the probabilities as functions of the conformal map from Ω to the unit disk. Our methods extend those of [Hon10] and [CHI13] which proved conformal invariance of energy densities and spin correlations for points fixed far apart from each other. We use discrete complex analysis techniques and construct a discrete multipoint fermionic observable that takes values related to pattern probabilities in the planar Ising model. Refined analysis of the convergence of the discrete observable to a continuous, conformally covariant function completes the result.

2015

Attacks on some post-quantum cryptographic protocols: The case of the Legendre PRF and SIKE

Novak Kaluderovic

EPFL2022