Theorem. In any triangle ABC, there is a correspondance between the length of a side and the measure of the angle opposite that side: The longest side is opposite the greatest angle, and vica versa: the greatest angle is opposite the longest side. The shortest side is opposite the smallest angle, and vica versa: the smallest angle is opposite the shortest side. So, the order between the three sides is the same as the order between the corresponding angles, and vica versa. We recommend that sides in triangles are tracked by their corresponding sides. This is because we can perceive the difference in angles much better than in side lengths.
Theorem. 2. The expected revenue of the sequence of deals found by the AAG algorithm with XXX is at least 1 CAW(S, I) with additive correlation.
Theorem. A-TGDH satisfies perfect forward secrecy, known- key security, and key authentication.
Theorem. In a right triangle, there can only be one right angle and it is the greatest angle in the triangle. Discussion
Theorem. The perimeter of a triangle ABC can be computed as P = a + b + c. When we reviewed the area formula for rectangles, we have mentioned that that was a very difficult formula to prove. It is often the case in mathematics that, once we worked very hard for a formula, we use that result over and over. When it comes to area, all we know are rectangles. In other words, every area formula was derived from the area formula of the rectangle. Every right triangle is half of a rectangle. In other words, given any right triangle, we can use two identical copies of it to form a rectangle. We know how to find the area of the rectangle: A = ab. Because the rectangle consists of two identical (also called congruent) right triangles, it naturally follows that each takes up half of the area of the rectangle. Thus the area of the right triangle is A = ab . .
Theorem. The area of a right triangle ABC, where c denotes the hypotenuse, is A = ab . What is unusual about this formula is that we don't need the length of the hypotenuse, only the lengths of the other two sides. This means that given the three sides of a right triangle, we need to know to only use the lengths of the two shorter sides.
Theorem. Let n be either an integer larger than 10400 or a prime power such that n > 11 and n ∈/ {33, 24, 25, 26}. Then, over C, all the automorphisms of a Cartan curve of level n are modular. For each Cartan or Cartan-plus subgroup H < GL2(Z/nZ), the group of modular automorphisms of the modular curve associated to H is easy to compute: it is either isomorphic to N'/H' × Z/2Z or to N'/H', where N' < SL2(Z/nZ) is the normalizer of H' := H ∩ SL2(Z/nZ). This is stated more precisely in Proposition 3.6.13. Remark
Theorem. Let n be a positive integer. Then the jacobian of a Cartan curve of level n is a quotient of the jacobian of the modular curve X0(n2). Using the last theorem and a result of Xxxxxxx characterizing the CM sub-abelian varieties of J0(n2), we prove that, for all but finitely many n, a large part of the jacobian of a Cartan curve does not contain any CM sub-abelian variety. This, using a result of Xxxxx, implies that all the automorphisms of a Cartan curve of level n are defined over a compositum of quadratic fields for all but finitely many n. The main result of chapter 3 then follows from Xxxxxxxxxx’x lower bound of the gonality of modular curves and the following criterion. Lemma. Let n be a positive integer and let X be the base change to C of a modular curve associated with a subgroup H < GL2(Z/nZ). Suppose that H contains the scalar matrices, that det(H) is the whole (Z/nZ)× and that there are two primes l1 < l2 not dividing n such that 5 ≤ l2 < 1 gon(X) − 1, with gon(X) the gonality of X. Then every automorphism of X which is defined over a compositum of quadratic fields is modular. For an automorphism u : X → X to be modular it is necessary and sufficient that u preserves the set of cusps, so that u restricts to an automorphism of the non-cuspidal locus Y , and preserves the set of elliptic points, namely the branch points of the map H → Y (C).
Theorem. There exists a probabilistic algorithm, described in Section 4.4, that solves the discrete logarithm problem in K× for all finite fields K of small characteristic (namely the fields Fpn with n > p) in expected time (log #K)O(log log #K) . Our algorithm uses some ideas of the algorithm in [19], whose running time is only heuristic, and adapts them to finite fields with a different type of presentation. Let Fq be a finite field with q > 2 elements, let E/Fq be an elliptic curve and let P1 be a point on E such that φ(P1) − P1 ∈ E(Fq), where φ : E → E is the q-th Frobenius. If K = Fq(P1), then the coordinates of P1 are generators of the extension Fq ⊂ K on which the q-th Frobenius acts “simply”. If this happens and if, moreover [K : Fq] > 2, the elliptic curve E and the point P1 give an elliptic presentation of K. Given the abundance of elliptic curves over Fq, for q big enough, it is easy to prove that every finite field of small characteristic can be embedded in a slightly larger field admitting an elliptic presentation such that q is small compared to #K. A more precise statement is given in Proposition 4.1.5. Given a finite field K with an elliptic presentation, we represent elements in K× as f (P1) with f varying among the rational functions in Fq(E) that are regular and non- vanishing on P1. Hence, we extend the discrete logarithm to these rational functions and, in a weak sense, to divisors on E. Notice that each divisor defined over Fq is a linear combination of irreducible divisors, namely those divisors that are the sum, with multiplicity 1, of all the GFq -conjugates of a point in E(Fq). Our algorithm is an index calculus using divisors: the idea is looking for linear re- lations among the discrete logarithm of h and the “discrete logarithms” of irreducible divisors of small degree; when many relations are found, we compute the discrete loga- rithm of h by solving a linear system. We find relations using a descent procedure, which, given an irreducible divisor D of degree 4d ≥ 320, computes irreducible divisors Di of degree dividing 2d such that the “discrete logarithm” of D is a linear combination of the “discrete logarithms” of the Di’s. Most of the last chapter is devoted to the description and the proof of the correctness of this descent procedure. It mainly uses the following equalities f (P1)q = fφ(φ(P1)) = fφ(P1 + P0) = fφ ◦ τP0 (P1) ,
Theorem. If n is odd, then log T res,β(ρ) = 0 ∀β ∈ Rn+1. If n is even, X log T res,β(ρ) is a smooth invariant if and only if β equals:
