Group of prime order

The second assumption used in the following is that groups H of prime order q exist where computing discrete logarithms is infeasible, i.e., an assumption slightly different from that mentioned in Section 2.4. (Some benefits of groups of prime order were already shown in Section 8.2.1.)... 

Construction 8.22. Let a generation scheme for subgroups of prime fields be given (see Definition 8.21a). The corresponding family of groups of prime order is defined by the following components ... 

This subsection considers the collision-intractability of tuple exponentiation. It is shown that it is infeasible to find exp -collisions for tuples of generators of groups of prime order where the discrete logarithm is hard. First, a related simpler notion, that of multiplicative relations between given generators, is defined. 

Theorem 8.41 (Collision-intractability of poly(ll)-tnple exponentiation in groups of prime order). Let a family of groups of prime order be given where the discrete logarithm is hard. For every probabilistic polynomial-time algorithm A (that tries to compute collisions) and every polynomial Qmu (determining the growth of /i as a function of k) ... 

Lemma 8.42 (Relation-intractability in groups of prime order). Let a... 

After all the mathematical and computational preparations, it is quite clear that pair exponentiation in a family of groups of prime order where the discrete logarithm is hard yields a collision-intractable family of bundling homomorphisms. (It was first used in this way in [HePe93].) It only remains to be decided how the security parameters are related, because the family of groups has only one, k, whereas the family of bundling homomorphisms has two, say k and T for distinction. The following facts are known ... 

Definition 8.48. A fixed-length encoding for a given family of groups (of prime order) has the following components ... 

Vol. 1789 Y. Sommerhauser, Yetter-Drinfel d Hopf algebras over groups of prime order (2002)... 

If p - 1 has a large prime factor q, then 2p has a rmique cyclic subgroup of prime order q. This group is called According to Section 8.1.1, it can be specified as... 

Every finitely generated abelian group can be written as a direct product of cyclic groups of prime-power order with a finite number of infinite cyclic groups. In this presentation, the summands are uniquely determined up to isomorphism and order. 

The torsion part can be represented as a direct product of cyclic groups of prime power order. The exponents of this groups are called the torsion coefficients. The nth Betti number and the torsion coefficients are uniquely determined by the group Hn A]Z). 

Let (A, A) be a principally polarized abelian variety over an algebraically closed field k. If the characteristic of k is not equal to the prime p, then the kernel of multiplication by p on A(k) is a finite group isomorphic to (Z/pZ)29. The polarization A induces a nondegenerate alternating pairing A k)[p] x A( )[p] -+ pp(k). Hence, we can try to classify principally polarized abelian varieties with a symplectic basis for the group of points of order p. However, this no longer works in characteristic p. 

Show that there can be only one group of order / , when h is a prime number. 

This group of order g is a cyclic group. Any cyclic group is Abelian. Further, if g is prime any group of order g is cyclic. 

A convenient procedure for determining the point group of a non-linear of non-cubic molecule is designed. The system requires finding out the axis of highest order, multiple axis of this order and additional twofold axis as prime step. 

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