Adic Expansion Research Articles

ℓ-adic digits and class number of imaginary quadratic fields

Motivated by the work of [K. Girstmair, A “popular” class number formula, Amer. Math. Monthly 101(10) (1994) 997–1001; K. Girstmair, The digits of [Formula: see text] in connection with class number factors, Acta Arith. 67(4) (1994) 381–386] and [M. R. Murty and R. Thangadurai, The class number of [Formula: see text] and digits of [Formula: see text], Proc. Amer. Math. Soc. 139(4) (2010) 1277–1289], we study the average of the digits of the [Formula: see text]-adic expansion of [Formula: see text] whenever [Formula: see text] is a product of two distinct primes or a prime power. More explicitly, if [Formula: see text] is an integer such that [Formula: see text] and suppose that [Formula: see text] is the [Formula: see text]-adic expansion of [Formula: see text] then we establish the average of the digits of the [Formula: see text]-adic expansion of [Formula: see text] in terms of [Formula: see text] and the “trace” of generalized Bernoulli numbers [Formula: see text] where [Formula: see text]’s are odd Dirichlet characters modulo [Formula: see text] As a consequence of these results, we recover two well-known results of Gauss and Heilbronn (see Theorems 1.6 and 1.7).

Primes with a missing digit: Distribution in arithmetic progressions and an application in sieve theory

AbstractWe prove Bombieri–Vinogradov type theorems for primes with a missing digit in their ‐adic expansion for some large positive integer . The proof is based on the circle method, which relies on the Fourier structure of the integers with a missing digit and the exponential sums over primes in arithmetic progressions. Combining our results with the semi‐linear sieve, we obtain an upper bound and a lower bound of the correct order of magnitude for the number of primes of the form with a missing digit in a large odd base .

Optimal Locally Recoverable Codes With Hierarchy From Nested F-Adic Expansions

Optimal Locally Recoverable Codes With Hierarchy From Nested <i>F</i>-Adic Expansions

The number of rational points of two parameter Calabi-Yau manifolds as toric hypersurfaces

The number of rational points in toric data are given for two-parameter Calabi-Yau $n$-folds as toric hypersurfaces over finite fields $\mathbb F_p$ . We find that the fundamental period is equal to the number of rational points of the Calabi-Yau $n$-folds in zeroth order $p$-adic expansion. By analyzing the solution set of the GKZ-system given by the enhanced polyhedron, we deduce that under type II/F-theory duality the 3D and 4D Calabi-Yau manifolds have the same number of rational points in zeroth order. Taking the quintic and its duality as an example, the number of rational points in some specific complex moduli are given by numerical calculation to support our results.

On the Lengths of Divisible Codes

In this article, the effective lengths of all $q^r$-divisible linear codes over $\mathbb{F}_q$ with a non-negative integer $r$ are determined. For that purpose, the $S_q(r)$-adic expansion of an integer $n$ is introduced. It is shown that there exists a $q^r$-divisible $\mathbb{F}_q$-linear code of effective length $n$ if and only if the leading coefficient of the $S_q(r)$-adic expansion of $n$ is non-negative. Furthermore, the maximum weight of a $q^r$-divisible code of effective length $n$ is at most $\sigma q^r$, where $\sigma$ denotes the cross-sum of the $S_q(r)$-adic expansion of $n$. This result has applications in Galois geometries. A recent theorem of N{\u{a}}stase and Sissokho on the maximum size of a partial spread follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.

Algorithms for Inversion Mod $p^k$

This article describes and analyzes all existing algorithms for computing $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) for a prime $p$ p , and also introduces a new algorithm based on the exact solution of linear equations using $p$ p -adic expansions. The algorithm starts with the initial value $c=a^{-1}\pmod {p}$ c = a - 1 ( mod p ) and iteratively computes the digits of the inverse $x=a^{-1}\pmod {p^k}$ x = a - 1 ( mod p k ) in base $p$ p . The mod 2 version of the algorithm is more efficient than all existing algorithms for small values of $k$ k . Moreover, it stands out as being the only one that works for any $p$ p , any $k$ k , and digit-by-digit. While the new algorithm is asymptotically worse off, it requires the minimal number of arithmetic operations (just a single addition) per step, as compared to all existing algorithms.

Arithmetic of $$\tau $$ τ -adic expansions for lightweight Koblitz curve cryptography

Koblitz curves allow very efficient elliptic curve cryptography. The reason is that one can trade expensive point doublings to cheap Frobenius endomorphisms by representing the scalar as a $$\tau $$ -adic expansion. Typically elliptic curve cryptosystems, such as ECDSA, also require the scalar as an integer. This results in a need for conversions between integers and the $$\tau $$ -adic domain, which are costly and hinder the use of Koblitz curves on very constrained devices, such as RFID tags, wireless sensors, or certain applications of the Internet of things. We provide solutions to this problem by showing how complete cryptographic processes, such as ECDSA signing, can be completed in the $$\tau $$ -adic domain with very few resources. This allows outsourcing conversions to a more powerful party. We provide several algorithms for performing arithmetic operations in the $$\tau $$ -adic domain. In particular, we introduce a new representation allowing more efficient and secure computations compared to the algorithms available in the preliminary version of this work from CARDIS 2014. We also provide datapath extensions with different speed and side-channel resistance properties that require areas from less than one hundred to a few hundred gate equivalents on 0.13- $$\upmu $$ m CMOS. These extensions are applicable for all Koblitz curves.

Some properties of $${\tau }$$ τ -adic expansions on hyperelliptic Koblitz curves

In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, Frobenius-and-add algorithm using the \({\tau }\)-adic non-adjacent form (\(\tau \)-NAF). The \(\tau \)-NAF has the three properties: (1) existence, (2) uniqueness, and (3) minimality of the Hamming weight. On the other hand, Gunther et al. (Speeding up the arithmetic on koblitz curves of genus two. LNCS, vol. 2012, pp. 106–117. Springer, Heidelberg, 2001) have proposed two generalizations of \(\tau \)-NAF for a family of hyperelliptic curves (hyperelliptic Koblitz curves) which have been proposed by Koblitz (J Cryptol 1(3):139–150, 1989). We call these generalizations \({\tau }\)-adic sparse expansion, and \({\tau }\)-NAF, respectively. To our knowledge, it is not known whether the three properties are true or not, especially, the existence must be satisfied for concrete cryptographic implementations. We provide an answer to the question. Our investigation shows that the \({\tau }\)-adic sparse expansion has only the existence and the \({\tau }\)-NAF has the existence and uniqueness. Our results guarantee the concrete cryptographic implementations of these generalizations.

Non-minimality of the width-$w$ non-adjacent form in conjunction with trace one $\tau $-adic digit expansions and Koblitz curves in characteristic two

Non-minimality of the width-$w$ non-adjacent form in conjunction with trace one $\tau $-adic digit expansions and Koblitz curves in characteristic two

On variants of the Halton sequence

A generalisation of the classical Halton sequence $$(\phi _{\beta }(n))_{n\in \mathbb {N}}$$ has emerged in recent years based on $$\beta $$ -adic expansions of elements of [0, 1). In the case where $$\beta $$ is a natural number greater than 1, this reduces to the classical Halton sequence. In this paper, we use ergodic and analytic methods to prove the uniform distribution of a sequence $$(\phi _{\beta }(k_j))_{j\in \mathbb {N}}$$ for the sequence of integers $$(k_j)_{j\ge 0}$$ , which is both Hartman uniformly distributed and good universal. This builds on earlier work of M. Hofer, M. R. Iaco and R. Tichy in the special case $$k_j =j \ (j=0,1, \ldots )$$ . Variants of this phenomenon are also studied.

Ramanujan-type congruences for overpartitions modulo 16

Let \(\bar{p}(n)\) denote the number of overpartitions of \(n\). Recently, Fortin–Jacob–Mathieu and Hirschhorn–Sellers independently obtained 2-, 3- and 4-dissections of the generating function for \(\bar{p}(n)\) and derived a number of congruences for \(\bar{p}(n)\) modulo 4, 8 and 64 including \(\bar{p}(8n+7)\equiv 0 \pmod {64}\) for \(n\ge 0\). In this paper, we give a 16-dissection of the generating function for \(\bar{p}(n)\) modulo 16 and show that \(\bar{p}(16n+14)\equiv 0\pmod {16}\) for \(n\ge 0\). Moreover, using the \(2\)-adic expansion of the generating function for \(\bar{p}(n)\) according to Mahlburg, we obtain that \(\bar{p}(\ell ^2n+r\ell )\equiv 0\pmod {16}\), where \(n\ge 0\), \(\ell \equiv -1\pmod {8}\) is an odd prime and \(r\) is a positive integer with \(\ell \not \mid r\). In particular, for \(\ell =7\) and \(n\ge 0\), we get \(\bar{p}(49n+7)\equiv 0\pmod {16}\) and \(\bar{p}(49n+14)\equiv 0\pmod {16}\). We also find four congruence relations: \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n) \pmod {16}\) for \(n\ge 0\), \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {32}\) where \(n\) is not a square of an odd positive integer, \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {64}\) for \(n\not \equiv 1,2,5\pmod {8}\) and \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {128}\) for \(n\equiv 0\pmod {4}\).

Intersections of multiplicative translates of 3-adic Cantor sets

This paper is motivated by questions concerning the discrete dynamical system on the 3 -adic integers \mathbb{Z}_{3} given by multiplication by 2 . The exceptional set \mathcal{E}(\mathbb{Z}_3) is defined to be the set of all elements of \mathbb{Z}_3 whose forward orbits under this action intersect the 3 -adic Cantor set \Sigma_{3, \bar{2}} (of 3 -adic integers whose expansions omit the digit 2 ) infinitely many times. It has been shown that this set has Hausdorff dimension at most \frac{1}{2} , and it is conjectured that it has Hausdorff dimension 0 . Upper bounds on its Hausdorff dimension can be obtained with sufficient knowledge of Hausdorff dimensions of intersections of multiplicative translates of Cantor sets by powers of 2 . This paper studies more generally the structure of finite intersections of general multiplicative translates S= \Sigma_{3, \bar{2}} \cap \frac{1}{M_1} \Sigma_{3, \bar{2}} \cap \dots \cap \frac{1}{M_n} \Sigma_{3, \bar{2}} by integers 1 &lt; M_1 &lt; M_2 &lt; \dots &lt; M_n . These sets are describable as sets of 3 -adic integers whose 3 -adic expansions have one-sided symbolic dynamics given by a finite automaton. This paper gives a method to determine the automaton for given data (M_1, \dots, M_n) and to compute the Hausdorff dimension, which is always of the form \log_{3}(\beta) where \beta is an algebraic integer. Computational examples indicate that in general the Hausdorff dimension of such sets depends in a very complicated way on the integers M_1, \dots, M_n . Exact answers are obtained for certain infinite families, which show as a corollary that a relaxed notion of generalized exceptional set has a positive Hausdorff dimension.

Further results on the compression maps of primitive sequences over Z=(&amp;lt;italic&amp;gt;p&amp;lt;/italic&amp;gt;&amp;lt;sup&amp;gt;&amp;lt;italic&amp;gt;e&amp;lt;/italic&amp;gt;&amp;lt;/sup&amp;gt;)

Let Z/( p e ) be the integer residue ring with p prime and e ≥ 2. Let f ( x ) be a primitive polynomial over Z/( p e ) with degree n and G ( f ( x ), p e ) the set of all primitive sequences over Z/( p e ) generated by f ( x ). For any sequence a∈ G ( f ( x ), p e ), it has a unique p -adic expansion a = a0 + a1 p +…+ a e -1 p e -1. Let o( x 0, x 1,…, x e -1) = g ( x e -1)+ μ ( x 0, x 1,…, x e -2) be a function from F p e to F p . Then o can induce a compression mapping from G ( f ( x ), p e ) to F p ∞. In recent years, Zhu, Tian and Qi have proved that the compression mapping is injective under the condition that p is an odd prime and f ( x ) is a strongly primitive polynomial. In this paper, we improve their results. More exactly, we prove that the compression mapping is also injective only under the condition that p is an odd prime and f ( x ) is a primitive polynomial. Of course we should ask that deg( g ( x )) is an odd number or g ( x ) = x k +Σi=0 k -2 c i x i.

Non-normal numbers with respect to Markov partitions

We call a real number normal if for any block of digits the asymptoticfrequency of this block in the $N$-adic expansion equals the expectedone. In the present paper we consider non-normal numbers and, inparticular, essentially and extremely non-normal numbers. We call areal number essentially non-normal if for each single digit thereexists no asymptotic frequency of its occurrence. Furthermore we calla real number extremely non-normal if all possible probability vectorsare accumulation points of the sequence of frequency vectors. Our aimnow is to extend and generalize these results to Markov partitions.

Normality of some decimals generated by primes in a residue class

Let F ( x ) be either a polynomial with real coefficients and with the leading coefficient rational or an entire function having logarithmic order α where 1 < α < 4 / 3 and taking real values at real x . Let q 1 , q 2 , … be the sequence of all the primes congruent to ℓ ( mod k ) with k > 1 a fixed integer. Let ( F ( q ) ) r denote the digits in the r -adic expansion of [ | F ( q ) | ] . We show the decimal . ( F ( q 1 ) ) r ( F ( q 2 ) ) r … , is normal to the base r .

On the zeroes of certain periodic functions over valued fields of positive characteristic

Let C be an algebraically closed field of positive characteristic p and complete with respect to a non-archimedean absolute value | . | and Λ ⊂ C a discrete F p -submodule. Suppose there exists an F p -basis { λ 0 , λ 1 , … } of Λ such that 0 < | λ 0 | < | λ 1 | < ⋯ and | λ i | → ∞ . For k ∈ N define the meromorphic function C k , Λ ( z ) = ∑ λ ∈ Λ 1 ( z − λ ) k on C . We show that all the zeroes x of C k , Λ satisfy (⁎) | x | = | λ i | for some i . Furthermore, the number (counted with multiplicities) of zeroes for which (⁎) holds depends only on i and the p -adic expansion coefficients of k , but not on Λ .

Dynamics of the $p$-adic shift and applications

There is a natural continuous realization of the one-sided Bernoulli shift on the $p$-adic integers as the map that shifts the coefficients of the $p$-adic expansion to the left.We study this map's Mahler power series expansion. We prove strong resultson $p$-adic valuations of the coefficients in this expansion, and showthat certain natural maps (including many polynomials) are in a sensesmall perturbations of the shift. As a result, these polynomials sharethe shift map's important dynamical properties. This provides a novel approach to an earlier result of the authors.

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ , we show that there is an integer c = c ( Φ ) such that dim Ext G 1 ( L , L ′ ) < c , for any reductive algebraic group G with root system Φ and any irreducible rational G -modules L , L ′ . There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Ext n , for any integer n ⩾ 0 , using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L = L ( λ ) , even allowing λ to vary, provided the p -adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.

Further Result of Compressing Maps on Primitive Sequences Modulo Odd Prime Powers

Let be the integer residue ring with odd prime and integer . For a sequence over , there is an unique -adic expansion , where each is a sequence over , and can be regarded as a sequence over the prime field GF naturally. Let be a strongly primitive polynomial over , and the set of all primitive sequences generated by over . Suppose that GF GF . It is shown that any function in is an injective map from to GF, and the derived sequences of different functions are also different. That is, if and only if and for and . These injective functions in can be considered as good candidates for the keys of a stream cipher.

Primitive recursive real numbers

AbstractIn mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure – Dedekind‐complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if “computable” is replaced by “primitive recursive” (p. r., for short), these definitions lead to a number of different concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b ‐adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)