Divisor function of the Gaussian integers weighted by the Kloosterman sum

In this paper, we investigate the multiplicative structure of the ring of complex valued arithmetic functions defined on the ring of Gaussian integers, and show that this ring is a unique factorization domain.

The results of the paper relate to the following general problem. Find natural finite generating sets of elements of a given linear group over a finitely generated commutative ring. Of particular interest are coefficient rings that are generated by a single element, for example, the ring of integers or the ring of Gaussian integers. We prove that a projective general linear group of dimension n over the ring of Gaussian integers is generated by three involutions two of which commute if and only if n is greater than 4 and 4 does not divide n. Earlier, M. A. Vsemirnov, R. I. Gvozdev, D. V. Levchuk and the authors of this paper solved a similar problem for the special and projective special linear groups.

In this paper, we extend Construction A of lattices to the ring of algebraic integers of a general imaginary quadratic field that may not form a principal ideal domain (PID). We show that such a construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. As an application, we then apply the proposed lattices to the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all PIDs. A novel scheme called adaptive compute-and-forward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive compute-and-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely.

We construct an asymptotic formula for a sum function for σ a (α), where σ a (α) is the sum of the ath powers of the norms of divisors of the Gaussian integer α on an arithmetic progression α ≡ α0 (mod γ) and in a narrow sector ϕ1 ≤ arg α < ϕ2. For this purpose, we use a representation of σ a (n) in the form of a series in the Ramanujan sums.

Whereas the homomorphic images of Z (the ring of integers) are well known, namely Z, {0} and Zn (the ring of integers modulo n), the same is not true for the homomorphic im-ages of Z[i] (the ring of Gaussian integers). More generally, let m be any nonzero square free integer (positive or negative), and consider the integral domain Z[ √m]={a + b √m | a, b ∈ Z}. Which rings can be homomorphic images of Z[ √m]? This ques-tion offers students an infinite number (one for each m) of investigations that require only undergraduate mathematics. It is the goal of this article to offer a guide to the in-vestigation of the possible homomorphic images of Z[ √m] using the Gaussian integers Z[i] as an example. We use the fact that Z[i] is a principal ideal domain to prove that if I =(a+bi) is a nonzero ideal of Z[i], then Z[i]/I ∼ = Zn for a positive integer n if and only if gcd{a, b} =1, in which case n = a2 + b2 . Our approach is novel in that it uses matrix techniques based on the row reduction of matrices with integer entries. By characterizing the integers n of the form n = a2 + b2 , with gcd{a, b} =1, we obtain the main result of the paper, which asserts that if n ≥ 2, then Zn is a homomorphic image of Z[i] if and only if the prime decomposition of n is 2α0 pα1 1 ··· pαk k , with α0 ∈{0, 1},pi ≡ 1(mod 4) and αi ≥ 0 for every i ≥ 1. All the fields which are homomorpic images of Z[i] are also determined.

A VLSI implementation of a bit-serial systolic architecture for a DFT processor has been developed which performs residue number system (RNS) processing over the ring of Gaussian integers. An architecture for a 128-point DFT using the chirp z-transform algorithm is described, and its use in an R2FFT architecture to obtain a 16,384-point transform is illustrated. Based on three custom-designed chips, the processor is capable of transforming data at a continuous 2 MHz rate. The use of RNS techniques and systolic arrays provides two dimensions of parallelism, resulting in hardware of low complexity and high speed. The overall system has great flexibility in dynamic range, and can be used in many signal processing applications.

M. C. Tamburini and P. Zucca proved that the special linear group of dimension greater than 13 over the ring of Gaussian integers is generated by three involutions, two of which commute (J. of Algebra, 1997). A similar result for projective special linear groups of dimension greater than 6 was established by D. V. Levchuk and Ya. N. Nuzhin (J. Sib. Fed. Univ. Math. Phys., 2008, Bulletin of Novosibirsk State Univ., 2009). We consider the remaining small dimensions. It is proved that the projective special linear group of dimension other than 5 and 6 over the ring of Gaussian integers if and only if is generated by three involutions, two of which commute when its dimension is greater than 6. For dimension 5 and 6, it was possible to find only generators triples of involutions without the condition that two of which commute.

The Montgomery multiplication is an efficient method for modular arithmetic. Typically, it is used for modular arithmetic over integer rings to prevent the expensive inversion for the modulo reduction. In this work, we consider modular arithmetic over rings of Gaussian integers. Gaussian integers are subset of the complex numbers such that the real and imaginary parts are integers. In many cases Gaussian integer rings are isomorphic to ordinary integer rings. We demonstrate that the concept of the Montgomery multiplication can be extended to Gaussian integers. Due to independent calculation of the real and imaginary parts, the computation complexity of the multiplication is reduced compared with ordinary integer modular arithmetic. This concept is suitable for coding applications as well as for asymmetric key cryptographic systems, such as elliptic curve cryptography or the Rivest-Shamir-Adleman system.

Modular arithmetic over integers is required for many cryptography systems. Montgomery reduction is an efficient algorithm for the modulo reduction after a multiplication. Typically, Montgomery reduction is used for rings of ordinary integers. In contrast, we investigate the modular reduction over rings of Gaussian integers. Gaussian integers are complex numbers where the real and imaginary parts are integers. Rings over Gaussian integers are isomorphic to ordinary integer rings. In this work, we show that Montgomery reduction can be applied to Gaussian integer rings. Two algorithms for the precision reduction are presented. We demonstrate that the proposed Montgomery reduction enables an efficient Gaussian integer arithmetic that is suitable for elliptic curve cryptography. In particular, we consider the elliptic curve point multiplication according to the randomized initial point method which is protected against side-channel attacks. The implementation of this protected point multiplication is significantly faster than comparable algorithms over ordinary prime fields.

We compute the analogue for Z [ i ] {\mathbf {Z}}[i] of Euler’s constant, that is δ = lim n → + ∞ δ n \delta = {\lim _{n \to + \infty }}{\delta _n} , where δ n = ( Σ 2 ⩽ k ⩽ n 1 / π r k 2 ) − log ⁡ n {\delta _n} = ({\Sigma _{2 \leqslant k \leqslant n}}1/\pi r_k^2) - \log n . For this purpose we give an estimate for \[ r k = min { r ⩾ 0 ; there exists z ∈ C such that card ( Z [ i ] ∩ D ¯ ( z , r ) ) ⩾ k } , {r_k} = \min \left \{ {r \geqslant 0;{\text {there exists}}\;z \in {\mathbf {C}}\;{\text {such that card}}({\mathbf {Z}}[i] \cap \bar D(z,r)) \geqslant k} \right \}, \] and we compute a great number of values of r k {r_k} .

Minimal Polynomials and Distinctness of Kloosterman Sums

Self-Similar Hex-Sums of Squares R. BREU ~ eading the very stimulating article by A. van der Poorten, et al. in the Mathematical Intelligencer [1], I realized that self-similar sums of squares have ~. ~ , a nice analogue in what I call self-similar hex-sums of squares, just as decomposing primes of the form 4q + 1 into sums of squares a 2 + b 2 o r factoring 4q + 1 in the ring of Gaussian integers has an analogue in the ring of Eisenstein integers, that is, decomposing primes of the form 3q + 1 into "hex-sums" of squares a 2 + b 2 a b . ("Hex," as the fundamental domain in the Eisenstein lattice is a hexagon, versus a square in the Gaussian case.) Eisenstein integers have the form a -+ b~0, where ~o = ( 1 + X/-Z-3)/2. Their n o r m a 2 -4b 2 + ab corresponds to the n o r m a 2 + b 2 o f Gaussian integers. Both signs can be chosen, as the "upper sign" version is just the mirrored version of the "lower sign," but then have to be used consistently throughout, that is, always the lower or always the upper sign. A norm equation of Gaussian integers, say (a 2 + b 2) (c 2 + d 2) = (ac + bd) 2 + (ad b c ) 2, translates into a similar norm equation of Eisenstein integers

Greg Dresden and Wayne M. Dymacek obtained factors of quotient rings over ring of Gaussian integers Z[i]={a+bi/a,b[is an element of a set] Z}. We generalize their idea to obtain factors of quotient rings of the ring of Eisenstein integers Z[omega]={a+b[omega]/a,b[is an element of a set]Z} where [omega]=(-1+[square root]-3)/2 and some other quadratic integer rings Z[omega]={a+b[omega]/a,b[is an element of a set]Z} for [omega]=[square root]d where d is an integer such that d [is equivalent to] 2, 3 (mod 4) or [omega] = (1+[square root]d)/2 where d is an integer such that d [is equivalent to] 1 (mod 4) and in both cases d is squarefree. Moreover, James T. Cross extended naturally the Euler [phi]-function of the ring of integers to the ring of Gaussian integers.We extend it to the ring of Eisenstein integers and some quadratic integers rings.

In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involed. It is shown that if [Formula: see text] and pε &lt; |B|, |C| &lt; |A| &lt; p1-ε, then |A + B| + |A · C| &gt; pδ (ε)|A|. Next we exploit the Szemerédi–Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from [Formula: see text] to [Formula: see text] satisfies |F(A,B)| &gt; pβ for some β = β (α) &gt; α whenever [Formula: see text] and |A| ~ |B|~ pα, 0 &lt; α &lt; 1. The exponential sum ∑x∈ A,y∈Bεp(axy+bx2y2), ab ≠ 0 ( mod p), may be estimated nontrivially for arbitrary sets [Formula: see text] satisfying |A|, |B| &gt; pρ where ρ &lt; 1/2 is some constant. From this, one obtains an explicit 2-source extractor (with exponential uniform distribution) if both sources have entropy ratio at last ρ. No such examples when ρ &lt; 1/2 seemed known. These questions were largely motivated by recent works on pseudo-randomness such as [2] and [3]. Finally it is shown that if pε &lt; |A| &lt; p1-ε, then always |A + A|+|A-1 + A-1| &gt; pδ(ε)|A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution ( mod p) of certain rational functions on the primes (cf. [12]).

We study coset weight distributions of binary primitive (narrow-sense) BCH codes of length n = 2 m (m odd) with minimum distance 8. In the previous paper [1], we described coset weight distributions of such codes for cosets of weight j = 1, 2, 3, 5, 6. Here we obtain exact expressions for the number of codewords of weight 4 in terms of exponential sums of three types, in particular, cubic sums and Kloosterman sums. This allows us to find the coset distribution of binary primitive (narrow-sense) BCH codes with minimum distance 8 and also to obtain some new results on Kloosterman sums over finite fields of characteristic 2.