Logarithmic diameter bounds for some Cayley graphs

Let S ⊂ GL n ⁢ ( Z ) S\subset\mathrm{GL}_{n}(\mathbb{Z}) be a finite symmetric set. We show that if the Zariski closure of Γ = ⟨ S ⟩ \Gamma=\langle S\rangle is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay ⁡ ( Γ / Γ ⁢ ( q ) , π q ⁢ ( S ) ) \operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S)) is O ⁢ ( log ⁡ q ) O(\log q) , where 𝑞 is an arbitrary positive integer, π q : Γ → Γ / Γ ⁢ ( q ) \pi_{q}\colon\Gamma\to\Gamma/\Gamma(q) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.

Evidence for and against Zauner's MUB conjecture in C^6

The problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, for composite dimensions $d$ which are not prime powers, remains completely open. In the first interesting case,~$d=6$, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in~$d=6$ by looking at their~`shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers~$q \equiv 5 \bmod 12$, however, we are able to show some curious evidence which --- at least formally --- points in the opposite direction. In $\CC^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.

Low latency Montgomery multiplier for cryptographic applications

In this modern era, data protection is very important. To achieve this, the data must be secured using either public-key or private-key cryptography (PKC). PKC eliminates the need of sharing key at the beginning of communication. PKC systems such as ECC and RSA is implemented for different security services such as key exchange between sender, receiver and key distribution between different network nodes and authentication protocols. PKC is based on computationally intensive finite field arithmetic operations. In the PKC schemes, modular multiplication (MM) is the most critical operation. Usually, this operation is performed by integer multiplication (IM) followed by a reduction modulo M. However, the reduction step involves a long division operation that is expensive in terms of area, time and resources. Montgomery multiplication algorithm facilitates faster MM operation without the division operation. In this paper, low latency hardware implementation of the Montgomery multiplier is proposed. Many interesting and novel optimization strategies are adopted in the proposed design. The proposed Montgomery multiplier is based on school-book multiplier, Karatsuba-Ofman algorithm and fast adders techniques. The Karatsuba-Ofman algorithm and school-book multiplier recommends cutting down the operands into smaller chunks while adders facilitate fast addition for large size operands. The proposed design is simulated, synthesized and implemented using Xilinx ISE Design Suite by targeting different Xilinx FPGA devices for different bit sizes (64-1024). The proposed design is evaluated on the basis of computational time, area consumption, and throughput. The implementation results show that the proposed design can easily outperform the state of the art

On the distribution of the Picard ranks of the reductions of a K3 surface

We report on our results concerning the distribution of the geometric Picard ranks of K3 surfaces under reduction modulo various primes. In the situation that $${\mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{K}}}}$$ rk Pic S K ¯ is even, we introduce a quadratic character, called the jump character, such that $${\mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{{\mathbb {F}}}}_{\!{{\mathfrak {p}}}}} > \mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{K}}}}$$ rk Pic S F ¯ p > rk Pic S K ¯ for all good primes at which the character evaluates to $$(-1)$$ ( - 1 ) .

Ideals Modulo a Prime

The main focus of this paper is on the problem of relating an ideal [Formula: see text] in the polynomial ring [Formula: see text] to a corresponding ideal in [Formula: see text] where [Formula: see text] is a prime number; in other words, the reduction modulo[Formula: see text] of [Formula: see text]. We first define a new notion of [Formula: see text]-good prime for [Formula: see text] which does depends on the term ordering [Formula: see text], but not on the given generators of [Formula: see text]. We relate our notion of [Formula: see text]-good primes to some other similar notions already in the literature. Then we introduce and describe a new invariant called the universal denominator which frees our definition of reduction modulo [Formula: see text] from the term ordering, thus letting us show that all but finitely many primes are good for [Formula: see text]. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.

On the Local Structure of Mahler Systems

This paper is a 1st step in the direction of a better understanding of the structure of the so-called Mahler systems: we classify these systems over the field $\mathscr{H}$ of Hahn series over $\overline{{\mathbb{Q}}}$ and with value group ${\mathbb{Q}}$. As an application of (a variant of) our main result, we give an alternative proof of the following fact: if, for almost all primes $p$, the reduction modulo $p$ of a given Mahler equation with coefficients in ${\mathbb{Q}}(z)$ has a full set of algebraic solutions over $\mathbb{F}_{p}(z)$, then the given equation has a full set of solutions in $\overline{{\mathbb{Q}}}(z)$ (this is analogous to Grothendieck’s conjecture for differential equations).

A note on prime divisors of polynomials P(Tk); k ≥ 1

Let F be a number field, OF the integral closure of ℤ in F, and P(T) ∈ OF[T] a monic separable polynomial such that P(0) ≠ 0 and P(1) ≠ 0. We give precise sufficient conditions on a given positive integer k for the following condition to hold: there exist infinitely many non-zero prime ideals 𝓟 of OF such that the reduction modulo 𝓟 of P(T) has a root in the residue field OF/𝓟, but the reduction modulo 𝓟 of P(Tk) has no root in OF/𝓟. This makes a result from a previous paper (motivated by a problem in field arithmetic) asserting that there exist (infinitely many) such integers k more precise.