We found that the work of Kim et al. [1] on trace representation of the Legendre sequence with the periods p ≡ ±3 (mod 8) can be improved by restricting the selection of the periods p while maintaining the form p ≡ ±3 (mod 8) unchanged. Our method relies on forcing the multiplicative group of residue classes modulo p, Zp*, to take 2 as the least primitive root. On the other hand, by relaxing the very strong condition in the theorem of these authors and by using the product among powers of the primitive root and powers of any quadratic residue element to represent an element in Zp*, we could extend Kim's formula so that it becomes a special case of our formula more general.

Variable-weight optical orthogonal codes (OOCs) were introduced by Yang for multi-media optical CDMA systems with multiple quality of service (QoS) requirements. Some works had been done on the existence of optimal (υ,W,1,Q)-OOCs for W ∈ {{3,4}, {3,5}, {4,5}}. In this paper, by using quadratic residues, seven new infinite classes of optimal (υ,{k,6},1,Q)-OOCs for k ∈ {3,4} are constructed.

This letter presents the field uniformity characteristics of a triangular prism reverberation chamber. A reverberation chamber that generally uses a stirrer to create a uniform electric field inside is an alternative to the semi-anechoic chamber for an electromagnetic compatibility test. To overcome the size and maintenance problems of a stirrer, we propose to replace it with a Quadratic Residue Diffuser which is commonly used in acoustics. To confirm that the diffuser is a valid alternative to the stirrer, a diffuser and an equilateral triangular prism reverberation chamber are designed and fabricated for 2.3-3.0 GHz operation. To investigate the field uniformity characteristics by varying the location of the transmitting antenna, both simulation and measurement in the triangular prism reverberation chamber were also done at its two positions, respectively. A commercial program XFDTD 6.2, engaging the finite difference time domain (FDTD), is used for simulation and a cumulative probability distribution, which the IEC 61000-4-21 recommends, is used to evaluate the field uniformity. Both simulation and measurement results show that the field uniformity in the chamber satisfies the international standard requirement of 6 dB tolerance and 3dB standard deviation, which means that a diffuser can be substituted for a stirrer.

It is necessary to perform arithmetic in Fp12 to use an Ate pairing on a Barreto-Naehrig (BN) curve, where p is a prime given by p(z)=36z4+36z3+24z2+6z+1 for some integer z. In many implementations of Ate pairings, Fp12 has been regarded as a 6th degree extension of Fp2, and it has been constructed by Fp12=Fp2[v]/(v6-ξ) for an element ξ ∈ Fp2 such that v6-ξ is irreducible in Fp2[v]. Such a ξ depends on the value of p, and we may use a mathematical software package to find ξ. In this paper it is shown that when z ≡ 7,11 (mod 12), we can universally construct Fp12 as Fp12=Fp2[v]/(v6-u-1), where Fp2=Fp[u]/(u2+1).

This paper presents the field uniformity characteristics in a triangular prism reverberation chamber that can be substituted for an open area test site or an anechoic chamber to measure electromagnetic interference. To improve size problems of a stirrer that is an official unit to generate a uniform field in the reverberation chamber, we suggest a diffuser of Quadratic Residue Sequence method. To validate the substitution of a diffuser for a stirrer, a diffuser is designed for 1-3 GHz, and three types of equilateral triangular prism reverberation chambers are modeled. Afterwards, the field distributions in these three reverberation chambers are both simulated and tested. Using XFDTD 6.2 of finite difference time domain method, field deviations of each structure are simulated and compared to each other. An evaluation of field uniformity is done by cumulative probability distribution which is specified in the IEC 61000-4-21. The result shows that the field uniformity in the chamber is within 6 dB tolerance and also within 3 dB standard deviation, which means a diffuser can satisfy the requirement of international standards.

This paper focuses on developing a square root (SQRT) algorithm in finite fields GF(p2d) (d 0). Examining the Smart algorithm, a well-known SQRT algorithm, we can see that there is some computation overlap between the Smart algorithm and the quadratic residue (QR) test, which must be implemented before a SQRT computation. It makes the Smart algorithm inefficient. In this paper, we propose a new QR test and a new SQRT algorithm in GF(p2d), in which not only there is no computation overlap, but also most of computations required for the proposed SQRT algorithm in GF(p2d) can be implemented in the corresponding subfields GF(p2d-i) for 1 i d, which yields many reductions in the computational time and complexity. The computer simulation also shows that the proposed SQRT algorithm is much faster than the Smart algorithm.

In 1998, Fan and Lei proposed a partially blind signature scheme that could reduce the computation load and the size of the database for electronic cash systems. In this Letter, we show that their scheme could not meet the untraceability property of a blind signature.

In a secure partially blind signature scheme, the signer assures that the blind signatures issued by him contains the information he desires. The techniques make it possible to minimize the unlimited growth of the bank's database which storing all spent electronic cash in an anonymous electronic cash system. In this paper we propose an efficient partially blind signature scheme for electronic cash. In our scheme, only several modular additions and modular multiplications are required for a signature requester to obtain and verify a signature. It turns out that the proposed scheme is suitable for mobile clients and smart-card applications because no time-consuming computations are required, such as modular exponentiation and inverse computations. Comparing with the existing blind signature schemes proposed in the literatures, our method reduces the amount of computations for signature requesters by almost 98%.

Multi-recast techniques make it possible for a voter to participate in a sequence of different designated votings by using only one ticket. In a multi-recastable ticket scheme for electronic voting, every voter of a group can obtain an m-castable ticket (m-ticket), and through the m-ticket, the voter can participate in a sequence of m different designated votings held in this group. The m-ticket contains all possible intentions of the voter in the sequence of votings, and in each of the m votings, a voter casts his vote by just making appropriate modifications to his m-ticket. The authority cannot produce both the opposite version of a vote cast by a voter in one voting and the succeeding uncast votes of the voter. Only one round of registration action is required for a voter to request an m-ticket from the authority. Moreover, the size of such an m-ticket is not larger than that of an ordinary vote. It turns out that the proposed scheme greatly reduces the network traffic between the voters and the authority during the registration stages in a sequence of different votings, for example, the proposed method reduces the communication traffic by almost 80% for a sequence of 5 votings and by nearly 90% for a sequence of 10 votings.

We propose an interactive identification scheme based on the quadratic residue problem. Prover's identity can be proved without revealing his secret information with only one accreditation. The proposed scheme requires few computations in the verification process, and a small amount of memory to store the secret information, A digital signature based on this scheme is proposed, and its validity is then proved. Lastly, analysis about the proposed scheme is presented at the end of the paper.