The PayWord Scheme, invented by Rivest and Shamir, is an efficient micropayment scheme utilizing a hash function. We point out that the scheme has the following problem: a malicious customer can damage the bank by purchasing in excess of the customer's credit which the bank has guaranteed by issuing a certificate. Generally, there are two positions of the bank with regard to the certificate. Position 1: the bank takes full responsibility for the certificate and compensates all payments created by the customer's purchases; and Position 2: the bank does not redeem payments exceeding a limit set for the customer and shares the loss with the shop if trouble occurs. In the PayWord Scheme, the bank can reduce its risk by adopting Position 2 rather than Position 1. However, this paper points out that the bank can damage the shop in Position 2 by impersonating an imaginary customer and making the shop share the loss with the bank. We propose a micropayment scheme (countermeasure) that overcomes these problems.

代数的整数論分野　Greenberg予想に関しては、総実代数体の相対p-拡大に於いて基礎体のλp＝μp＝0が拡大体に“遺伝”するための必要条件を与えた。そして具体的に与えられた素数pに対して、λp＝μp＝0となる総実代数体のパラメータ付きの無限族を発見した。また、虚数乗法論の代数的整数論への応用に関しては、虚二次体のある種のアーベル拡大体上のZp-拡大が正規p整数基底を持つことを楕円曲線に対する虚数乗法論（志村理論）を用いて示すことに成功した。保型関数論分野　ある種のＱ上のアーベル多様体に対してこの問題を解決し、その応用として代数体上のＱ-curveと呼ばれる楕円曲線や、種数２の代数曲線で、ヤコビ多様体が四元数乗法をもつもの（ＱＭ-curve）について、谷山・志村予想を証明した。さらに、この結果を用いて１パラメータ付きのモジュラーＱ-curveの無限族の具体的な構成も行なった。楕円曲線論分野　本年度は、24個の実二次体に対し、その上で至るところgood reduction を持つ楕円曲線の非存在を示し、８個の実二次体に対しては、その上で至る所good reduction を持つ楕円曲線を全て決定した。決定された８個の実二次体にはＱ（√37）が含まれるが、その決定の過程で、４x4－37y2＝－1（x, y∈Z）を解く必要が生じたが、それを含むより一般的な問題も解決した。研究成果の発表M. Ozaki, On the cyclotomic unit group and the ideal class group of a real abelian number field, Journal of Number Theory, vol. 64, 1997, pp. 211-222.M. Ozaki, On the cyclotomic unit group and the ideal class group of a real abelian number field II, Journal of Number Theory, vol. 64, 1997, pp.223-232.M. Ozaki, Kummer's lemma for Zp-extensions over totally real number fields, Acta Arith., vol. 81, 1997, pp.37-44.M. Ozaki, The class group of Zp-extensions over totally real number fields, TÔ　hoku Math. Journal, vol. 49, 1997, pp.431-435.M. Ozaki, On the cyclotomic unit group and the p-ideal class group, Tokyo Journal of Math., vol. 20, 1997, pp. 475-480.T. Fukuda, K. Komatsu, M. Ozaki and H. Taya, On Iwasawa λp-invariants of relative real cyclic extensions of degree p, Tokyo Jounal of Math., vol. 20, 1997, pp. 475-480M. Ozaki and H. Taya, On the Iwasaki λ2-invariant of certain families of real quadratic fields, Manuscripta Math., vol. 94, 1997, pp. 437-444.M. Kida and T. Kagawa, Nonexistence of elliptic curves with good reduction everywhere over real quadratic fields, J. Number Theory, vol. 66, 1997, pp.201-210.T. Kagawa, Determination of elliptic curves with everywhere good reduction over Ｑ（√37）, Acta Arith., vol. 83, 1998, pp. 253-269.T. Hibino and N. Murabayashi, Modular equations of hyperelliptic X0（N） and an application, Acta Arith., vol. 82, 1997, pp.279-291.T. Hibino and A. Umegaki, Families of elliptic Q-curves defined over number fields with large degrees, Proc. Japan Acad., vol. 74, Ser. A, No 1, 1998, pp. 20-24.G. Yamamoto, On the vanishing of Iwasawa in variants of certain (p, p)-extensions of Q, Proc. Japan Acad., vol. 73, Ser. A, No 3, 1997, pp. 45-47.T. Ito, A construction of normal bases over the Hilbert p-class field of imaginary quadratic field, Proc. Japan Acad., vol. 74, Ser. A, No 1, 1998, pp.25-28.K. Hashimoto, Q-curve of degree 5 and jacobian surfaces of GL2-type, 理工総研Technical Report No. 97-8.K. Hashimoto and Ko. Miyake, Inverse Galois Problem for Dihedral Groups, 理工総研 Technical Report No. 98-4.