Cryptography and Lattices: International Conference, CaLC by Miklós Ajtai, Ravi Kumar, Dandapani Sivakumar (auth.),

By Miklós Ajtai, Ravi Kumar, Dandapani Sivakumar (auth.), Joseph H. Silverman (eds.)

This e-book constitutes the completely refereed post-proceedings of the foreign convention on Cryptography and Lattices, CaLC 2001, held in windfall, RI, united states in March 2001. The 14 revised complete papers offered including an outline paper have been conscientiously reviewed and chosen for inclusion within the booklet. All present features of lattices and lattice aid in cryptography, either for cryptographic development and cryptographic research, are addressed.

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Additional info for Cryptography and Lattices: International Conference, CaLC 2001 Providence, RI, USA, March 29–30, 2001 Revised Papers

Sample text

The corresponding lattice L2 has dimension 2d and determinant N −d B 2d(2d−1)/2 . The enabling condition becomes c2 (d) N −d B 2d(2d−1)/2 1/(2d) < 1 , 2d or equivalently B ≤ c2 (d)N 1/(2d−1) . The exponent of N has improved from 2/d(d + 1) to 1/(2d − 1). The improvement came about because the dimension of L increased, while its determinant decreased. We obtain a second improvement by considering higher powers of the modulus N , along with a still larger collection of polynomials. Fix a positive integer h.

Xd ) MF (X1 , . . , Xd )T . ÙÒ Ø ÓÒ size(F ) ÒÓØ × Ø Ò ÖÝ Ò Ó Ò Ð Ò Ø Ó MF º ÌÛÓ ÓÖÑ× d×d F Ò G Ö ÕÙ Ú Ð ÒØ Ø Ö Ü ×Ø× ÙÒ ÑÓ ÙÐ Ö Ñ ØÖ Ü U ∈ Z Û Ø T MG = U MF U º Ï × Ý Ø Ø U ØÖ Ò× ÓÖÑ× F ÒØÓ Gº Ì ÒÙÑ Ö ∆F = det MF Ì × Ø Ë Ø ÖÑ Ò ÒØ Ó Ø ¸ º º¸ ¾ d×d ÓÖ ÑÓÖ ÓÖѺ Ì ÓÒ Ø Ø Ø ÖÑ Ò ÒØ × ÒÚ Ö ÓÖÝ Ó ÕÙ Ö Ø ÒØ ÙÒ ÓÖÑ׺ Ì Ö ÕÙ Ú Ð Ò º Ó ÒØ Ñ ØÖ Ü T ØÓÖ Þ Ø ÓÒ MF = R DR¸ T MF ∈ Z × ÙÒ ÕÙ R DR ØÓÖ Þ Ø ÓÒ¸ º ¸ d×d Û Ö R ∈ Q × Ò ÙÔÔ Ö ØÖ Ò ÙÐ Ö Ñ ØÖ Ü Û Ø ÓÒ × ÓÒ Ø ÓÒ Ð Ò D × ÓÒ Ð Ñ ØÖ Üº Ì Ñ ØÖ Ü R × ÙÒ ÕÙ ÒÓÖÑ Ð Þ Ø ÓÒ R = RU ¸ ÓÒ Ð Ò Û Ö U × ÙÒ ÑÓ ÙÐ Ö Ò R × ÙÔÔ Ö ØÖ Ò ÙÐ Ö Û Ø ÓÒ × ÓÒ Ø Ð Ñ ÒØ× ÓÚ Ø ÓÒ Ð Ò Ø Ö Ò (− 12 , 12 ]º Ì ÓÖÖ ×ÔÓÒ Ò Ñ ØÖ Ü T R DR Ò × ÓÖÑ F Û × ÕÙ Ú Ð ÒØ ØÓ F º Ì ÓÖÑ F × ÐÐ Ø Ö Ñ¹Ë Ñ Ø ÒÓÖÑ Ð Þ Ø ÓÒ Ó F º Ì × × Ø ÒÓÖÑ Ð Þ Ø ÓÒ ×Ø Ô Ó Ø ÄÄÄ Ð ÓÖ Ø Ñ × ÓÒ¸ Ø ¸ ØÖ Ò×Ð Ø Ö Ñ¹Ë Ñ ÒØÓ Ø Ð Ò Ù Ø ÒÓÖÑ Ð Þ Ø ÓÒ Ó Ó ÕÙ ÓÖÑ Ö Ø F Ó ÓÖÑ׺ ÁÒ × Þ s Ò Ü Ñ Ò¹ ÓÑÔÙØ ×Ø Ê Û Ø ÓÒ×Ø ÒØ ÒÙÑ ÓÔ Ö Ø ÓÒ׺ Ï Ò Ø Ö Ø Ñ Ø Ø Ò ÖÝ × ÓÒ Ð Ð ÓÔ Ö Ø ÓÒ׸ Ò γ ¹Ö Ù Ø ÓÒ Ñ ÒØ× Ó MG Ó Ò F¸ × Ö Ø G ÓÖÑ× ¿ Û Ø O(M (s)) × ÕÙ Ú Ð ÒØ ØÓ Ø¹ γ ∆F º Ø ÑÓ×Ø f × ÓÖ ÓÖÑ Ò ØÛÓ Ú Ö gº Ì Ò ÖÝ ÓÖÑ f Р׺ Ï × ÒÓØ Ö Ù Ò ÖÝ ÓÖÑ× Û Ø a11 a12 a12 a22 × Ø × Mf = ÐÓÛ Ö × a11 ≤ a22 |a12 | ≤ 12 a11 .

0≤k

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