I. Blake, G. Seroussi, and N. P. Smart, Eds. Advances in Elliptic Curve Cryptography, London Mathematical Society Lecture Note Series 317, Cambridge University Press, 2005.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....log q) which is polynomial in d but exponential in log p. In particular, the zeta function Z(C d ; T ) can be computed in polynomial time if p = O( d log q) This example is important because of its many applications in number theory and cryptography, see Koblitz [19] and Blake et al. [4]. For special types of curves in small characteristic, more practical versions of various p adic algorithms have been designed by a number of authors, see Satoh [32] Kedlaya [17] Lauder Wan [23] Denef Vercauteren [10] and the references listed in those papers. Restricting Problem 2.9 to plane ....

I.F. Blake, G. Seroussi, N.P. Smart, Elliptic Curves in Cryptography, London Mathematical Society Lecture Note Series, 265, Cambridge

....for commercial web tra#c for the foreseeable future. We chose to represent elements of GF (2 ) in polynomial basis, i.e. polynomials a = am 1 t a 1 t a 0 are represented as bit strings (a m 1am 2 . a 1 a 0 ) For a detailed mathematical background on ECC the reader is referred to [3]. 5.1 Architectural Overview We developed a programmable processor optimized to execute ECC point multiplication. The data path shown in Figure 3 implements a 256 bit architecture. Parameters and variables are stored in an 8kB data memory DMEM and program instructions are contained in a 1kB ....

Blake, I., Seroussi, G., Smart, N.: Elliptic Curves in Cryptography. London Mathematical Society Lecture Note Series 265, Cambridge University Press, 1999.

....1 illustrates the number of operations for a scalar multiplication on a 32 bit processor depending on the MI ratios. The scalar multiplication with an n bit scalar is realized by the sliding window method with an approximated cost of n doublings 0:2 n additions for a 4 bit window size [BSS99]. Figure 1 allows to estimate the eciency of an ECC or a HECC built on top of a given eld library by comparing the di erent MI ratios. In general we can draw the following conclusions from this comparison: 1. ECC with projective coordinates is in almost all cases the most ecient cryptosystem. ....

I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic Curves in Cryptography. London Mathematical Society Lecture Notes Series 265. Cambridge University Press, Reading, Massachusetts, 1999.

....the for group doubling. The following comparison is based on the assumption that a scalar multiplication with an n bit scalar is realized by the sliding window method. Hence, the approximated cost of the scalar multiplication is n doublings 0:2 n additions for a 4 bit window size [BSS99] For arbitrary curves over elds of general characteristic, we achieve a 24 improvement compared to the results presented in [Nag00] In the case of HEC over F 2 n with h(x) x, we can reach an improvement up to 60 compared to the best known formulae for genus 4 curves. When comparing ....

I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic Curves in Cryptography. London Mathematical Society Lecture Notes Series 265. Cambridge University Press, Reading, Massachusetts, 1999.

....Figure 1 illustrates the number of operations for a scalar multiplication on a 32 bit processor depending on the MI ratios. The scalar multiplication with an n bit scalar is realized by the sliding window method with an approximated cost of n doublings 0. 2 additions for a 4 bit window size [BSS99]. Figure 1 allows to estimate the e#ciency of an ECC or a HECC built on top of a given field library by comparing the di#erent MI ratios. Table 5. Total number of atomic operations for ECC and HECC ECC a#ne Jacobian projective new projective (mixed) Addition (2 k) 7] 15 ....

I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic Curves in Cryptography. London Mathematical Society Lecture Notes Series 265. Cambridge University Press, Reading, Massachusetts, 1999.

....libraries one inversion takes less than six multiplications. The times needed for multiplication, squaring and inversion by the di erent libraries are listed in the appendix. Addition formulae for elliptic curves can be found in almost any textbook on this subject, e.g. Blake, Seroussi, and Smart [1], Silverman [17] or Koblitz [5] To add two distinct points one needs one inversion, one squaring and two multiplications whereas doubling takes one more squaring. ECC, IF q ell HEC, IF qhyp I S M I S M Addition 1 1 2 1 2 23 Doubling 1 2 2 1 5 22 m fold 6 10 12 6 24 134 log 2 m = ....

I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic curves in cryptography. London Mathematical Society Lecture Note Series. 265. Cambridge University Press, 1999.

....characteristic can be solved in polynomial time using Schoof s algorithm [33] and its improvements due to Atkin [2] and Elkies [7] An excellent F.W.O. research assistant, sponsored by the Fund for Scienti c Research Flanders (Belgium) account of the resulting SEA algorithm can be found in [3] and [20] For nite elds of small characteristic, Satoh [30] described an algorithm based on p adic methods which is asymptotically faster than the SEA algorithm. Skjernaa [34] and Fouquet, Gaudry and Harley [9] extended the algorithm to characteristic 2 and Vercauteren [36] presented a memory ....

I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic curves in cryptography. London Mathematical Society Lecture Note Series. 265. Cambridge University Press., 1999.

....curve E de ned over a nite eld F q , a base point P 2 E(F q ) of order n and two points generated by P , xP and yP (where x and y are integer) nd xyP . This problem is closely related to the well known elliptic curve discrete logarithm problem (ECDLP) given E(F q ) P; n and xP , nd x)[7] and there is strong evidence that the two problems are computationally equivalent (e.g. see [8] and [16] All protocols in this paper have been described in the setting of the group of points on an elliptic curve de ned over a nite eld. The following abbreviations are used for clear ....

I. F. Blake and G. Seroussi, Elliptic Curves in Cryptography, London Mathematical Society Lecture Note Series 265, Cambridge University Press, 1999.

....it is applied to groups whose order is very dicult to compute. This is not the case for groups of elliptic curves over nite elds GF(q) q = 2 k or q a large prime) because of the polynomial time algorithm of Schoof [32] and recent improvements by Atkins, Couveignes, Elkies, Lercier and Morain [1, 4, 5, 8, 18, 19, 23]. The expected run time of the best of these algorithms is O( log q) 6 ) Our notation and assumptions are described in x2, and the new algorithms for determining a cryptosystem, signing and verifying a message are given in x3. The new algorithms are compared with the standard ElGamal ....

I. F. Blake, G. Seroussi and N. P. Smart, Elliptic Curves in Cryptography, London Mathematical Society Lecture Note Series, Vol. 265, Cambridge University Press, 1999.

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Ian F. Blake, Gadiel Seroussi, and Nigel P. Smart. Elliptic Curves in Cryptography, vol. 265 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2000.

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Ian F. Blake, Gadiel Seroussi, and Nigel P. Smart. Elliptic Curves in Cryptography, vol. 265 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2000.

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Blake, I. F.; Seroussi, G.; Smart, N. P.: Elliptic curves in cryptography. Reprint of the 1999 original. London Mathematical Society Lecture Note Series, 265. Cambridge University Press, Cambridge, 2000. CMP 2000:15

....noted that these promises do not necessarily realize when using cards with Java because there is no direct access to hardware. There are many variations of both of the aforementioned fields; many properties can be carefully chosen to affect either performance or ease of implementation. See e.g. [19], 21] Of these we chose the odd prime field largely for practical reasons. As mentioned before, we had a possibility to use an existing debugged implementation. This implementation was based on the finite field arithmetic being an odd prime field. That existing code base and architecture ....

....card 10 inversion routine is already near the optimal and major improvements are unlikely. Inversion routine also quite heavily uses division which spends a significant amount of time calling multiplication. Our division and multiplication routines are not optimal and could be improved [18] [19]. 6 Future Work Many things still remain undone if one considers implementing cryptographic operations for a smart card in software. Here we would like to take the opportunity to suggest directions for future research. To produce a more widely usable implementation than our prototype, speed of ....

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Ian F. Blake, G. Seroussi, Nigel P. Smart, Elliptic Curves in Cryptography, London Mathematical Society Lecture Note Series 165, 1999.

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I. Blake, G. Seroussi, and N. P. Smart, Eds. Advances in Elliptic Curve Cryptography, London Mathematical Society Lecture Note Series 317, Cambridge University Press, 2005.

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I. Blake, G. Seroussi, and N. P. Smart. Elliptic Curves in Cryptography. London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.

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I. Blake, G. Seroussi, and N. P. Smart. Elliptic Curves in Cryptography. London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.

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I. Blake, G. Seroussi and N. Smart. Elliptic Curves in Cryptography. London Mathematical Society Lecture Notes Series, 265, Cambridge Univ. Press, 1999.

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Blake, I. F., Seroussi, G., and Smart, N. P. Elliptic Curves in Cryptography, vol. 265 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.

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I. F. Blake, G. Seroussi, and N. P. Smart, Elliptic curves in cryptography, London Mathematical Society Lecture Note Series, vol. 265, Cambridge University Press, Cambridge, 2000.

No context found.

I. Blake, G. Seroussi, and N. Smart, Elliptic curves in cryptography, London Mathematical Society Lecture Note Series, vol. 265, Cambridge University Press, 1999.

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I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic curves in cryptography. London Mathematical Society Lecture Note Series. 265. Cambridge University Press., 1999.

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Ian Blake, Gadiel Seroussi and Nigel Smart, Eliptic Curves in Cryptography , London Mathematical Society Lecture Note Series 265, Cambridge Press.

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I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic curves in cryptography. London Mathematical Society Lecture Note Series. 265. Cambridge University Press., 1999.

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I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic curves in cryptography. London Mathematical Society Lecture Note Series. 265. Cambridge University Press., 1999.

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I. Blake, G. Seroussi, and N. Smart, Elliptic curves in cryptography, London Mathematical Society Lecture Note Series, vol. 265, Cambridge University Press, 1999. MINIMAL EXPANSIONS IN REDUNDANT NUMBER SYSTEM 15

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