Use "elliptic" in a sentence
elliptic example sentences
elliptic
1. Then she got out her own program for elliptic curve factorization and set about cracking the file she had downloaded from the N
2. The best she could think of was, “Do you know about elliptic curves?”
3. But what was so remarkable was that August had started with the random number she had given him at Ingarö and taken that further to develop a considerably better elliptic curve than the one she herself had made
4. In contrast, elliptic curves offer the same functionality but are not affected by advances in factoring integers; therefore, shorter keys can be used (a 256-bit key in ECC is believed to offer comparable security to a 1024-bit key in an RSA-type scheme)
5. Public/private key pairs are generated by choosing points on these elliptic curves that are mathematically related to each other
6. However, with elliptic curves addition has a special meaning and is defined as follows:
7. Figure 7-3 shows an example elliptic curve with two points A and B and the resulting point C created by following these addition rules
8. Repeatedly jumping between points on the elliptic curve using this method of addition can help you obscure your private key in a digital signature system in the same way, as you’ll soon see
9. Similar to lines, you can use a geometric trick to calculate the sum of adding two points on elliptic curves (i
10. Then draw a vertical line starting at this point of intersection and see where else it crosses the elliptic curve
11. Of course, if we want to “drive around” our elliptic curve, as in our conceptual example involving a car, it’s somewhat awkward that we need two points to generate every new point: As with a car, it would be ideal to go from a single point to another single point so our “car” only has to be in one place at a time
12. Fortunately, this is possible with elliptic curves as well by using a form of multiplication to multiply a point on the curve by an integer, which is the same as adding a point to itself multiple times
13. So far we’ve been depicting elliptic curves as smooth, continuous functions that extend into infinity
14. For practical implementations, only integer-valued points on elliptic curves are allowed, and modular arithmetic is used to keep all of the points within certain bounds (from 0 to 512, for example)
15. Let’s first choose the same elliptic curve that Bitcoin uses, which is called a Koblitz curve (Figure 7-7), using the parameters a = 0 and b = 7
16. We then choose a prime modulo p so that the elliptic curve satisfies this equation:
17. Figure 7-8: On the left is a standard picture of the elliptic curve in the form we’re familiar with (the bold curve) with additional curves that are drawn by using other multiples of p (the thin curves
18. The zero point is important because it is valid output from an addition or point multiplication operation and therefore needs to be carefully accounted for (see the sample code in “Pseudocode for Elliptic Point Summation and Point Multiplication” on page 158)
19. In this elliptic curve point summation (ECPS) pseudocode, which allows you to add two points on the elliptic curve to generate a third point, we first check whether A or B is the zero point ➊ (recall that this is the single weird point that’s part of an elliptic curve that is essentially at infinity)
20. For elliptic curve multiplication, we simply run the ECPS function repeatedly ➊
21. From the private key, Bitcoin wallet programs can calculate the associated public key (which is a point on an elliptic curve; see Chapter 7 for the cryptographic details), which in turn can be converted into a Bitcoin address by applying the RIPEMD160 and SHA256 hash functions
22. Recall that both G and Q are points on the elliptic curve, but that G is publicly known to everyone and is a hard-coded constant in the Bitcoin protocol (whereas Q is unique to you)
23. Nor is it consistent with itself: thus the boa-constrictor has rudiments of hind limbs and of a pelvis, and if it be said that these bones have been retained "to complete the scheme of nature," why, as Professor Weismann asks, have they not been retained by other snakes, which do not possess even a vestige of these same bones? What would be thought of an astronomer who maintained that the satellites revolve in elliptic courses round their planets "for the sake of symmetry," because the planets thus revolve round the sun? An eminent physiologist accounts for the presence of rudimentary organs, by supposing that they serve to excrete matter in excess, or matter injurious to the system; but can we suppose that the minute papilla, which often represents the pistil in male flowers, and which is formed of mere cellular tissue, can thus act? Can we suppose that rudimentary teeth, which are subsequently absorbed, are beneficial to the rapidly growing embryonic calf by removing matter so precious as phosphate of lime? When a man's fingers have been amputated, imperfect nails have been known to appear on the stumps, and I could as soon believe that these vestiges of nails are developed in order to excrete horny matter, as that the rudimentary nails on the fin of the manatee have been developed for this same purpose
24. The woman’s orbit—in a civil state—is, like that of other celestial bodies, either annular or elliptic
25. —, on the elliptic polarization of light, reflected from various substances, xlvi, 390
26. —, theory of the variations of the arbitrary constants in elliptic motion, xxx, 248
