1+ day, 1+ hour ago (521+ words) A zero knowledge proof (ZKP) answers a question without revealing anything more than answer. For example, a digital signature proves your possession of a private key without revealing that key. You can think of Fermat's primality test as a zero knowledge proof. For example, I can convince you that the following number is composite without telling you what its factors are. Fermat's little theorem'says that if'n'is a prime and'b'is not a multiple of'n, then A number'b such that bn1" 1 (mod'n) is a proof that'n is not prime, i.e. n is composite. So, for example,'b = 2 is a proof that'n above is composite. This can be verified very quickly using Python: I tried the smallest possible base [1] and it worked. In general you may have to try a few bases. And for a few rare numbers (Carmichael numbers) you won't be able to…...

1+ day, 20+ hour ago (228+ words) Monero has a way of generating new addresses analogous to the way HD wallets generate new addresses for Bitcoin. In both cases, the recipient's software can generate new addresses to receive payments that others cannot link back to the recipient. Monero users have two public/private keys pairs: one for viewing and one for spending. Let Ks and ks be the public and private spending keys, and let Kv and kv be the public and private viewing keys. Then the user's ith subaddress is given by Here'G is a generator for the elliptic curve Ed25519 and'H is a hash function. The hash function output and kv are integers; the public keys, denoted by capital'Ks with subscripts and superscripts, are points on Ed25519. The corresponding private keys are As with hierarchical wallets, the user scans the blockchain to see which of his addresses…...

2+ day, 18+ min ago (236+ words) In spherical geometry, the interior angles of a triangle add up to more than ". And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds ". On a sphere of radius 1, the area equals the triangle excess Area = E = interior angle sum " ". Small triangles have interior angle sum near ". But you could, for example, have a triangle with three right angles: put a vertex on the north pole and two vertices on the equator 90" longitude apart. In hyperbolic geometry, the sum of the interior angles of a triangle is always less than ". In a space with curvature "1, the area equals the triangle defect, the difference between " and the angle sum. Area =D = " " interior angle sum. Again small triangles have an interior angle sum near ". Both spherical and hyperbolic geometry are locally Euclidean. The…...

2+ day, 3+ hour ago (352+ words) Let " be the upper half plane, the set of complex real numbers with positive imaginary part. When we measure distances the way we've discussed in the last couple posts, the geometry of " is hyperbolic. What is a circle of radius r in "? The same as a circle in any geometry: it's the set of points a fixed distance r from a center. But when you draw a circle using one metric, it may look very different when viewed from the perspective of another metric. Suppose we put on glasses that gave us a hyperbolic perspective on ", draw a circle of radius'r centered at'i, then take off the hyperbolic glasses and put on Euclidean glasses. What would our drawing look like? In the previous post we gave several equivalent expressions for the hyperbolic metric. We'll use the first one here. Here…...

3+ day, 4+ hour ago (329+ words) The previous post described a metric for the Poincar" upper half plane. The development is geometrical rather than analytical. There are also analytical formulas for the metric, at least four that I've seen. It's not at all obvious that the four equations are equivalent, or that any of them matches the expression in the previous post. There are equations for expressing arcsinh, arccosh, and arctanh in terms of logarithms and square roots. See the bottom of this post. You could use these identities to show that the metric expressions are equal, but I don't know of a cleaner way to do this than lots of tedious algebra. Before diving into the calculations, you might want some assurance that you're trying to prove the right thing. Here's some Python code that generates random pairs of complex numbers and shows that the…...

3+ day, 23+ hour ago (305+ words) One common model of the hyperbolic plane is the Poincar" upper half plane ". This is the set of points in the complex plane with positive imaginary part. Straight lines are either vertical, a set of points with constant imaginary part, or arcs of circles centered on the real axis. The real axis is not [] The post Hyperbolic metric first appeared on John D. Cook. One common model of the hyperbolic plane is the Poincar" upper half plane ". This is the set of points in the complex plane with positive imaginary part. Straight lines are either vertical, a set of points with constant imaginary part, or arcs of circles centered on the real axis. The real axis is not part of ". From the perspective of hyperbolic geometry these are ideal parts, infinitely far away, and not part of the plane itself. We…...

5+ day, 16+ hour ago (303+ words) The opening line of William Gibson's novel Neuromancer is famous: The sky above the port was the color of a television, tuned to a dead channel. When I read this line, I knew immediately what he meant, and thought it was a brilliant line. Later I learned that younger readers didn't know what he was saying. My mind went to an old black-and-white television, one that received analog broadcasts, and that displayed "snow" when tuned to a channel that had no broadcast signal. Someone whose earliest memories of television are based on digital color broadcast might imagine the sky above the port was solid blue rather than crackly gray. Gibson discusses how his book has aged in a preface to a recent edition. He says that science fiction that is too prescient would be received poorly. Imagine a novel from…...

6+ day, 1+ hour ago (438+ words) Suppose Alice runs a confidential restaurant [1]. Alice doesn't want there to be any record of who visited her restaurant but does want to get paid for her food. She accepts Monero, and instead of a cash register there are two QR codes on display, one corresponding to her public view key'A and the other corresponding to her public spend key'S. A customer Bob walks into the restaurant and orders a burger and fries. When Bob pays Alice, here's what's going on under the hood. Bob is using software that generates a random integer'r and multiplies it by a point'G on an elliptic curve, specifically ed25519, obtaining the point on the curve. The software also multiplies Alice's view key'A, a point on the same elliptic curve, by'r, then runs a hash function'H on the produce'rV that returns an integer'k. Finally, Bob's software…...

1+ week, 2+ day ago (160+ words) I was reading about Shackleton's incredible expedition to Antarctica, and the Weddell Sea features prominently. That name sounded familiar, and I was trying to remember where I'd heard of Weddell in math. I figured out that it wasn't Weddell exactly but Weddle I was thinking of. The Weddell Sea is named after James Weddell (17871834). Weddle's integration rule is named after Thomas Weddle (18171853). I wrote about Weddle's integration rule a couple years ago. Weddle's rule, also known as Bode's rule, is as follows. Let's try this on integrating sin(x) from 1 to 2. If we divide the interval [1, 2] into 6 subintervals,h = 1/6. The 8th derivative of sin(x) is also sin(x), so it is bounded by 1. So we would expect the absolute value of the error to be bounded by 9 / (69 " 1400). Let's see what happens in practice....

1+ week, 2+ day ago (139+ words) The previous post includes code for solving the equation i.e. finding the value of'n for which the'nth harmonic number is the closest to'm. It works well for small values of'm. It works for large'm in the sense that the solution is very close to'm, but it's not necessarily the best solution. For example, set'm = 100. The code returns and indeed for that value of'n, Hn "100 " 3 " 10"15 and that's as much as we could hope for with IEEE 754 floats. n = exp(m ") is very good for large values of'm. Using Mathematica we can find the exact value of'n. A similar process can find the solution to In case you're wondering whether my function for computing harmonic numbers is accurate enough, it's actually overkill, with error O(1/120n4)....