It is an aphrodisiac that encourages passion and intimacy between you and your partners Awakening sensual creativity and tender fantasies are well known rose quartz powers.Strengthening love or romantic relationship.Delicate in color but strong in presence, the benefits of rose quartz are quite magnificent. The presence of rose quartz in your home brings emotional, physical and spiritual healing. Coined as the “Heart Stone,'' it is an important gem that draws new love, keeps family ties together and nurtures friendship. Powerful vibrations that echo into your ambiance is how the rose quartz benefits affect your life. Rose Quartz Healing Properties and Benefits ![]() ![]() The rich history of this semi-precious stone and its popularity makes it a true gem over the centuries and it is cherished worldwide. The characteristics and the difference between the two have been developed over time and the meaning of rose quartz has been identified. Rose Quartz’s color is always even and does not form a single crystal while pink quartz’s color is unstable and normally forms a single crystal. In East Asia, it was used as a token of love and had been widely used and honored in their carvings and lucky charms.ĭuring the '90s, mineralogists discovered that rose quartz and pink quartz are different by examining unique crystalline specimens. Rose quartz crystal meaning for them is akin to personal growth and well-being. On the other hand, Americans connect with the crystal to achieve balance and peace. It was being used in facial masks and cosmetic products as an anti-aging component. The Assyrians of Mesopotamia were the first people who have utilized and embodied rose quartz spiritual meaning in their practices and ceremonies.įor Romans and Egyptians, rose quartz stone meaning was beauty. It was extensively part of the arts and crafts of the Assyrians as early as 7000 B.C. The use of rose quartz can be dated back to ancient times. In another Greek story, Cupid and Eros brought the rose quartz to humans to sow love and hope. Zeus brought Adonis back from war out of empathy and made the meaning of rose quartz, love rekindled or renewed. The rose quartz crystal meaning in legends suggests that Aphrodite’s blood-stained on white quartz when she cut herself on a thorny bush to save her love Adonis from Are’s war. The name rose quartz came from the Greek word “hyalos” meaning glass. Whenever you're seeking more love in your life, this is the go-to stone.Early civilizations used to call rose quartz, pink quartz or hyaline quartz. Its properties, according to Heather Askinosie, crystal expert and author of CRYSTA元65 and Crystal Muse, are "equally powerful for manifesting new relationships and deepening existing ones-whether it's the relationship you have with yourself, a romantic partner, or friends and family. It's typically a soft pink, though there are some varieties that are a richer, darker pink, as well as light purple or lavender varieties. In ancient Egyptian, Chinese, and Tibetan culture, rose quartz has long been thought of as a stone for beauty, as well as a symbol of love. In another tale, Cupid used rose quartz to inspire love between people. ![]() When their blood mixed together, it stained quartz a subtle pink, and rose quartz was born. As the story goes, Aphrodite, the goddess of love, once cut herself by accident while trying to save her lover, Adonis. Rose quartz has a long history dating back to Greek mythology. ![]() And if we're getting technical, it's actually a mineral ( like many other popular crystals). Today, this crystal can be found all over the world, including in parts of the U.S., India, Brazil, Australia, Sweden, and more. Rose quartz, in particular, is a bit more rare. As its name suggests, rose quartz is a type of quartz (which is fairly common).
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* reaConverter 7.6.x – dead for both preview and render. * SoThink SWF Decompiler – sort of, now shows no previews at all, but the saved SVG image loads in Inkscape. It’s a whole image renderer, not a shape decompiler. * Kurst’s SWF Renderer – works fine to PNG, no update needed. * JPEXS Free Flash Decompiler – works fine to PNG, there’s a new version from a few weeks ago. SWF reader software needed to read and convert old creative Flash assets? ĪeeFromString generates unnecessary GetMember actionĪctiveX and PlayerGlobal fail to download (during 4.1.Adobe finally forcibly killed Flash. flash player projector content debugger path doesn't work)ĩ-slice scaling support (DefineScalingGrid)Ī bit meta: This bug tracker page's filter and sort aren't workingĪ lot of space after adding/replacing characters in fontsĪ question regarding command line replacement for a scriptĪ way to mass-import shapes, sprites, text, etc.Ībility of adding new tags in the SystemClass of a SWFĪbout client.swf that release version can read it but nightly cant. usr/bin/ffdec: line 30: [: too many argumentsġ00% CPU usage on building script tree phase longer and longerrġ7.x.x is not showing all ActionScript filesģ) PlayerGlobal (.swc) path where do i find that? ( 2. regexp/ compilation in AS3 direct editation selectid does not work on sprites and buttons ![]() replace commanline arg gives NullPointer on script problem format argument is not working in commandline enabledebugging -injectas3 supplied debug information results in incorrect line numbers during debugging (AS3) Exception when removing any entry from DefineSceneAndFrameLabelData (2 issues) 1: White simple User Interface despite oceanic user interface skin | 2: definevideo and definevideostream ''this.'' placements in appropriate areas. #566 weird bug with external flash player window in latest nightly after FFDec closing #218 bug is not fixed yet (link not wrapped) "true" statements (with automatic deobfuscation, too) "This Application requires Java Runtime Environment 1.8.0" "Replacing" Image/Shape and then Saving SWF will corrupt SWF. "Replace references with other character ID" not working on Images "register" parameter names instead "_loc*_" "printGraph max recursion level reached" exception "Needed characters / Dependent characters" font unreadable in most themes "Go to document class" does nothing when the document class name is obfuscated "Export selection" button is grayed out for AS2 classes "Dependent frames" field not updated after removing symbols "DefineVideoStream " stuck on desktop after closing program mid-export. "Curly brace on new line" doesn't affect starting method brace "Check for updates" is not working with nighty build but is working with official release "Cannot load SWF file" in the new 1.2.0 version "-zoom" command line parameter for Frame export Possibly the parentheses () are missing after this function reference. Warning: Function value used where type Boolean was expected. ![]() Ribbon's menu does not fit into the screen (1280px) Problem when saving SWF file, all files are saved with extension. Returned: Program changes were made but user is not ![]() Ignored: Developer decided to take no action on this Made changes to the program and new versionĬlosed: This means the user is satisfied with issue Upgraded: Issue is in upgraded state when developer Now, for some reason it cannot be implemented Postponed: This means developer is not working on it Opened: Opened state means developer started working Quick by label: ABC addFrameScript Adding script AIR AS1/2 AS3 AVI BinaryData Blend modes Broken JPEG Broken SWF Build Bundles BUTTONCOND Canvas export Charset Clipping Commandline Cyclic tags Debugger Decompilation DefineInSprite Deobfuscation Direct Editation Documentation Export External Viewer ExternalLink FFDec source Filters FLA export Flash Viewer FlashCC Flex Fonts Formatting Frames Free transform GFX Google Code GUI HaXe Header Iggy Images Import Import scripts ImportAssets Initializers Installation Java Java 9 JSyntaxPane Large SWF Library LineEndings Linux Loader LZMA Mac Memory search Morphshapes Namespaces Obfuscated on(xxx) P-code PDF Performance Progress Proxy Raw Editation Read only SWF Rename identifiers RTL Scaling strokes Script execution Search Search Memory Shapes Site feature Sound Special Sprites Startup SVG SwfMovie SWFTools Tag position Tags Tamil Text Tools Translation Unicode Update system Value Too Large Video VirusTotal XML Zoom Id ![]() I have no idea what email address it wants in this field, as nothing I’ve tried is valid. This page has a message that there is no phone number associated with my email, but when I try to provide that info I can’t because I also need to provide an “alias email”. Next to my main email address there are 2 links: “Check email” and “Delete”, and next to “Account Status” it says “Active”. Here I can see my email address/account (and 2 sub accounts, that I also can’t access). When I login to myAT&T (where I can see my active wireless account), if I go under My Profile I can get to the section for my “linked accounts”. No one can tell me how or why it was deleted. Though my email address exists, they all tell me my account has been deleted/disabled which is why they can’t help get my access back. I have made numerous calls to the various AT&T support numbers, including Digital Assistance and ConneTech (AT&T’s paid support service). ![]() But you can’t know what you’re working with until you contact EarthLink.I have an account with a 22 year-old address that I haven’t been able to access sine 12/9/17. ![]() Your area probably has download speeds in the ballpark of at least one plan spelled out below, and your package will fall in line with that price point. It basically rents bandwidth from better-connected internet providers (like AT&T or CenturyLink) and sells it to you, so it can’t control your internet connection quality.Įarthlink may try to offer the best speeds it can for the DSL or fiber network it’s borrowing, but there’s only so much it can do.Īt least EarthLink tries to even out prices across the US. See, EarthLink doesn’t own its own internet network. While unlimited data is a perk you’ll find in nearly every EarthLink plan (except its LTE home internet plans), consistent prices, speeds, and availability are not. You won’t have to worry about EarthLink throttling your speeds or slapping you with fines because you overindulged in a Netflix marathon. we highlighted EarthLink’s customer service reputation and dragged its pricing, so what’s left? It looks like it’s time to bring up the joys of unlimited data and the trouble with internet plans that vary by location.ĮarthLink joins the elite ranks of internet providers that don’t arbitrarily limit your internet use with crummy data caps. And if you can get through a potentially turbulent installation day, you’ll be cradled back in the capable arms of EarthLink’s customer service team. You may end up scheduled last for your technician’s appointments that day or get rescheduled if an emergency pops up.Īll this hassle is a bummer, especially since EarthLink charges you $79.95 for installation.īut your installation day could always go off without a hitch. These contracted technicians will probably install your internet just fine, but EarthLink customers aren’t their priority. They may not know much about EarthLink as a company, and they might even claim their company owns EarthLink (it doesn’t). That means on installation day, your installation technician may show up late in an AT&T van (or CenturyLink, Verizon, etc.). EarthLink doesn’t have direct control of your technician’s schedule, branding, or training. Like how EarthLink contracts with other internet providers to use their networks, EarthLink also contracts to use those providers’ technicians. “What’s with the enigmatic warning about EarthLink installation in this article’s intro?” you ask. See John Phillips, The Reformation of Images: Destruction of Art in England, 1535-1660 (Berkeley and Los Angeles: University of California Press, 1973) Margaret Aston, England's Iconoclasts, vol. Sally-Beth MacLean, Chester Art: A Subject List of Extant and Lost Art Including Items Relevant to Early Drama, Early Drama, Art, and Music Reference Series 3 (Kalamazoo: Medieval Institute Publications, 1982).Ĥ. Anderson, Drama and Imagery in English Medieval Churches (Cambridge: Cambridge University Press, 1963).ģ. Hildburgh, "English Alabaster Carvings as Records of the Medieval Religious Drama," Archaeologia 93 (1949): 51-101 M. Richard Beadle (1994 reprint, Cambridge: Cambridge University Press, 1996), 109-33.Ģ. See also David Mills, "The Chester Cycle," in The Cambridge Companion to Medieval English Theatre, ed. 3, 9 (London: Oxford University Press, 1974-86) citations to the Chester cycle in this article are to this edition and refer to play and line numbers. Martin Stuchfield, and Uli Wunderlich for providing some of the illustrations.ġ. ![]() Clough, Miriam Gill, Fred Kloppenborg, Martine Meuwese, David Mills, Malcolm Jones, Clifford Davidson, and James Stokes for their valuable comments and suggestions on earlier drafts of this paper, and to Allan B. Louis, Missouri, in 1998, while I was a Ph.D. The original idea for the present article was first presented at the Midwest Modern Language Association conference in St. Reider fully examines their cultural history, multifaceted roles, and complex significance as "others" to the Japanese. Noriko Reiderýs book is the first in English devoted to oni. Oni appear frequently in various arts and media, from Noh theater and picture scrolls to modern fiction and political propaganda, They remain common figures in popular Japanese anime, manga, and film and are becoming embedded in American and international popular culture through such media. ![]() Usually male, their female manifestations convey distinctivly gendered social and cultural meanings. There has been much ambiguity in their character and identity over their long history. Characteristically threatening, monstrous creatures with ugly features and fearful habits, including cannibalism, they also can be harbingers of prosperity, beautiful and sexual, and especially in modern contexts, even cute and lovable. Oni, ubiquitous supernatural figures in Japanese literature, lore, art, and religion, usually appear as demons or ogres. ![]() You don’t need to go to Google Playstore and install the game. If you have an APK file, then there is an option in Bluestacks to Import APK file. You can use the App the same way you use it on your Android or iOS smartphones. Now you can just double click on the App icon in bluestacks and start using NeoRouter VPN Professional App on your laptop. You can find the App under list of installed apps in Bluestacks. ![]()
![]() Find Reviews, Ratings, Directions, Business Hours, Contact Information and book online appointment. You can view 1 entry, complete with personal details, location history, phone numbers, relatives and locations for John Stratton. One of the best Family Practice, Healthcare business at 8 Morton Ave, Ridley Park PA, 19078 United States. Just as Friday had begun to appear on the. John - 'Right Place Wrong Time' from the album In The Right Place. Traverso is Board Certified by the American Board of Surgery and he is currently a member of numerous prestigious medical societies including the American Society for Surgery of the Hand, the American Association for Hand Surgery, the American College of Surgeons and the American Medical Association. John Stratton lives in Northport, MI Below are the results we could find for John Stratton. Youre listening to the official audio for Dr. Traverso completed his training in Microsurgery at Columbia University in New York.ĭr. Traverso gained expertise in the care of the hand and wrist. Get contact info, maps, medical practice history, affiliated hospitals & more. Traverso completed his hand and wrist surgery training at the University of Connecticut under the world renowned Dr. where he was honored as a ‘Chief Resident’ in 2013 and received the Howard University Chairman of Surgery Award for Most Outstanding Resident in 2012 as well as Intern of the Year in 2008.Īfter residency, Dr. Malcolm John Rebennack (born November 20, 1941, New Orleans, Louisiana, USA died June 6, 2019), better known by his stage name Dr. He completed his residency training at Howard University Hospital in Washington D.C. Traverso has done research in stem cell transplantation as well as many areas of treatment of hand and wrist injuries. Lazaro Pepen MD at READING FAMILY MEDICAL CENTER Reading, PA. Platelet rich plasma (PRP) protein injectionsĭr. View John Traverso’s profile on LinkedIn, the world’s largest professional community.Traverso operates on both national and international patients, including Trinidad and the Caribbean.ĭr. Dr Traverso is a Board Certified, Fellowship trained, hand and microvascular surgeon with subspecialty certificate in surgery of the hand. Dr. Traverso has special expertise in the treatment of challenging problems of the forearm, wrist and hand. Purnell Traverso specializes in hand and wrist surgery in adults. ![]() World of Warships is not a classical pay-to-win game. As you progress in the game, however, your positioning, communication, and synchronization will improve along with your abilities for precise shots. All shots are typically lucky shots, and players tend to focus on what’s in front of them and don’t pay much attention when they’ve isolated themselves from their allies. In the lowest tier of play from the early stages of the game, there is no goal or direction, as players simply move their ships around. However, this emphasis on cooperation may not be so clear to a novice. A decent player can hit their target from 4 kilometers away half of the time, and a great player will have set up their position so that they can shoot without fear of retaliation. Each ship has a critical role to play, no matter what type of ship it is. This being said, positioning, team collaboration, and map knowledge are very important in the game. In a situation when there is so little room for error, every action and movement must be carefully planned. There is a difference of a single millimeter between a direct successful hit and a pricey dud. With miles of open space between them, players engage in cross-play long-range warfare. Multiplayerīattles in World of Warships require players to use their brains and work together as a team in order to be successful. Burning steel is illuminated by good lighting and water effects, and the sinking of a battleship is dramatic. Options like “Sea Rendering Quality” or “Sky and Clouds Quality” sound like specialized settings in any other game, but in World of Warships, half of what the game renders is either water or the sky, so they play a great role in the overall video-gaming experience. A variety of native resolutions are supported, as well as multiple monitors. The game comes packed with visual choices that should tone down enough for less powerful machines to run it. It’s hard to believe how good these warships look. With a ship’s weapons mounted in the body, it gives you the thrill to think about the ideal way to face an enemy and unload on it. Some players find the art of angling in Warships to be a pleasure that keeps their brain occupied with precise mathematical and geographical calculations that ensure they don’t miss a target. Sneaking up behind a ship and landing a series of devastating blows while the target’s cannons turn to counter the attacker is a particularly successful tactic. Captains have more time to prepare for battle when they anticipate their enemy’s attacks and make the necessary preparations. Bringing weapons out to face an attacker takes time, especially in bigger boats. To really enjoy playing World of Warships, you must have a long-term strategy in place. When the captain is preoccupied with other matters, the rudder and throttle controls are designed to be set and ignored. Wargaming continues to excel at creating user-friendly keyboard controls for complex battle machines. The captain has a bird’s-eye view of the ships as if he were perched twenty feet above the central control tower. Repairing ships and reloading guns also necessitate the usage of in-game currency. Free Warships at the lowest stages of the game provide a tactical challenge if you want to earn more points. As you progress through the game, you’ll be able to unlock use more powerful and advanced vessels. Increasing your Tier by winning matches and performing well in combat gives you points. Some are faster, more powerful, or equipped with smoke screens and devastating torpedo spreads, others are all three. There are four main ship types (destroyer, cruiser, carrier, and battleship) with each of them having its own sailing style and abilities. Game-PlayĪs you join World of Warships, a wide range of tools and weapons are available to play with. Slow and vulnerable, the battleships lack the mobility of land and air vehicles, thus, they lay an even greater emphasis on teamwork and strategic positioning. The ships of World of Warships engage in combat across all the oceans, from South Pacific archipelagos all the way to the Alaskan glacier waters. ![]() The game is a polished combination of ponderous war vessels and massive weapons. In 2015, Wargaming released World of Warships, a free-to-play naval battle simulation video game that dives players into semi-historical naval machine battles. World Of Warships using Boot Camp Assistant.How to download World of Warships on Mac.Play World of Warships through GeForce Now.Play World of Warships through Boosteroid. ![]() Disney has been able to make this accessible to their younger audience. These additional elements on the show have cause it to delve into deeper themes that are important for the youth today to watch. It’s really cool to be embodying that in a TV show,” said Ruth Righi, who plays Sydney on the show. They still make it through together and they learn from each other they go through experiences that are hard to explain. “Not every family is the same, and they’re not perfect. The show is perfect for all kinds of families, however, since it gravitates towards the aspects of growing up, the struggles of parenting, and the importance of family. With a strong focus on both the kids and the parents, the show is something that families with single parents can relate to. I can understand those connections, and I think it’s important to see,” said Caroline Rhea, who plays Sydney’s grandmother on the show. The show not only captures the essence of finding yourself in middle school but also the struggle of growing up with a single parent, or raising a child as a single parent. While the show contains Disney’s special charm, appealing to young audiences who commonly watch the channel, they have also been able to add elements that reach the lives of a broader audience. Whether it’s living with a single parent, struggling with self identity, or trying to smoothly transition into teenage-hood, Sydney and her father, Max, live relatively similar lives. Through many flashbacks, the show parallels Sydney’s life to the one of her father in the 1990s. ![]() Disney Channel’s new original show “Sydney to the Max” follows the life of Sydney, a young girl entering into the complicated world of middle school, while living with her single father and grandmother. ![]() #6 Shadow of Mordor Crashes on Changing Video Settings You can also try lowering other graphics settings like Shadow Quality, etc to make it run better. If you are getting poor FPS, first thing you can try is disable Vsync. I am sure that many of you would have hoped to run the game at maxed out settings, but since it’s a resource-intensive game, you might have to tone down a few settings. If you are getting a disk write error during your installation, you should consult this troubleshooting thread to fix your problem. If you are downloading the package, and it gets stuck or gives you an error, just pause the download and restart it after a few moments. #3 Shadow of Mordor Download Stuck Due to Some Reason If Shadow of Mordor is having the same problem, you can give Far Cry 3 profile a go which some users have reported to be working fine. Sometimes, it’s hard to get the maximum out of both your GPUs using SLI. #2 Shadow of Mordor Working SLI Profile for Nvidia Users Furthermore, if you are on a laptop, make sure that you are running the game using dedicated graphics memory. The only way to fix it is that you purchase a better GPU with DirectX 11 support. You might have installed it on your PC but the hardware might be unable to utilize it. ![]() Shadow of Mordor has high needs when it comes to hardware so if you are experiencing this error, it’s most probably that your current GPU does not support DirectX 11. However, for your assistance, we have this list of most common issues being faced by the game with workarounds to fix them. That being said, the game is huge, which also means chances that the game will have more problems than usual, especially around the launch are high. If you are below minimum requirements in any regard, the game most probably isn’t going to work at your end. It’s not that most of us weren’t planning to play it already on PC, but impressive reviews by most of the sites must have added more people to the list. Fortunately, the game offers a 30fps cap option if running the game at the highest quality possible is your goal, though you'll still want as much GPU memory as possible for a good experience.įor a comparison of the Xbox One, PS4, and PC (running without the Ultra textures) versions, check out our video below.Middle Earth: Shadow of Mordor has been released to masses with a bang. You might as well, right?"Īccording to Digital Foundry, high-end artwork can actually be run on GPUs with less memory, but you'll run into issues without capping the framerate. "It makes sense to get it out into the world there-we have it, we built it that way to look as good as possible. "Then obviously there's going to be that boundary where our monster development PCs are running it OK-but why not give people the option to crank it up?" Roberts continued. Roberts said the game's artists built the game's assets at "an outrageously high fidelity" before then optimizing them "to fit onto, to fit onto PCs at high-end specs." It is, however, what's making me so sad about having spent a ton of money on a 3 GB video card not so long ago.ĭigital Foundry spoke with lead designer Bob Roberts about why this option is even being offered, given how few people have the necessary hardware. In addition to a 64-bit processor, you'll need at least 8 GB of RAM and a DirectX 11-compatible video card with at least 6 GB of memory. Installing the HD content is now easy enough, but you'll need a fairly ridiculous rig in order to take advantage of this if you follow the recommended requirements. Earlier this week, the pack was only obtainable through an unintuitive process. The option for these Ultra textures are built into the game, but in order to actually activate them, you need to download a free DLC pack from Steam. "For the most part, we found that the high setting-in itself placing a significant video RAM requirement on the user-offers the bulk of the ultra experience," writes Digital Foundry. However, it's not the kind of leap over High that you might be hoping for, particularly because the most-improved areas are hard to notice while playing the game in a normal manner. You can also check out the video below for a comparison of the PS4 and PC (with Ultra textures) versions.Įxpectedly, it's very pretty when running the game at Ultra. In the very likely case that you're unable to take advantage of these yourself, Digital Foundry has provided some impressions of what they're like. Gamers with extremely high-end computers can download a special HD content pack for Middle-earth: Shadow of Mordor that enables the game's Ultra-level textures. ![]() While we’ve made efforts to make the model refuse inappropriate requests, it will sometimes respond to harmful instructions or exhibit biased behavior.Instead, our current models usually guess what the user intended. Ideally, the model would ask clarifying questions when the user provided an ambiguous query.These issues arise from biases in the training data (trainers prefer longer answers that look more comprehensive) and well-known over-optimization issues. The model is often excessively verbose and overuses certain phrases, such as restating that it’s a language model trained by OpenAI.For example, given one phrasing of a question, the model can claim to not know the answer, but given a slight rephrase, can answer correctly. ChatGPT is sensitive to tweaks to the input phrasing or attempting the same prompt multiple times.Fixing this issue is challenging, as: (1) during RL training, there’s currently no source of truth (2) training the model to be more cautious causes it to decline questions that it can answer correctly and (3) supervised training misleads the model because the ideal answer depends on what the model knows, rather than what the human demonstrator knows. ChatGPT sometimes writes plausible-sounding but incorrect or nonsensical answers.We performed several iterations of this process. Using these reward models, we can fine-tune the model using Proximal Policy Optimization. We randomly selected a model-written message, sampled several alternative completions, and had AI trainers rank them. To collect this data, we took conversations that AI trainers had with the chatbot. To create a reward model for reinforcement learning, we needed to collect comparison data, which consisted of two or more model responses ranked by quality. We mixed this new dialogue dataset with the InstructGPT dataset, which we transformed into a dialogue format. We gave the trainers access to model-written suggestions to help them compose their responses. We trained an initial model using supervised fine-tuning: human AI trainers provided conversations in which they played both sides-the user and an AI assistant. We trained this model using Reinforcement Learning from Human Feedback (RLHF), using the same methods as InstructGPT, but with slight differences in the data collection setup. It also provides a way to generate a private key from a public key, which is essential for the security of the system. In this way, Fermat’s Little Theorem allows us to perform modular exponentiation efficiently, which is a crucial operation in public-key cryptography. ![]() To decrypt the message, the recipient simply computes m = c b m o d n m = c^b \bmod n m = c b mod n, which (by Fermat’s Little Theorem) is equivalent to m = ( m a ) b m o d n = m ( a b ) m o d n = m 1 m o d n = m m o d n m = (m^a)^b \bmod n = m^(ab) \bmod n = m^1 \bmod n = m \bmod n m = ( m a ) b mod n = m ( ab ) mod n = m 1 mod n = m mod n. To encrypt a message with the user’s public key ( n, a ) (n, a) ( n, a ), we first convert the message into a number m m m (using some agreed-upon scheme), and then compute the encrypted message c c c as c = m a m o d n c = m^a \bmod n c = m a mod n. ![]() This means that when we multiply a a a and b b b together, the result is congruent to 1 1 1 modulo n n n. The user’s private key would be the pair ( n, b ) (n, b) ( n, b ), where b b b is the modular multiplicative inverse of a modulo n n n. The user’s public key would then be the pair ( n, a ) (n, a) ( n, a ), where aa is any integer not divisible by p p p or q q q. We might choose two large prime numbers, p p p and q q q, and then compute the product n = p q n = pq n = pq. For example, suppose we want to generate a public-key cryptography system for a user with the initials “ABC”. One way to generate these keys is to use prime numbers and Fermat’s Little Theorem. In a public-key cryptography system, each user has a pair of keys: a public key, which is widely known and can be used by anyone to encrypt a message intended for that user, and a private key, which is known only to the user and is used to decrypt messages that have been encrypted with the corresponding public key. One of the most common applications is in the generation of so-called “public-key” cryptography systems, which are used to securely transmit messages over the internet and other networks. Fermat’s Little Theorem is used in cryptography in several ways. |
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