Lesson 15 - Strength in numbers

Lesson 15 Strength in numbers Let me see: four times five is twelve, and four times six is thirteen, and four times seven is fourteen—oh dear! I shall never get to twenty at this rate!   Numbers are an essential part of our everyday life. Large numbers, however, aren’t something most of us are too familiar with. The largest numbers we might encounter in everyday life are in the range of millions, billions, or trillions. We might read about millions of people in poverty, billions of dollars spent on bank bailouts, and trillions of national debt. Even though it’s hard to make sense of these headlines, we are somewhat comfortable with the size of those numbers. Although we might seem comfortable with billions and trillions, our intuition already starts to fail with numbers of this magnitude. Do you have an intuition how long you would have to wait for a million/billion/trillion seconds to pass? If you are anything like me, you are lost without actually crunching the numbers. Let’s take a closer look at this example: the difference between each is an increase by three orders of magnitude: 10?, 10?, 10¹². Thinking about seconds is not veryuseful, so let’s translate this into something we can wrap our head around: 10?: One million seconds was 1½ weeks ago. 10?: One billion seconds was almost 32 years ago. 10¹²: One trillion seconds ago Manhattan was covered under a thick layer of ice. As soon as we enter the beyond-astronomical realm of modern cryptography, our intuition fails catastrophically. Bitcoin is built around large numbers and the virtual impossibility of guessing them. These numbers are way, way larger than anything we might encounter in day-to-day life. Many orders of magnitude larger. Understanding how large these numbers truly are is essential to understanding Bitcoin as a whole. Let’s take SHA-256, one of the hash functions used in Bitcoin, as a concrete example. It is only natural to think about 256 bits as “two hundred fifty-six,” which isn’t a large number at all. However, the number in SHA-256 is talking about orders of magnitude?—?something our brains are not well-equipped to deal with. While bit length is a convenient metric, the true meaning of 256-bit security is lost in translation. Similar to the millions (10?) and billions (10?) above, the number in SHA-256 is about orders of magnitude (2²??). So, how strong is SHA-256, exactly? “SHA-256 is very strong. It’s not like the incremental step from MD5 to SHA1. It can last several decades unless there’s some massive breakthrough attack.”