Exponential

by JM Addison

  *

  She cut him off, “Oh! Don’t worry about me. Please go right ahead and eat your dinner. I know it was rude of me to just drop by, but I was really hoping you could help. Please, go ahead and eat. I would be happy to wait as long as you need.”

  “It’s no problem, I usually eat right there with the news.” He pointed to an easy chair in front of the television. He then retreated into what she assumed was the kitchen to get his food. Her heart raced at the comment he had made about the apparent interest in the Tomb. Finally! She might begin to untangle this mystery. While he was gone, she took a moment to examine the room more closely. One entire wall was bookshelves, overloaded with books. Most of the books had to do with math and science. Adjacent to the bookshelves was a large table covered with articles and magazines piled around a couple of large computer monitors. One monitor was running with one of those ‘Search for new proteins, or amino acid’ screen savers displayed, apparently going through data supplied by some bioresearch agency or government lab that collected and distributed amino acid sequence structures over the internet. The computer itself was on the floor, surprisingly large and apparently quite new. The small table next to the easy chair was littered with various scientific articles and periodicals. The same with the end table next to the sofa. She never realized there was so much information available on the subject of math in common periodicals. He came back in carrying a steaming bowl of something and two cups of tea on a tray with a bit of bread.

  “I made you a cup of tea. I hope you don’t mind,” he said as he sat the tray precariously on the pile of paper. He handed her the cup and said, “you looked like you would probably take sugar, so I took the liberty of adding a little.”

  She loved the way he spoke with a bit of British accent softened by the years spent in America. There was also a hint of Irish lilt in the gravely but cheerful voice. He settled himself in the chair, looked at her with bushy eyebrows raised and asked, “So what’s this talk about the Tomb that’s brought you to my door this evening?”

  She began to outline the events of the past few days. Her missing brother. The kidnapping attempts. The death of her mother. The search for ‘Mara the fugitive’ by the police. After practically running out of breath, she paused to collect her thoughts. Professor Tattinger interrupted, “Well, it sounds to me like I should be calling the police, wouldn’t you say?” She was quite unsettled, but relaxed a bit when she caught his wolfish grin and understood he wasn’t serious.

  She also realized some key elements were left out of her story. She explained both she and Chris worked at Viiradium. This seemed to interest the old professor greatly. She began to explain that they were in the business of producing security products for data transfer and computer network communication, but he interrupted her by explaining he knew all about what Viiradium did.

  She described the circumstance of the anonymous e-mail from Chris and the apparent foul play and involvement of Viiradium. She finally explained that the simple e-mail message only alerted her that the Tomb of Rivest had been discovered at Viiradium. Since she had no idea what this meant she could only hope to learn by doing more research.

  “Yes. I suppose you might eventually happen across some text that mentions the Tomb.” He began. “I haven’t heard anyone use that phrase in about twenty five years.”

  She was eager to clear up her cloudy understanding of the situation and pressed him for more information. “Well, tell me! Where is it? How can it help me determine what’s going on? Why are there people trying to kill me!?”

  “Slow down, young lady. I need to give you a little background so you will understand first.” He paused and had a bit of his soup and some tea. She used the time to sip some of the tea herself and was surprised by how good it was. “It’s not so much as ‘where’ is it, but rather, ‘what’ is it. It really began back during the war. I’m talking about World War II of course. The Americans and other allies needed to send messages and information to each other. You couldn’t simply send strategic offensive plans on a broadcast radio station. The enemy would hear and be able to mount a defense. You couldn’t write the plans down and send them by courier because if the courier was captured by the enemy, the plans would be revealed.

  “Hence, the age of encryption was born. Of course, it wasn’t really ‘born’ then since even the ancient Romans were known to have encrypted messages carried by courier. But the idea remains the same. By applying some sort of secret to your message, you could scramble it enough so that it would be unintelligible and too difficult to de-scramble if the message were intercepted by the enemy. This worked very well except that there was one problem. The person receiving the message also had to know the secret for decrypting it. This led to all sorts of fancy code-book and password schemes for decoding messages but in each case, you still had to get the secret to unlock the original message somehow and this always proved to be a big obstacle.

  “Well, along came computers which were very good at scrambling messages so well that they became virtually impossible to decode no matter what. And with computer networks, especially with a network like the internet, messages could be routed to the proper person very rapidly – and unfortunately, secretly intercepted by the adversary too. And still, that same old problem of knowing the secret to de-scrambling the message remained. You couldn’t send the secret with the scrambled message for any ill intended individual who might be able to intercept your computer coded message would have both the encrypted message and the secret key to decode it.

  “Am I making sense so far?”

  “Yes, go on!” she eagerly replied. She wanted to get to the ‘good part’ of the story.

  “Well that changed in the 1970’s. Someone discovered a way to send the key or the secret to decode the message safely along with the message. That might sound impossible, but it turned out to be not all that difficult for a computer.

  “To illustrate how this works, let’s say that we want to leave messages for each other but we don’t want to just pin them up on the bulletin board because they are private. So we put them in a box that is locked. However, this particular lock is unique in that it has two keys: a red one and a blue one. Each key can only unlock the box if the opposite key locked it.

  “So if you put a message in the box and lock it with your red key, only my blue key can unlock it. Not even your red key can unlock the box if the red key was used to lock it. Since my blue key unlocks it only if your red key locked it, I know that you must have locked it so I am positive the message is indeed only from you since the only person with the red key is you.

  “You might wonder, why not have a simple box that has only one kind of key? Say a silver one? We would both have the same key so we can use the box freely to message each other. Well, here’s where it becomes interesting. In order to get you this copy of my silver key, I would have to have one made at the locksmith and then personally deliver it to you so that I can be sure that you indeed have it. If we must meet personally, why bother with the key at all, I’ll just hand you the message.

  “Well the dual key method creates a way to privately send messages without ever actually having to meet! Let’s say I don’t have a blue key to match your red one. So you lock the message in the box with your red key and just leave a few blue keys laying there so that if I happen by, I can just pick up a blue key and look in the box and find your message. This may seem silly, but if there is a message there, I at least know one thing for sure. It had to be from you because one of the blue keys worked, thus the box must have been locked with the red key and since you have the only red key, only you could have put the message in the box. If I respond with a reply message and lock it with one of the blue keys, only your red key can unlock it now – none of the other blue keys will work once the box is locked with a blue key – remember the rule, each key can unlock the box when the other key locks it.

  “So now messages from you are not private since anyone can unlock the box with one of th
e blue keys that you left once you have locked it with your red key. However, all messages To you are private. Once someone locks a message in the box with a blue key, only you can unlock it since you have the only red key. Again, remember the rule, if the box is locked with a blue key, another blue key will not unlock it, this is a unique box where the keys work in pairs.

  “Now we have a ‘one way’ method of getting private messages from anyone to you. In fact we’ll call this your box for anyone to use to leave a message for you. If you want to send me a private message you can not use your box because anyone with a blue key could open and read the message, so we have to have another box, say one that works with a green and yellow pair of keys. We’ll call this ‘my’ box and set it right next to your box. I leave some yellow keys next to the box so you can lock a message inside for me. Since there is only one green key and I have it, you can now send me a private message with my box and by using your box with the blue key you have provided, I can send you a private response to your message.

  “So far, so good – right? Now, suppose there are fifty of us and we want to leave messages for any of us among ourselves. We simply need 50 boxes, one for each person, each with a set of paired keys. One of the keys is duplicated many times and left behind freely for the taking while there is only one of the matching key pair and each person holds that private key for his box to himself. Now all 50 of us can communicate, but in total privacy and without ever have to physically meet!”

  He paused to let this sink in while he ate some more of his soup. Mara was struggling a bit at this nonsense but there was a certain amount of bizarre logic to it. “Then I still have to have a key ring with 50 keys in it” She said. “I have my own key, the red one, and one key from each of the 49 other people. Wouldn’t it just be easier for each of us to carry 50 keys, one for each of the 50 boxes?”

  “Ah! But you miss the point. If that were the case and I leave a private message for you, the other 48 people have a key to your box and could read the message that wasn’t intended for them. These are 50 people that don’t trust each other.”

  “O.K. But how does this tie into computers and the Tomb?”

  “Oh Gosh!” he began rather excitedly, “let’s say I wanted to send you a private message over a public network like the internet. All you have to do is send your ‘blue’ key to me first. We’ll consider the blue key the ‘public’ key. Once I lock or encode the message using the blue key, only you can unlock it because you are the only one with the corresponding red ‘private’ key. If anyone intercepted a copy of the blue key you sent to me, who cares! All they could do with it is lock a message that only you could unlock! The only problem with all of this is that there is no such magical box that works like that with pairs of keys. That is until the 1970’s when somebody invented one.”

  He stopped to have a little more of his meal. Mara said, “I haven’t heard of such a contraption. It seems a little impossible to contrive such an odd system of dual keys.”

  “Ah, but it’s not a mechanical contraption but only a virtual box because it exists just as a mathematical theory. However, being mathematical, it could be implemented using computers, and it has indeed become a reality and a very popular one. It is the basis of practically all private data encryption practices in use today. It is very difficult, if not impossible to overcome by the malicious, mysterious interceptor of messages. For the most part it was invented right here at MIT by a team of three individuals. Mathematician types. You know the kind. And that’s where we get to you, young lady. Viiradium and other companies use this same method today to produce the encryption products that everyone is using.

  “What’s interesting is how it works and this is where the math gets a little droll. To understand it, you have to know a bit about prime numbers and their relationship with each other.”

  Mara was familiar with prime numbers, but a ‘relationship’? “I didn’t think prime numbers were really related except for the fact that they all share the common characteristic of being prime numbers.”

  “Well you see, the trick is to pick two prime numbers, multiply them together to produce a product. Using the same two prime numbers you also choose an exponent. The exponent must meet certain qualifications. It must not share any factors with the product of the first prime minus one and the second prime minus one. Another exponent must be computed which is the multiplicative inverse of the first exponent. The original prime numbers are thrown away.” He made a gesture of tossing something in the waste can. “The product of the original primes and the first exponent together make up the public key. The second exponent is used as the private key.”

  Even with her math background, that was much too quick to follow. She felt like she just walked into a cave with no lantern.

  “Now to encrypt your information, treat each piece of information as a number. For a computer that’s easy since any information is represented as strings of numbers anyway. Each number is raised to the power of the exponent and divided by the product of the original primes. The remainder of the division operation becomes the encrypted version of the original piece of information. The quotient is not needed and simply ignored.

  “To decrypt, take the encrypted information represented as a number, raise it to the power of the second exponent – the private one – and again divide by the product of the primes. This new remainder is the original number. Doing arithmetic where you are only interested in the remainder instead of the quotient is called ‘modular division’.”

  Mara’s head was spinning. She simply looked at Professor Tattinger with a blank expression of bewilderment while he finished his dinner. “Well if one number – the public one – is some sort of ‘inverse’, as you put it, of the private one, it seems like someone could determine the private number given the public one. Wouldn’t this be a weakness?”

  “Very Good! You see, that’s the key to the whole thing! You understand this more than you know. The numbers we are using are very large. So large in fact that factoring them into the original pair of prime numbers is what we mathematicians call a ‘hard problem’. In fact, it is so difficult that in practice, it would take teams of today’s computers years just to factor one such set of numbers to determine the original primes, thus revealing the private key.”

  Mara tried to comprehend some of the details again in her mind as to how this worked but her exhaustion was catching up. Without seeing an example written on a piece of paper… She stopped trying to analyze it and simply believed what the professor was saying. “So what does this have to do with my situation?” she asked.

  “The original three men that discovered or at least developed this technique were Ron Rivest, Adi Shamir and Leonard Adleman. Back when they originally published their work, it was viewed as an interesting curiosity, especially given the computer power needed to handle the decryption process. Back in the 70’s, with the limited power of computers, the process was quite slow. There were those that thought that a factoring algorithm could be developed that would be able to handle the difficult problem of discovering the original primes and when this algorithm was fully developed, it would mean the death of this encryption technique and thus became nick-named the ‘Tomb of Rivest’.

  “However, the three of them went off and started their own company and the rest, shall we say, is history. No such algorithm has ever been discovered and in fact, there have been numerous corporate challenges in effect offering to pay a reward to anyone who can rapidly decrypt such an encrypted message.

  “This really was only the beginning. What evolved was today’s world of safe private data transfer across the public network. In practice, even with today’s computers, the whole process of encrypting pages of data using private and public key pairs is too slow for typical amounts of data. But, this technique solves a very old problem. The problem of sending a secret without having to physically meet. If the secret message was very short, then it would not be too difficult to use this method to send the secret. W
ell, that is the case in our example. The secret we really need to send is that silver key. Once both sides have the same silver key, they can just use that to lock or encrypt their message instead. Using an old fashioned key to encrypt a message – where the key is the same on both sides –is very fast, in fact, it takes almost no time at all. But how do you get the secret key, the ‘silver’ one, to the other person? The method of using the paired keys solves this. Now I can use the paired key technique to send, not a message, but an old fashioned encryption key, that both parties can then use to encrypt or decrypt messages from each other. Once this third key has been sent, now I can quickly encrypt a message for you and since you already received the key through the paired key method, you can quickly decode it.

  “In fact, this is exactly how your internet web browser works today. When you go to a ‘secure’ site on the web, the site sends you the public key of a public / private pair of keys. Your web browser generates a symmetric encryption key – the silver one – and encrypts the key with the site’s public key and sends it back to the site. The site can decode the message with its private key and now both the site and your browser know the encryption key. Tada! Now rapid encryption and decryption can take place using the same key at both ends throughout the remainder of your browser session with that web site.”

  Mara finished the last of her tea and said, “Well, let’s suppose that someone has finally discovered this ‘Tomb of Rivest’ and that someone is likely Viiradium. But I don’t see why this is such a big deal? People are getting kidnapped and killed over this. Do you have any idea why?”

  He gave her a hard stare that made her uncomfortable, then he started to chuckle which grew into a good old fashioned belly laugh. “My dear”, he said, “This is the next best thing to a time machine that you’ll ever get!”

  “What do you mean, Time Machine?”

  “Well, what if you knew accurately, just one day in advance, what would happen tomorrow? Suppose you had the financial section of tomorrow’s paper today? What would that be worth? Billions! You would have all of tomorrow’s prices on hundreds of stocks, commodities, metals, exchange rates and everything. Well, if someone had discovered the Tomb and was using it in some systematic fashion, they could gather private information on anyone or anything, any market, any development simply by watching, collecting and illegally decoding private information. Especially in the corporate world where companies communicate between branch locations in different cities or even in different countries.

  “A large organization could systematically collect and categorize all this data on hundreds of companies and use it for profit. They could snoop and sell anything they learned. And now, if this little operation were about to be blown because of you, I’m sure they wouldn’t think twice about making sure you go away!”

  Suddenly his face became ashen. His expression carried a perplexity as though he had just been licked by a llama. “Dear me!” he said. “You need to leave. What you have just revealed to me is enough to make me dangerous to them, whoever ‘they’ are!”

  “I didn’t mean to cause you any trouble...” Mara began. “I simply had nowhere else to go.” She still had many unanswered questions, so she went on. “How could this be? Wouldn’t this method, or ‘Tomb’ if you prefer, be something well known? Wouldn’t it be the end of secure data transmission as we know it? What you’re saying is that such a discovery would be huge!”

  “That right. It would be ‘huge’ as you put it. So huge, that it’s best if kept quiet. If this were to get out, the whole house of cards would come crashing down. If you happened to be a company like Viiradium whose products depend on this knowledge remaining secret, you might do anything to make sure it stays a secret.

  “What’s fascinating is that someone could have accomplished this! It has never been proven, but always believed that such factoring of primes would be impossible. The only known possibility to date would be to use quantum computing techniques, but those are still in the future. In fact the only place where such a computer exists is in laboratories at places like MIT and Hewlett-Packard. And those machines are still far too crude to be able to be put to any practical application yet.”

  He stood up and walked toward the door. Clearly, the discussion was over. He seemed very nervous and peeked through the curtain out into the street. “Who else knows you’re here?” His voice cracked a bit with his question.

  “No one. I came here directly from the MIT Library. I only came over on a whim and the fact that I remembered you from when I was a student here.” She got up and collected her purse and light jacket.

  As she approached the door, he stopped and gave her a stern warning: “This is bigger that you think. Get some help. The police or government or something. If they find you, I’m afraid they would feel no remorse at ‘removing’ you. Be careful. Think of what happened to your brother and your mom.” He opened the door enough for her to squeeze out and then closed it firmly behind her. She could hear the sound of the locks being engaged. She cautiously approached her car keeping an eye out for anything suspicious. She was pretty anxious before, now she was downright terrified.