Two Keys for Gisela

It had taken Uwe a while to understand that. He understood it at roughly the moment KevIn read him the reviews of PhysioPremium Ruhr, one after another, all in the same tone, until after the forty-seventh Uwe said “Stop.”

Now it was Monday, half past eight, and Gisela was sitting in a bright waiting room in Herne, in her good coat, with her handbag on her lap like a shield. Sabine sat beside her. Something was playing quietly on the radio that Gisela called “lift music”.

“Mrs Brettschneider?”

A woman in her early forties stood in the doorway, in a navy polo shirt, with short hair and forearms that showed she had spent twenty years getting people moving again.

“Ayşe Kurt. You’re the lady who fell over a vacuum cleaner.”

“A robot vacuum,” said Gisela, standing up. “He was on strike.”

Ayşe Kurt raised an eyebrow. “On strike.”

“He wanted Sundays off. And he got them. Ever since, he stands around getting in the way on his days off.”

“I know that from my husband,” said Ayşe Kurt. “Come on in.”

Balance

The first session consisted mainly of questions. How Gisela had fallen. What had hurt since. How many stairs there were at home. Whether she got up at night. Whether there were rugs, runners, “trip hazards”.

“A dachshund,” said Gisela. “And the robot vacuum. And my son, when he’s watching football.”

Then Ayşe Kurt took Gisela’s right arm and lifted it slowly, forwards, to the side, backwards, until Gisela said “Stop.” “That’s where it’s rusted up,” said Ayşe Kurt. “We’ll get that sorted. But not in a week.”

Then Gisela had to stand on one leg. First on the left, then on the right, then with her eyes closed. With her eyes closed it didn’t last long.

“After a fall, a lot of people stop trusting their bodies,” said Ayşe Kurt. “Then they move less, and then they really do fall. We want to turn that round. You’ll get exercises for home. Ten minutes every day. Not twenty, not tomorrow, not ‘when I’ve got time’. Ten, every day.”

“I’m eighty-two,” said Gisela. “I haven’t got time for ‘when I’ve got time’.”

Ayşe Kurt laughed. It was a loud, honest laugh, and Sabine thought that you couldn’t buy a laugh like that. Not even for 4.9 stars.

A Letterbox with a Slot

At home Gisela wanted to send Herbert a photo. Of Wilhelm’s notebook, the page with the Rhine. On Sunday Tobi had added her to the family group, “Brettschneiders + Herbert”, in which so far it had mainly been Uwe sending photos of the barbecue and Herbert sending voice messages eleven minutes long.

“KevIn,” said Gisela, holding the phone as far away from herself as if it might bite. “If I send that in there, can everyone read it? The company? You?”

“Not me. And the company that runs the app, going by everything it describes publicly, can’t either. The group is end-to-end encrypted. That means: your phone encrypts the photo before it goes off, and only Herbert’s phone decrypts it again. In between it’s gibberish. Even on the company’s computers.”

“But then Herbert needs the key. Like with Papa. You said that’s the big problem.”

Tobi, who was lying on the sofa, looked up from his phone. “He was going to explain that to me yesterday. He said tomorrow.”

“Today is tomorrow,” said Gisela.

“That’s true,” said KevIn. “All right. Imagine a letterbox. A perfectly ordinary one, with a slot at the top. Anyone can post a letter in it. The postman, the neighbour, Marlies.”

“Marlies doesn’t post anything at mine.”

“In theory. Anyone can post something. But only one person has the key to open it. You. The letterbox is, so to speak, your public key. You can show it around everywhere; it isn’t secret. And the little key in your apron is your private key. You don’t show that to anyone.”

“And how does that work in the phone?”

“In the phone it’s exactly the same, only with maths. Every phone has two keys that belong together. A public one, which it gives to everyone. And a private one, which never leaves the phone. Whatever is locked with the public one can only be opened by the private one. Herbert’s phone sends you its public key. Your phone uses it to lock the photo in. And only Herbert’s private key can open it. That’s in Herbert’s pocket.”

“Everything’s in Herbert’s pocket,” said Gisela. “And he never finds anything.”

“That’s a different problem. We’ll come to that.”

Tobi sat up. “So you don’t have to hand the key over secretly any more?”

“That’s exactly the trick. Anyone may see the public key. It’s no use to the eavesdropper who reads it. He can lock with it, but not unlock. Until the seventies that was unthinkable. For thousands of years people believed you needed a secret in order to share a secret.”

Gisela sent the photo. Three minutes later a voice message arrived from Herbert. It lasted fourteen minutes. The first twelve were about his blood pressure.

Paint

On Wednesday Sabine decided to paint the hallway. On his free Sundays Horst had left a trail of grey streaks along the skirting board, and Sabine felt that if you had to paint anyway, you might as well change the colour. She came back from the DIY store with six little sample pots and put them on the kitchen table, together with a stack of empty yoghurt pots.

“That’s perfect,” said KevIn.

“What’s perfect?”

“I was just about to explain to Tobi how two phones agree on a shared key without an eavesdropper finding it out. It works best with paint. Tobi, Gisela, may I borrow the pots for a moment?”

Sabine sighed and handed him the pots. Uwe joined them, because he thought it was about Schalke blue.

“Right,” said KevIn. “Tobi, you’re Herbert. Gisela, you’re Gisela. Uwe, you’re the eavesdropper.”

“Why me?”

“Because you’re nosy and you’re listening anyway.”

Uwe thought about it and couldn’t find a counter-argument.

“Step one. Gisela and Herbert agree openly, in front of everyone, on a shared base colour. Let’s say yellow. Uwe is welcome to know that.”

Tobi filled two pots with yellow. Uwe nodded importantly.

“Step two. Each of you secretly picks your own secret colour and doesn’t tell anyone. Gisela, choose one.”

Without anyone watching, Gisela took a spoonful from the pot labelled “Country House Red” and stirred it into her yellow. Tobi took “Dove Blue”.

“Step three. You swap the mixtures. Completely openly. Uwe may look at the pots.”

Gisela gave Tobi her orange pot, Tobi gave Gisela his greenish one. Uwe bent over them as if he were with forensics.

“Step four. Each of you stirs your own secret colour once more into the pot you’ve been given.”

Gisela stirred Country House Red into Tobi’s greenish yellow. Tobi stirred Dove Blue into Gisela’s orangey yellow. Then they held the pots side by side.

It was the same murky brown, twice.

“It’s the same colour!” said Gisela.

“Yellow plus red plus blue,” said KevIn. “In both. Just mixed in a different order. Now you both have the same secret shade. And Uwe?”

“I saw: yellow. Then orange. Then a sort of green.” Uwe stared at the pots. “I can’t get the brown. I don’t know what they put in.”

“And you can’t unmix paint. You’ll never get the red back out of the orange. Just like with the mincer. That brown is now the shared key. Gisela and Herbert use it to encrypt everything else, quite normally, with AES.”

“And phones do that with paint?” asked Gisela.

“With numbers. With calculations that are easy in one direction and practically impossible in the other. But the idea is exactly this one. Two mathematicians called Whitfield Diffie and Martin Hellman published it in 1976¹. The method is named after them. It was a small revolution.”

“And the paint?” asked Sabine, holding up the two pots of brown.

Everyone looked towards the hallway.

“No,” said Sabine.

Ninety-One

That evening, at dinner, Tobi came back to it. “And the letterbox thing? How does that work with numbers?”

“The best-known method is called RSA, after its three inventors, Rivest, Shamir and Adleman. 1977². It rests on something every child knows and that is still astonishing. Tobi, what’s 7 times 13?”

“91.”

“How long did that take you?”

“Two seconds.”

“Uwe, is 91 a prime number?”

Uwe put down his fork. “Course. It’s odd. You can’t divide it by anything.”

“By 7,” said Gisela, without looking up. “And by 13. The boy just said so.”

Uwe picked his fork up again.

“That’s exactly the point,” said KevIn. “Multiplying is easy. Working out what a number is made of is hard. Not with 91. But with numbers six hundred digits long, it is. With RSA you multiply two enormous prime numbers. The result is part of the public key. The two primes are the secret. Whoever knows them can unlock. Whoever only knows the product would have to break it down, and today’s computers would need longer for that than humans have existed.”

“Today’s,” said Tobi.

KevIn’s light pulsed once. “Well spotted. We’ll talk about that another time.”

“And why do you need to know all this?” asked Uwe, who had taken prime numbers as a personal insult since Year Nine.

“Because you use it every day. Every time your browser shows a little padlock. With online banking. When you order shoes. Your life is full of letterboxes with slots. You just don’t see them.”

Marlies and the Signature

On Thursday Gisela came back from bowling in a bad mood.

“Marlies,” she said, and she didn’t really need to say any more. But she said more anyway. “She claimed I signed in 1982 that the club funds would pay for her trip to Winterberg. I never did. She forged my signature. Even back then. Everybody knows it.”

“Since 1982?” asked Sabine.

“Since 1982. That’s why we don’t speak any more.”

KevIn saw an opportunity and took it. “Gisela, may I tell you something that would annoy Marlies?”

“Always.”

“You can also sign with the same two keys. Only the other way round. With the letterbox, anyone locks with the public key, and only you unlock with the private one. With a signature it’s the reverse: only you can sign with your private key. And anyone can use your public key to check whether the signature is genuine.”

“And Marlies?”

“Marlies can’t forge it. She doesn’t have your private key. She can try as hard as she likes, she won’t manage a signature that passes the check. And something else: the digital signature belongs to exactly this one text. If Marlies changes so much as a comma afterwards, turns ‘Winterberg no’ into ‘Winterberg yes’, the signature doesn’t fit any more. The check fails.”

“Like the mincer,” said Tobi. “One letter different, everything different.”

“Exactly. Because you don’t sign the whole text, you sign its fingerprint, the hash³. If the text changes, the fingerprint changes, and the signature no longer fits.”

Gisela had gone quiet and looked very satisfied. “And that sort of thing really exists?”

“Everywhere. Phone updates are signed, so nobody can slip you a fake one. ID cards have chips with signatures. One modern method for it is called Ed25519⁴. A dreadful name, I know. It was published in 2011, it’s fast, the signatures are short, and it’s used a great deal these days.”

“Ed,” said Gisela. “As in Eduard.”

“If you like.”

“Then from now on I’ll sign with Eduard.” She got up to hang up her coat. “Let Marlies try forging that.”

The Sixth Session

After five sessions Gisela could reach the top kitchen cupboard with her right hand again and could stand on one leg for twenty seconds with her eyes open, and eight seconds with them closed. She did her exercises every day, ten minutes, and KevIn counted out loud, which every time set Rudi barking along. During the exercises Horst was sent to his charging station. He accepted it.

Tobi came along to the sixth session, because Sabine had to work. He sat in the waiting room, listened to the lift music and tried not to be bored.

When Gisela had finished, they went to reception to make the next appointment. Next to the appointment book stood a little wooden stand. On it was a QR code and underneath, quite small:

“Were you happy? Or not? Tell us honestly. Reviews of our practice can only be left here, on site.”

“What’s that?” asked Gisela.

Ayşe Kurt was just coming out of the treatment room, drying her hands. “That’s for McGesund. A directory for practices. If you like, you can review us there. You don’t have to.”

“And if I write something bad?”

“Then it’ll say something bad.” Ayşe Kurt shrugged. “I don’t ask anyone for five hearts. I ask for honest ones. I’ve got twenty-three reviews in four years. Others have eight hundred in one.”

“I know,” said Gisela. “They all sound like KevIn on a bad day.”

“Who’s KevIn?”

“Our speaker.”

Ayşe Kurt nodded as if this were a perfectly normal piece of information. In her line of work you heard a lot.

Tobi took Gisela’s phone and opened the camera. “Just point it at it, Grandma.”

Gisela held the phone up to the stand, as far away as possible, then as close as possible, then away again. On the third attempt it beeped and a page opened.

“There are hearts instead of stars,” said Gisela. “Five of them. Can you give six?”

“No,” said Tobi.

“KevIn gave me six.”

“KevIn cheated.”

Gisela tapped five hearts. Then she wrote. She wrote for a long time. Tobi watched the waiting-room door open and close twice, the next patient being called and the streetlights coming on outside. Gisela typed with one finger, with great emphasis and no regard for the autocorrect, which she considered a cheek.

When she had finished, she let Tobi read it.

“Mrs Kurt tells you what you don’t want to hear, and she’s right. I can reach the top cupboard again and stand on one leg for twenty seconds. Before, it was three. The exercises are a nuisance and they work. She laughs loudly and for real. The music in the waiting room is terrible. Gisela B., 82, Wanne-Eickel.”

“Grandma, that bit about the music …”

“Is true. She said honest.”

Ayşe Kurt, who had been reading over Tobi’s shoulder, laughed her loud laugh again. “My husband’s been telling me that about the music for years. At least now it’s on the internet.”

Gisela pressed “Submit”. A little tick appeared on the screen and the sentence: “Your review was submitted on site and signed.”

“Signed,” said Gisela. “With Eduard?”

“Ask KevIn,” said Tobi.

What Actually Happened

At home she asked KevIn. The whole family was sitting at supper, and Uwe had just been telling them that someone at the utility company where he worked had received an invoice by email that claimed to be from the utility company and wasn’t from the utility company.

“What does ‘signed’ mean?” asked Gisela. “Have I signed something now?”

“So to speak,” said KevIn. “I can’t see inside the system, but McGesund describes publicly how it works, and I can explain what basically happens. There are three steps.”

“Three is good,” said Gisela. “I like three.”

“First: the QR code. It hangs in the practice. Whoever scans it proves that they were there, or at least that they had the code in front of them. Someone sitting in a review factory at the other end of the world, writing eight hundred texts a day, isn’t standing at Mrs Kurt’s reception desk.”

“Unless he photographs the code and sends it round,” said Tobi.

“Well spotted. A stand on its own wouldn’t be proof yet; a system like that has to check more than just a picture of a code. How exactly McGesund does it, I don’t know, and I haven’t made anything up since Bielefeld. But the principle is: reviews are created where the treatment took place. That doesn’t make forgery impossible. But it makes mass forgery very laborious. You’d have to send eight hundred people to Herne.”

“Eight hundred people in Herne,” said Uwe. “People would notice.”

“Second: the signature. Your review is digitally signed, exactly as we discussed with Marlies. With Ed25519, so with Eduard, if you like. And additionally with two further methods that are meant to hold up in the future too. I’ll tell you what that means exactly when we talk about quantum computers. That’ll be a little while yet.”

“And what does the signature do?” asked Sabine.

“If someone changes your text afterwards, turns ‘the music is terrible’ into ‘the music is great’, or five hearts into one, the signature no longer fits. That gets noticed. And anyone who wants to can use the public key to check that the review came about by this route and hasn’t been changed since. You don’t have to simply believe it. You can do the sums yourself.”

“And third?” asked Tobi.

“Third, something else happens with the review’s fingerprint. But that’s a story in itself. It has to do with Bitcoin.”

“With what?” asked Uwe.

“Later.”

What a Signature Can’t Do

Uwe leaned back. “Well, then it’s proven. Then everything it says is true.”

“No,” said KevIn. “And this matters to me. A signature proves where something comes from and that it hasn’t been changed. It doesn’t prove that it’s true.”

“How d’you mean?”

“If Gisela writes that the music is terrible, the signature proves that Gisela wrote that on site and that nobody changed it afterwards. It doesn’t prove that the music is terrible. Maybe Mrs Kurt’s husband secretly likes it after all.”

“He doesn’t like it,” said Gisela. “She said so.”

“Then that’s a signed opinion about an opinion about an opinion. What I mean is: even a genuine review can be unfair. A person can have a bad day. A person can be wrong. A person can be unjust. No mathematics can prevent that. What mathematics can prevent is someone inventing eight hundred people who never existed. And someone secretly rewriting afterwards what real people said.”

Sabine nodded slowly. “That’s really all I wanted. When I was looking for Mum. I didn’t want anyone to guarantee me the truth. I just wanted to know that real people were writing.”

“That’s already a great deal,” said KevIn. “In a world where I could write eight hundred reviews in a second that all sound like me on a bad day, that’s a great deal.”

For a moment the table was quiet. Then Gisela said: “KevIn, do I get two keys as well? A public one and a private one?”

“You’ve already got some. In your phone. Since Tobi set it up. You just don’t notice.”

“But I want to see them.”

Two Keys

The next morning a yellow sticky note was stuck to the fridge, right between the weekly rota that nobody followed and a photo of Rudi in a Santa hat. On it, in Gisela’s neat handwriting:

“GISELA. PUBLIC KEY. Anyone may look. Even Marlies.”

Underneath she had drawn a little letterbox, with a slot.

Uwe stood in front of it with his coffee. “And where’s the private one?”

“I’m not telling you that,” said Gisela. “It’s private.”

KevIn’s light pulsed. Somewhere between Dublin and Frankfurt, a fan spun a little faster. “Gisela. I have to ask. It’s not under the keyboard, is it?”

Gisela smiled the smile she otherwise only wore at the bowling alley, just before she knocked down all nine pins.

Tobi went to look. He lifted the keyboard. Underneath was a new yellow sticky note.

It said: “Not here. Fooled you.”

And underneath, smaller: “Eduard.”