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In the beginning, there was symmetric encryption.

In the beginning, there was symmetric encryption. If you had the key you could encrypt …. If you had the key you could encrypt … Message: Key: +3 Ciphertext :. ATTACK AT DAWN ↓↓↓↓↓↓ ↓↓ ↓↓↓↓ DWWDFN DW GDZQ. If you had the key you could decrypt ….

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In the beginning, there was symmetric encryption.

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  1. In the beginning, there was symmetric encryption.

  2. If you had the key you could encrypt …

  3. If you had the key you could encrypt … Message: Key: +3 Ciphertext: ATTACK AT DAWN ↓↓↓↓↓↓ ↓↓ ↓↓↓↓ DWWDFN DW GDZQ

  4. If you had the key you could decrypt …

  5. If you had the key you could decrypt … Message: Key: +3 Ciphertext: ATTACK AT DAWN ↑↑↑↑↑↑ ↑↑ ↑↑↑↑ DWWDFN DW GDZQ

  6. … and some people were happy.

  7. Then, there was asymmetric encryption.

  8. Some people encrypted …

  9. Some people encrypted … … others decrypted.

  10. E-commerce ensued …

  11. E-commerce ensued … … and more people were happy.

  12. The first and most used asymmetric cipher was RSA.

  13. The first and most used asymmetric cipher was RSA.

  14. Some people noticed the algebraic structure …

  15. Some people noticed the algebraic structure …

  16. Some people noticed the algebraic structure … Ergo …

  17. Some people noticed the algebraic structure … Ergo …

  18. Some people noticed the algebraic structure … Ergo …

  19. Some people noticed the algebraic structure … Ergo …

  20. Some people noticed the algebraic structure … Ergo …

  21. They looked for interesting applications …

  22. They looked for interesting applications … … and they failed.

  23. People mused …

  24. People mused … … if only RSA worked additively …

  25. People mused … … if only RSA worked additively … we could compute sums …

  26. People mused … … if only RSA worked additively … we could compute sums … and averages …

  27. People mused … … if only RSA worked additively … we could compute sums … and averages … and tally elections …

  28. I was one of those musing.

  29. I was one of those musing. An additive encryption homomorphism …

  30. I was one of those musing. An additive encryption homomorphism …

  31. The product of encryptions of two messages is an encryption of the sum of the two messages.

  32. I used this to build verifiable election systems …

  33. I used this to build verifiable election systems … … and I was really happy …

  34. I used this to build verifiable election systems … … and I was really happy … and few others cared.

  35. What people really wanted was the ability to do arbitrary computing on encrypted data…

  36. What people really wanted was the ability to do arbitrary computing on encrypted data… … and this required the ability to compute both sums and products …

  37. What people really wanted was the ability to do arbitrary computing on encrypted data… … and this required the ability to compute both sums and products … … on the same data set!

  38. People tried to do this for years …

  39. People tried to do this for years … … and years …

  40. People tried to do this for years … … and years … … and years …

  41. People tried to do this for years … … and years … … and years … … with no success.

  42. WHY does ADD AND MULTIPLY help?

  43. WHY does ADD AND MULTIPLY help? XOR (add mod 2) AND (mult mod 2) 0 0 0 XOR 0 0 AND 0 1 XOR 0 1 1 AND 0 0 0 XOR 1 1 0 AND 1 0 1 XOR 1 1 AND 1 0 1

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