The Math That Holds the Internet Together

Written By Morgan Burns

# The Math That Holds the Internet Together

Public-key cryptography made a promise: strangers could establish a shared secret over a public channel, with no prior meeting and no trusted courier. That promise isn't backed by secrecy of method — the algorithms are public, published, and studied by everyone, including attackers. It's backed entirely by **math problems that are easy to compute in one direction and brutally hard to reverse.** This article looks at the three problems that, between them, secure nearly every encrypted connection on Earth.

## The Asymmetry Principle

Every public-key system relies on what's called a **trapdoor function** — an operation that's cheap to perform forward but, without some secret piece of information, effectively impossible to undo in reasonable time. This isn't the same as a classical cipher's "hidden method," because the *function itself* is public. What's hidden is a private number that acts as a shortcut back through the trapdoor. Without it, an attacker is stuck brute-forcing a problem specifically chosen because brute force doesn't work.

Three such problems underpin almost all of modern encryption.

## 1. Prime Factorization (RSA)

Multiply two large prime numbers together, and you get their product instantly. Given only that product, however, finding the two original primes is extraordinarily slow once the numbers get large enough — there's no known shortcut, only exhaustive or near-exhaustive search through candidate factors.

RSA keys are built from this asymmetry. The public key is derived from the product of two huge primes; the private key requires knowing the primes themselves. A 2048-bit RSA key involves numbers hundreds of digits long — factoring one with today's best-known algorithms and hardware would take far longer than the age of the universe.

## 2. Discrete Logarithms (Diffie-Hellman, DSA)

Take a number, raise it to some power within a specific modular arithmetic system, and the result is easy to compute. Given only the result, though, working backward to find the original exponent — the "discrete logarithm" — is believed to be computationally intractable for well-chosen numbers.

This is the problem behind the original Diffie-Hellman key exchange and related signature schemes. Two parties can each pick a secret exponent, exchange values computed from it publicly, and combine them to arrive at a shared secret — while an eavesdropper watching the exchange is stuck trying to reverse a discrete logarithm to figure out either secret.

## 3. Elliptic Curve Discrete Logarithms (ECC)

**Elliptic curve cryptography** uses a geometric variant of the same discrete-log idea, defined over points on a specific type of curve rather than plain numbers. The appeal is efficiency: elliptic curve problems are considered as hard to reverse as RSA's factoring problem, but achieve that security with much smaller keys — a 256-bit elliptic curve key offers roughly the security of a 3072-bit RSA key. Smaller keys mean faster computation and less bandwidth, which is why ECC has become the backbone of most modern TLS connections, including much of the traffic behind HTTPS today.

## "Hard" Is a Bet, Not a Proof

Here's the uncomfortable truth underlying all three: **none of these problems have been mathematically proven to be hard.** They're believed to be hard because brilliant mathematicians have tried to find efficient solutions for decades and failed. That's strong empirical evidence, but it isn't the same as Shannon's proof of perfect secrecy from the last article. It's a bet — a very well-tested one, but a bet nonetheless.

This is precisely why cryptographers watch these problems so closely, and why any breakthrough in factoring or discrete logarithms would be front-page news across the security world — it would mean the ground underneath enormous portions of global infrastructure had shifted.

## The Threat Already on the Horizon

There's one scenario where this bet is already known to fail: a sufficiently powerful **quantum computer**. In 1994, mathematician Peter Shor showed that a quantum algorithm could solve both prime factorization and discrete logarithms efficiently — collapsing the very asymmetry RSA, Diffie-Hellman, and ECC depend on. Today's quantum computers aren't yet powerful or stable enough to run this attack against real-world key sizes, but the theoretical break already exists on paper, which is a very different situation from "nobody has found a classical shortcut yet."

This isn't a distant academic curiosity — it's actively reshaping which cryptographic standards get adopted today, because encrypted data captured now could be decrypted retroactively once sufficiently powerful quantum computers exist.

*Next in this series: how cryptographers are responding before that day arrives — the new families of post-quantum algorithms built on entirely different hard problems, and why standards bodies are already rolling them out.*


  1. Alien Magic — William Benjamin — https://books2read.com/u/4EdvBl

  2. All The Flowers In The Rainbow — Ronald Bartholomew — https://books2read.com/u/4ArqDN

  3. Blackwood Mountain's Secret — Jack Wilson — https://books2read.com/u/bxxWWv

  4. Bomb Iran — Jack Wilson — https://books2read.com/u/mqjV66

  5. Choke Point — Garrison West — https://books2read.com/u/3LPpd7

  6. Commander in the Chief House — James Essex — https://books2read.com/u/mdZpRw

  7. Cyber Dracula on the Moon — Stephen Jacobs — https://books2read.com/u/47KzBE

  8. Draconicum Astralis — Frank Jackson — https://books2read.com/u/38NWqB

  9. Dragon Beast 2: The Global Resonance — Orion Graves — https://books2read.com/u/mgMPo0

  10. Dragon Beast: Magic of the Machine — Orion Graves — https://books2read.com/u/31ND1W

  11. Dull World — Jack Wilson — https://books2read.com/u/mgDRgv

  12. Echoes of Eternity — Frank Jackson — https://books2read.com/u/3kXX7g

  13. Echoes of the Machine — Blake Edwards — https://books2read.com/u/bO5Gjo

  14. Elysium: The AI Conundrum — Buddi T: GAL — https://books2read.com/u/brdvvM

  15. Enhancing Cyber Culture — Stan Bradley — https://books2read.com/u/4N5O7x

  16. Enhancing Tech Theory — T. T. Samuels — https://books2read.com/u/3112aa

  17. Existential Reflections — Jean Francois Enrique — https://books2read.com/u/bwgPgP

  18. From Khartoum to Kharg Island — Liam Conrad — https://books2read.com/u/mB0Qkv

  19. Galaxy Outlaws — Jack Wilson — https://books2read.com/u/bQEeVP

  20. He Who — Morgan Burns — https://books2read.com/u/mZGQ92

  21. Heady Days — Barnaby Finch — https://books2read.com/u/mZ6JxE

  22. History in Time: Middle East — Neville St Claire — https://books2read.com/u/4DvDlg

  23. I'm You — Morgan Burns — https://books2read.com/u/3L7ljJ

  24. I(You!) — Morgan Burns — https://books2read.com/u/m0DwlV

  25. Introduction to Algorithms — Harold Morrison — https://books2read.com/u/m08kOl

  26. Magic Aliens — Marvin Hamner — https://books2read.com/u/3G7EqK

  27. Memory Days — Yug Gohan — https://books2read.com/u/4DvOQr

  28. Mirror of Memories — Steven Jacobs — https://books2read.com/u/bwALVv

  29. Neon Sanctuary — Marvin Hamner — https://books2read.com/u/b6glaZ

  30. Ocean Moon — Rane Corvus — https://books2read.com/u/3LPpdX

  31. Peaks — Robin Crystal — https://books2read.com/u/mYlZRY

  32. Portals — Chester Craig — https://books2read.com/u/4EOrRO

  33. Practical Mysticism I — Brent Newman — https://books2read.com/u/b5yp17

  34. Practical Mysticism II — Brent Newman — https://books2read.com/u/3L7QpM

  35. Practical Mysticism III — Brent Newman — https://books2read.com/u/4XpxK9

  36. Practical Mysticism IV — Brent Newman — https://books2read.com/u/4EV2Wz

  37. Practical Mysticism V — Brent Newman — https://books2read.com/u/479q77

  38. Practical Mysticism VI — Brent Newman — https://books2read.com/u/b5yKgG

  39. Quade's Cosmos — Blake Edwards — https://books2read.com/u/brjVoY

  40. Raid Island — Frank Jackson — https://books2read.com/u/bQ8Kx6

  41. Silicon Cross — Rane Corvus — https://books2read.com/u/mvAWee

  42. Singularity — Harold Morrison — https://books2read.com/u/m279l1

  43. Skin to Sea — Brooks Miller — https://books2read.com/u/mZ9q0R

  44. Soot and Sapphire — Garth Toxo — https://books2read.com/u/bx2Yeo

  45. Souls Beneath — Zoules The Magnificent — https://books2read.com/u/mgQdyD

  46. Sovereign Intelligence — Jack Wilson — https://books2read.com/u/479LKR

  47. Sovereign Sync — Rane Corvus — https://books2read.com/u/3JzZVg

  48. Sovereign Sync II — Rane Corvus — https://books2read.com/u/3kRLK8

  49. Starman — Chad Winthrop — https://books2read.com/u/mKgZZ5

  50. System Restore — Emit Jugal — https://books2read.com/u/3ndwOx

  51. The 9/11 War — Liam Conrad — https://books2read.com/u/b66x8M

  52. The Aldous Huxley Compendium — Brent Newman — https://books2read.com/u/3LPQoD

  53. The Celestial Nomad — Russell Lavine — https://books2read.com/u/m2nqwj

  54. The Chrono Accord — Frank Jackson — https://books2read.com/u/mdBa1l

  55. The Cosmic Awakening — Stan Bradley — https://books2read.com/u/bPPvA7

  56. The Dream Navigator — William Benjamin — https://books2read.com/u/bpvXR6

  57. The Elaraeon — Garth Toxo — https://books2read.com/u/38A8wL

  58. The Fabric of the Hill — Preston Ashcroft — https://books2read.com/u/meYZxl

  59. The Forgotten Enclave — Lyle Davenport — https://books2read.com/u/bO2koA

  60. The Game of Win — Barnaby Finch — https://books2read.com/u/mYNrPM

  61. The Glass Garden — Jonathan Smith — https://books2read.com/u/3n9kpo

  62. The Good Ghost — Ronald Bartholomew — https://books2read.com/u/bzElEn

  63. The Hard Handover — Tomison Peters — https://books2read.com/u/meWP2g

  64. The Infinite Spiral — Steven Jacobs — https://books2read.com/u/499ddW

  65. The Mnemosyne Protocol — Harold Morrison — https://books2read.com/u/4AqgV0

  66. The Node Seven Sync — Rane Corvus — https://books2read.com/u/3GnWLn

  67. The Perpetual Twins — Ronald Bartholomew — https://books2read.com/u/3RENkp

  68. The Soot of Oakhaven — Garth Toxo — https://books2read.com/u/3REVpx

  69. The Temporals — Morgan Burns — https://books2read.com/u/m29EaG

  70. The Visionary — Preston Ashcroft — https://books2read.com/u/4jZ5wl

  71. The War That Rewrote the Middle East — Liam Conrad — https://books2read.com/u/3L7o95

  72. The Zeus Mandate — Russell Lavine — https://books2read.com/u/m0yVYP

  73. They Are Here — Yug Gohan — https://books2read.com/u/3RW5Zv

  74. Venus Rising — Marvin Hamner — https://books2read.com/u/38Nr76

  75. Verses Through the Ages — Morgan Burns — https://books2read.com/u/baENGP

  76. Visions Through Time — Lyle Davenport — https://books2read.com/u/mvAYVe

  77. War and Wisdom — Liam Conrad — https://books2read.com/u/mgMjwK

  78. Whispers from the Stars — Blake Edwards — https://books2read.com/u/4X19rL

  79. Whispers of the Old World — Steven Jacobs — https://books2read.com/u/bz89g2

  80. William Tell — Liam Conrad — https://books2read.com/u/49AM6M

  81. X-9 — The Robotic Van Helsing — Marvin Hamner — https://books2read.com/u/mvgwxX

  82. XOXO — Garth Toxo — https://books2read.com/u/4XWRp5

  83. Zephyrz — Garth Toxo — https://books2read.com/u/47L2PN

  84. Zooz — Frank Jackson — https://books2read.com/u/3y0WOZ


  1. Alien Magic — William Benjamin — https://books2read.com/u/4EdvBl

  2. All The Flowers In The Rainbow — Ronald Bartholomew — https://books2read.com/u/4ArqDN

  3. Blackwood Mountain's Secret — Jack Wilson — https://books2read.com/u/bxxWWv

  4. Bomb Iran — Jack Wilson — https://books2read.com/u/mqjV66

  5. Choke Point — Garrison West — https://books2read.com/u/3LPpd7

  6. Commander in the Chief House — James Essex — https://books2read.com/u/mdZpRw

  7. Cyber Dracula on the Moon — Stephen Jacobs — https://books2read.com/u/47KzBE

  8. Draconicum Astralis — Frank Jackson — https://books2read.com/u/38NWqB

  9. Dragon Beast 2: The Global Resonance — Orion Graves — https://books2read.com/u/mgMPo0

  10. Dragon Beast: Magic of the Machine — Orion Graves — https://books2read.com/u/31ND1W

  11. Dull World — Jack Wilson — https://books2read.com/u/mgDRgv

  12. Echoes of Eternity — Frank Jackson — https://books2read.com/u/3kXX7g

  13. Echoes of the Machine — Blake Edwards — https://books2read.com/u/bO5Gjo

  14. Elysium: The AI Conundrum — Buddi T: GAL — https://books2read.com/u/brdvvM

  15. Enhancing Cyber Culture — Stan Bradley — https://books2read.com/u/4N5O7x

  16. Enhancing Tech Theory — T. T. Samuels — https://books2read.com/u/3112aa

  17. Existential Reflections — Jean Francois Enrique — https://books2read.com/u/bwgPgP

  18. From Khartoum to Kharg Island — Liam Conrad — https://books2read.com/u/mB0Qkv

  19. Galaxy Outlaws — Jack Wilson — https://books2read.com/u/bQEeVP

  20. He Who — Morgan Burns — https://books2read.com/u/mZGQ92

  21. Heady Days — Barnaby Finch — https://books2read.com/u/mZ6JxE

  22. History in Time: Middle East — Neville St Claire — https://books2read.com/u/4DvDlg

  23. I'm You — Morgan Burns — https://books2read.com/u/3L7ljJ

  24. I(You!) — Morgan Burns — https://books2read.com/u/m0DwlV

  25. Introduction to Algorithms — Harold Morrison — https://books2read.com/u/m08kOl

  26. Magic Aliens — Marvin Hamner — https://books2read.com/u/3G7EqK

  27. Memory Days — Yug Gohan — https://books2read.com/u/4DvOQr

  28. Mirror of Memories — Steven Jacobs — https://books2read.com/u/bwALVv

  29. Neon Sanctuary — Marvin Hamner — https://books2read.com/u/b6glaZ

  30. Ocean Moon — Rane Corvus — https://books2read.com/u/3LPpdX

  31. Peaks — Robin Crystal — https://books2read.com/u/mYlZRY

  32. Portals — Chester Craig — https://books2read.com/u/4EOrRO

  33. Practical Mysticism I — Brent Newman — https://books2read.com/u/b5yp17

  34. Practical Mysticism II — Brent Newman — https://books2read.com/u/3L7QpM

  35. Practical Mysticism III — Brent Newman — https://books2read.com/u/4XpxK9

  36. Practical Mysticism IV — Brent Newman — https://books2read.com/u/4EV2Wz

  37. Practical Mysticism V — Brent Newman — https://books2read.com/u/479q77

  38. Practical Mysticism VI — Brent Newman — https://books2read.com/u/b5yKgG

  39. Quade's Cosmos — Blake Edwards — https://books2read.com/u/brjVoY

  40. Raid Island — Frank Jackson — https://books2read.com/u/bQ8Kx6

  41. Silicon Cross — Rane Corvus — https://books2read.com/u/mvAWee

  42. Singularity — Harold Morrison — https://books2read.com/u/m279l1

  43. Skin to Sea — Brooks Miller — https://books2read.com/u/mZ9q0R

  44. Soot and Sapphire — Garth Toxo — https://books2read.com/u/bx2Yeo

  45. Souls Beneath — Zoules The Magnificent — https://books2read.com/u/mgQdyD

  46. Sovereign Intelligence — Jack Wilson — https://books2read.com/u/479LKR

  47. Sovereign Sync — Rane Corvus — https://books2read.com/u/3JzZVg

  48. Sovereign Sync II — Rane Corvus — https://books2read.com/u/3kRLK8

  49. Starman — Chad Winthrop — https://books2read.com/u/mKgZZ5

  50. System Restore — Emit Jugal — https://books2read.com/u/3ndwOx

  51. The 9/11 War — Liam Conrad — https://books2read.com/u/b66x8M

  52. The Aldous Huxley Compendium — Brent Newman — https://books2read.com/u/3LPQoD

  53. The Celestial Nomad — Russell Lavine — https://books2read.com/u/m2nqwj

  54. The Chrono Accord — Frank Jackson — https://books2read.com/u/mdBa1l

  55. The Cosmic Awakening — Stan Bradley — https://books2read.com/u/bPPvA7

  56. The Dream Navigator — William Benjamin — https://books2read.com/u/bpvXR6

  57. The Elaraeon — Garth Toxo — https://books2read.com/u/38A8wL

  58. The Fabric of the Hill — Preston Ashcroft — https://books2read.com/u/meYZxl

  59. The Forgotten Enclave — Lyle Davenport — https://books2read.com/u/bO2koA

  60. The Game of Win — Barnaby Finch — https://books2read.com/u/mYNrPM

  61. The Glass Garden — Jonathan Smith — https://books2read.com/u/3n9kpo

  62. The Good Ghost — Ronald Bartholomew — https://books2read.com/u/bzElEn

  63. The Hard Handover — Tomison Peters — https://books2read.com/u/meWP2g

  64. The Infinite Spiral — Steven Jacobs — https://books2read.com/u/499ddW

  65. The Mnemosyne Protocol — Harold Morrison — https://books2read.com/u/4AqgV0

  66. The Node Seven Sync — Rane Corvus — https://books2read.com/u/3GnWLn

  67. The Perpetual Twins — Ronald Bartholomew — https://books2read.com/u/3RENkp

  68. The Soot of Oakhaven — Garth Toxo — https://books2read.com/u/3REVpx

  69. The Temporals — Morgan Burns — https://books2read.com/u/m29EaG

  70. The Visionary — Preston Ashcroft — https://books2read.com/u/4jZ5wl

  71. The War That Rewrote the Middle East — Liam Conrad — https://books2read.com/u/3L7o95

  72. The Zeus Mandate — Russell Lavine — https://books2read.com/u/m0yVYP

  73. They Are Here — Yug Gohan — https://books2read.com/u/3RW5Zv

  74. Venus Rising — Marvin Hamner — https://books2read.com/u/38Nr76

  75. Verses Through the Ages — Morgan Burns — https://books2read.com/u/baENGP

  76. Visions Through Time — Lyle Davenport — https://books2read.com/u/mvAYVe

  77. War and Wisdom — Liam Conrad — https://books2read.com/u/mgMjwK

  78. Whispers from the Stars — Blake Edwards — https://books2read.com/u/4X19rL

  79. Whispers of the Old World — Steven Jacobs — https://books2read.com/u/bz89g2

  80. William Tell — Liam Conrad — https://books2read.com/u/49AM6M

  81. X-9 — The Robotic Van Helsing — Marvin Hamner — https://books2read.com/u/mvgwxX

  82. XOXO — Garth Toxo — https://books2read.com/u/4XWRp5

  83. Zephyrz — Garth Toxo — https://books2read.com/u/47L2PN

  84. Zooz — Frank Jackson — https://books2read.com/u/3y0WOZ

Morgan Burns
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Post-Quantum Cryptography: Rebuilding Before the Break

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One Key or Two? Symmetric Ciphers, Public Keys, and the Problem Nobody Could Solve