CryptologyDay 4: Going public

MAT 140: Introduction to the Mathematical Sciences24 September 2008

Robert Talbert, PhDAssociate Professor of Mathematics and Computing Science rtalbert@franklincollege.edu

Recap of Day 3

•Bits, bytes, and binary representation of text using ASCII

•The XOR operation

•The Simple XOR cipher

•Perfect security and one-time pads

Alice

Alice Bob

Alice Bob

Plaintext

Dear Bob - The meeting will be at

the embassy.

Alice Bob

Plaintext

Dear Bob - The meeting will be at

the embassy.

Alice Bob

Enc

rypt

ion

func

tion

Plaintext

Dear Bob - The meeting will be at

the embassy.

Ciphertext

Qrne Obo - Gur zrrgvat jvyy or ng gur rzonffl.

Alice Bob

Enc

rypt

ion

func

tion

Plaintext

Dear Bob - The meeting will be at

the embassy.

Ciphertext

Qrne Obo - Gur zrrgvat jvyy or ng gur rzonffl.

Alice Bob

Enc

rypt

ion

func

tion

Plaintext

Dear Bob - The meeting will be at

the embassy.

Ciphertext

Qrne Obo - Gur zrrgvat jvyy or ng gur rzonffl.

Alice Bob

Enc

rypt

ion

func

tion

Dec

rypt

ion

func

tion

Plaintext

Dear Bob - The meeting will be at

the embassy.

Ciphertext

Qrne Obo - Gur zrrgvat jvyy or ng gur rzonffl.

Original plaintext

Dear Bob - The meeting will be at the embassy.

Alice Bob

Enc

rypt

ion

func

tion

Dec

rypt

ion

func

tion

Plaintext

Dear Bob - The meeting will be at

the embassy.

Ciphertext

Qrne Obo - Gur zrrgvat jvyy or ng gur rzonffl.

Original plaintext

Dear Bob - The meeting will be at the embassy.

Alice Bob

Enc

rypt

ion

func

tion

Dec

rypt

ion

func

tion

How are these communicated?

The key distribution problem

The key distribution problem

The key distribution problem

The key distribution problem

The key distribution problem

Insecure channel

The key distribution problem

Insecure channel

The key distribution problem

Insecure channel

The key distribution problem

Insecure channel

The key distribution problem

Insecure channel

CENSORED

The key distribution problem

Insecure channel

CENSORED

Anonymity

The key distribution problem

Insecure channel

CENSORED

Anonymity

The key distribution problem

Insecure channel

CENSORED

Anonymity

The key distribution problem

Insecure channel

CENSORED

Anonymity

The key distribution problem

Insecure channel

CENSORED

Anonymity

The key distribution problem

Insecure channel

CENSORED

Anonymity

The key distribution problem

Insecure channel

CENSORED

Anonymity

Many-to-many communication

Activity: A “public-key” cipher

Why does Kid Crypto work? Can an adversary find the private key from the public key? How does this solve the key distribution problem?

Alice Bob

Alice Bob

Alice Bob

Alice Bob

One-way trapdoor function

Alice Bob

Public(e,n)

PrivatedOne-way trapdoor function

Plaintext

Dear Bob - The meeting will be at

the embassy.

Alice Bob

Public(e,n)

PrivatedOne-way trapdoor function

Plaintext

Dear Bob - The meeting will be at

the embassy.

Alice Bob

Public(e,n)

PrivatedOne-way trapdoor function

Plaintext

Dear Bob - The meeting will be at

the embassy.

Alice Bob

Enc

rypt

ion

func

tion

(ex)

mod

n

Public(e,n)

PrivatedOne-way trapdoor function

Plaintext

Dear Bob - The meeting will be at

the embassy.

Ciphertext

Qrne Obo - Gur zrrgvat jvyy or ng gur rzonffl.

Alice Bob

Enc

rypt

ion

func

tion

(ex)

mod

n

Public(e,n)

PrivatedOne-way trapdoor function

Plaintext

Dear Bob - The meeting will be at

the embassy.

Ciphertext

Qrne Obo - Gur zrrgvat jvyy or ng gur rzonffl.

Alice Bob

Enc

rypt

ion

func

tion

(ex)

mod

n

Public(e,n)

PrivatedOne-way trapdoor function

Plaintext

Dear Bob - The meeting will be at

the embassy.

Ciphertext

Qrne Obo - Gur zrrgvat jvyy or ng gur rzonffl.

Alice Bob

Enc

rypt

ion

func

tion

(ex)

mod

n

Dec

rypt

ion

func

tion

(dy)

mod

nPublic(e,n)

PrivatedOne-way trapdoor function

Plaintext

Dear Bob - The meeting will be at

the embassy.

Ciphertext

Qrne Obo - Gur zrrgvat jvyy or ng gur rzonffl.

Original plaintext

Dear Bob - The meeting will be at the embassy.

Alice Bob

Enc

rypt

ion

func

tion

(ex)

mod

n

Dec

rypt

ion

func

tion

(dy)

mod

nPublic(e,n)

PrivatedOne-way trapdoor function

RSA

Alice

Choose two large prime numbers p, qCalculate n = pq and m = (p-1)(q-1)

Choose e such that gcd(e, (p-1)(q-1)) = 1Choose d such that ed = 1 mod (p-1)(q-1)

Publicize e, n. Keep d private.

To encrypt x: calculate y = xe mod n

Bob

To decrypt y: calculate z = yd mod n.

Fermat’s Little Theorem: xed = x mod n.

74037563479561712828046796097429573142593188889231289084936232638972765034028266276891996419625117843995894330502127585370118968098286733173273108930900552505116877063299072396380786710086096962537934650

563796359

This number is a product of two primes:

Modulus (1024 bits):9d 88 94 32 62 30 a3 3b 0f 5b 30 7c d5 69 75 21 25 99 0f e1 a8 c2 7d 5b 2b e9 24 72 48 fa 13 47 7c 53 8b 6a 56 be 66 5a 93 7c 3b bb 84 d4 90 4e ab d5 aa f1 33 de 93 2a 1c 80 e3 f5 19 25 a8 69 d3 55 44 fd 7f 76 bb 28 f5 90 62 69 57 eb 82 ba dd 73 40 c0 dd fb 7e 2b ea 86 8e 83 eb e1 5f cc f4 a5 a4 be 99 4f 90 ff 66 79 b1 44 d4 13 60 96 80 00 3d f8 49 b5 e7 82 54 b3 9d 17 56 cc 90 bb

Exponent (24 bits):01 00 01

Cutting edge

Design/analysis of block ciphers

Elliptic curve cryptography

Quantum cryptography

Cryptology at Franklin College

MAT 135 MAT 140

MAT 142Calculus II

MAT 240Methods of

Problem Solving

MAT 233Linear Algebra

MAT 361Modern Algebra

CMP 141Computing I

MAT 300Topics in

Mathematics

MAT 490Individualized

Study

Careers: NSA and other government; network security

and CISSP license; more

For more information

http://delicious.com/robert.talbert/cryptohttp://www.slideshare.net/rtalbert

http://en.wikipedia.org/wiki/Portal:Cryptography