CSCI 7000 - Cryptanalysis - Spring 2005

Problem Set #1

Due: Feb 1st, 2005 at 11am

Problem 1. (Difficulty: 4) The ciphertexts below were encrypted with a blockcipher with domain and range {a,...,z}. The key is a random permutation of the domain (and there are too many keys for exhaustive key-search to make sense). Please write a python program that deciphers these texts. Note: each ciphertext has a different key, so attack them separately. All are English text.

You may use any techniques you like. Frequency analysis, index of coincidence, stats on diphthongs, triphthongs, start letters, end letters, double-letters, etc.

Your program should recover the plaintext with 100% accuracy with no human intervention. To achieve this, use a list of English words, and go with the substitution that yields a sufficiently-high number of English words in the text. There is a word list you might use.

Here is the program used to generate the ciphertexts:

#!/usr/local/bin/python

def perm(s, key):
    ct = ''
    a = ord('a')
    for c in s.lower():
        if 'a' <= c <= 'z':
             ct += chr(key[ord(c)-a]+a)
        else:
             ct += c
    return ct


p1 = """
Sample plaintext 1
"""

p2 = "Sample 2"

import random

key = range(26)
random.shuffle(key)

for pt in (p1,p2):
    ct = perm(pt, key)
    print ct

And here are the ciphertexts:

0. wqp dyee akifkj fq quy qd fgy ieollki xepunycl. fgy jqlf dojqpl kl "uyayc myf kuaqeayn ku o eoun soc ku olko." xpf quew lekmgfew eyll syee ruqsu kl fgkl: "uyayc mq ku omokulf o lkikekou sgyu nyofg kl qu fgy ekuy."

1. hsen owsj sud, zq't erkvzkwztj; hsen owsj qas, zq't wdtdkwhp.

2. y zyufa cbhvxa cw rna oawr uahcqa ivu nyqqcbaww c alau nayuj vi.

3. jk cwjv sqzhe, kqcwjkp jv lozcmjk txc eomcw mke cmfov.

4. kqozy kfq vzzx hq rz zmius kahf ozy sulx uorahaqy.

5. oltp wsfy qsp wfypdsz ef, ethfy ef ypdsqmfd.

6. ugsxqxai xi qez wzyqsz pvq gj wzqqxyw fgqzi jvgl qez uggv pyh aplupxwy jdyhi jvgl qez vxae om uvglxixyw qg uvgqzaq zpae jvgl qez gqezv.

7. tfble snowpnywzv, vnf lgoykwpz vnf. tfble skvvtfwzv, wp'z utzp prl kookzwpl.

8. ugcvivyl fmce vi kzie hcdji emj gjhjhogcyaj wjcg.

9. brwqhlfkw hjk qibwk qicedw zbg wkk mike zbg qhsk zbgj kzkw buu zbgj dbhfw.

10. rka pbrgsgxr rkgjnx rkgx gx rka zaxr pt uii ypfiex. rka baxxgsgxr taufx rkgx gx rfda.

Turn in your program along with a program-run that deciphers each of the ciphertexts above.

Problem 2. (Difficulty: 1)

Implement, once again in python, functions to perform the following tasks:

• gcd(a, b) finds the gcd of a and b. Use Euclid's algorithm.
• modx(base,exp,mod) performs the modular exponentiation of the integer base to the power exp modulo the integer mod. Using the repeated squaring algorithm described in class.
• pseud(base, start) finds the first integer starting at the positive integer start that passes the Fermat test with base base.
• inv(a, n) finds the inverse of a modulo n. Use the extended Euclidean algorithm, and return -1 if a has no inverse modulo n.

Problem 3. (Difficulty: 1)

Using your functions above, answer the following questions:

• What is the first (probable) prime after 81359181681933678136913918193493? (Set variable p to this value.)
• What is the first (probable) prime after 72691876913962323962357607903641? (Set variable q to this value.)
• What is pq? (Set variable n to this value.)
• What is phi(n)? (Set variable phi to this value.)
• What is the smallest odd prime that is relatively prime (coprime) to phi? (Set variable e to this value.)
• What is the inverse of e mod phi? (Set variable d to this value.)
• Set variable M to 8718157196969696969223571. Show that M is in Z_n^*.
• What is the RSA encryption of M with the above parameters? (Set variable C to this value.)
• Decrypt C and show that you get M back again.