Advanced Cryptography
Spring 2020

Assignment 0, Tuesday, January 21

Final exam from the first cryptography course in Fall 2019. This is to see how much of the common background this group has. Do not study for this exam, possibly just review what you had in earlier cryptography related courses. This way I will be able to see better what to focus on in class.

Assignment 1, due Tuesday, February 4

Decrypt the ciphertext from the table 7.4 page 305 (6.3 page 278 in the third edition), which was obtained by an application of the ElGamal Cryptosystem 7.1 page 257 (6.1 page 235). The parameters of the system are p = 31847 = 1 + 2*15923 (15923 is prime), alpha=5, a=7899 and beta=18074. Each element of Zp in the range <0,17575> represents three alphabetic characters as in Exercise 6.13 page 247 (5.12 page 227). You have to use square-and-multiply algorithm for modular exponentiation, and the Extended Euclid Algorithm or other not-by-force algorithm for calculating modular inverses. You may use parts of the code from previous course assignments,

What are the secret values of parameter k used for encryption? Use both Shanks' algorithm and brute force (for verification) to find them. Note, that k's are not needed for the decryption. In this toy example they can be found with the help of any discrete logarithm algorithm. Find the first 30 values of k.

Submit in class a hardcopy of the following (or in special situations a single pdf or txt document by email):

Sample solution by Hannah Miller.

Assignment 2, due Monday, February 10

Complete the first step of the term-long research paper and presentation project. We will review all proposals in class on 2/11.

Assignment 3, due Tuesday, February 18

This task is for everybody.

Choose one of the two following tasks involving Pollard-PHO algortihm. The double pointers to chapters and pages are as in editions 3 and 4 of the textbook, respectively.

Hint: in all three parts above go to more bits if you can.

Sample solution by Tom Arnold.

Assignment 4, Galois fields, due Tuesday, March 3

The due date is postponed till Thursday, March 5.

Solve parts 1 and 2 by hand, use computer help to solve 3 and 4. In all exercises explain what you did and show the details of your work. Attach source code as applicable.

  1. The polynomial x4 + x + 1 is irreducible in Z2[x]. Compute x15 mod x4 + x + 1 in Z2[x], i.e. in the Galois field GF(24). Use two approaches: standard square-and-multiply for exponent 15, and for the exponent written as (16 - 1).

  2. Find all irreducible polynomials in Z2[x] of degree 5. You can assume that the polynomial x2 + x + 1 is the only irreducible binary quadratic (you do not need to show that).

  3. Solve exercise 6.12 pages 277/278 (7.12 pages 305/306). You can use this representation of the Galois field GF(27)

  4. Let p=131. First show that (x2+1) is irreducible in Zp[x] - this can be done by hand using the Euler criterion for quadratic residuosity. Next, represent GF(p2) by polynomials modulo (x2+1). Use naive algorithm to find the number of elements of each order in GF(p2), and list 10 smallest monic primitive (generators with coefficient 1 in the highest degree term) elements. Illustrate the computation of discrete logarithm of (x+101) with base equal to the smallest such generator using Shanks' algorithm.

Sample solution by Thomas Bottom.

Assignment 5, due Thursday, March 19

Due date has been postponed till March 26.

The break at RIT has been extended by one week, until March 23. Thus, the due date of the current assignment is also delayed by a week until March 26. Please submit your work by sending a pdf in an email (can be a scan of a document produced by any means).

Exploring elliptic curves.

For edition 4 of the textbook, use chapter 7 (instead of 6), same exercise numbers, pages 306/307.

  1. Solve exercise 6.13 page 278. Note that the answer in (c) must be a divisor of (a).
  2. Solve exercise 6.14 page 279.
  3. Solve exercise 6.15 page 279.
  4. Solve exercise 6.16 page 279.

  5. Proving associativity of point addition on elliptic curves is quite complicated. In this exercise you will do just a special case of it. Suppose that points P=(p1,p2) and Q=(q1,q2), p1 not equal to q1, are on an elliptic curve E (either real or modular). It is obvious that ((-P) + P) + Q = Q. Prove that (-P) + (P + Q) = Q by

Sample solution by Thomas Bottom, except exercise 6.16b
Solution to exercise 6.16b

Assignment 6, due Tuesday, April 7

EC and NAF

Submit pdf by email, it can be a scan of documents produced by other means.
  1. Let Bk, k ≥ 2, consist of all 0-1 strings of length k with both ends equal to 1 (there are 2k-2 of them). Show explicitly two bijections: between 16 strings in B6 and their NAF representations, and 32 strings in B7 and their NAF representations.

  2. Solve exercise 6.17 page 279 (ECIES). In (a) show the intermediate values of variables. This exercise is not in edition 4 of the textbook. Edition 4 does not include the ECIES scheme, but a similar to it cryptosystem 7.2 EC ElGamal. For this problem use the ECIES slide discussed in class.

    Let E be the elliptic curve y^2=x^3+2x+7 defined over Z_31. It can be shown that #E=39 and P=(2,9) is an eleement of order 39 in E. The simplified ECIES defined on E has Z_31^* as its plaintext space. Suppose the private key is m=8.

    (a) Compute Q = mP.

    (b) Decrypt the ciphertext ((18,1),21), ((3,1),18), ((17,0),19), ((28,0),8).

    (c) Assuming that each plaintext represents one alphabetic character, convert the plaintext into an English word. Use the correspondence A<->1, ..., Z<->26, because 0 is not allowed in a plaintext ordered pair.

  3. Solve exercise 6.18 page 279 (7.19 page 307).

    Sample solution for questions 1-3 by Hannah Miller.

  4. (Optional) Prove that the NAF representation is unique. You need to show that two distinct NAF strings cannot encode the same integer.

    A nice solution at crypto stackexchange.

    Interesting papers on generalizations of NAF to more digits and positions: Redundant tau-adic expansions I: non-adjacent digit sets and their applications to scalar multiplication (2011), and Minimality of the Hamming Weight of the tau-NAF for Koblitz Curves and Improved Combination with Point Halving (2006), by Roberto Avanzi, Clemens Heuberger, Helmut Prodinger.

Assignment 7, due Tuesday, April 21

Digital signatures

Submit a single pdf by email.

  1. Solve exercise 7.6 page 319 (8.6 page 335/336).
  2. Solve exercise 7.7 page 319 (8.7 page 336).
  3. Solve exercise 7.9 page 320 (8.10 page 337).
    In the SHA-3 competition NIST requested that the new hash has to be 0-preimage resistant.
  4. Solve exercise 7.13 page 320 (8.14 page 338).

Two submissions complementing each other:
sample solution by Devin Kott, sample solution by Tom Arnold.

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