EIE553代做、代写EIE553程序语言

日期：2025-03-01 07:32

EIE553 Lab 1 1

EIE553 Security in Data Communication

Lab 1: RSA Public-Key Encryption

and Signatures

Report Deadline: 11:59 pm, Mar. 2, 2025 HKT

(Credits: SEED Labs 2.0 by Prof. Du, Wenliang)

1 Overview

RSA (Rivest–Shamir–Adleman) is one of the first public-key cryptosystems and is widely used for secure

communication. The RSA algorithm first generates two large random prime numbers, and then use them

to generate public and private key pairs, which can be used to do encryption, decryption, digital signature

generation, and digital signature verification. The RSA algorithm is built upon number theories, and it can

be quite easily implemented with the support of libraries.

The learning objective of this lab is for students to gain hands-on experiences on the RSA algorithm.

From lectures, students should have learned the theoretic part of the RSA algorithm, so they know math ematically how to generate public/private keys and how to perform encryption/decryption and signature

generation/verification. This lab enhances student’s understanding of RSA by requiring them to go through

every essential step of the RSA algorithm on actual numbers, so they can apply the theories learned from

the class. Essentially, students will be implementing the RSA algorithm using the C program language. The

lab covers the following security-related topics:

• Public-key cryptography

• The RSA algorithm and key generation

• Big number calculation

• Encryption and Decryption using RSA

• Digital signature

• X.509 certificate

Lab environment: The SEED Lab series (including this one) has been tested on the SEED Ubuntu 20.04

VM. In our lab at CF105, the VM is pre-built and you can connect to it via:

1. Run Hyper-V Virtual Machine Connection

2. Select “SEED Ubuntu 20.04”

EIE553 Lab 1 2

3. Click Start

4. Input username: seed and password: dees

NOTE: The Ubuntu 20.04 VM is not strictly necessary. You can complete or implement the tasks

below using your preferred IDE (on your own PC) and programming language (though C/C++

is recommended).

NOTE: The PC might REBORN AFTER REBOOT AND SHARED WITH OTHER STUDENTS.

Save your work in an external drive and back up your files before rebooting or shutting down.

You also can download a pre-built image from the SEED website, and run VM on your own PC.

The setup can be found: https://seedsecuritylabs.org/labsetup.html (for either Intel/Apple/AMD CPU)

How to build SEED VM: https://github.com/seed-labs/seed-labs/blob/master/manuals/vm/seedvm-from scratch.md

A step-by-step guideline (prepared by TAs) on how to build SEED VM on a Windows PC has been

uploaded to Blackboard for your reference.

2 Background

The RSA algorithm involves computations on large numbers. These computations cannot be directly con ducted using simple arithmetic operatorsin programs, because those operators can only operate on primitive

data types, such as 32-bit integer and 64-bit long integer types. The numbers involved in the RSA algorithms

are typically more than 512 bits long. For example, to multiple two 32-bit integer numbers a and b, we just

EIE553 Lab 1 3

// Assign a value from a decimal number string

BN_dec2bn(&a, "12345678901112231223");

// Assign a value from a hex number string

BN_hex2bn(&a, "2A3B4C55FF77889AED3F");

// Generate a random number of 128 bits

BN_rand(a, 128, 0, 0);

// Generate a random prime number of 128 bits

BN_generate_prime_ex(a, 128, 1, NULL, NULL, NULL);

void printBN(char *msg, BIGNUM * a)

{

// Convert the BIGNUM to number string

char * number_str = BN_bn2dec(a);

// Print out the number string

printf("%s %s\n", msg, number_str);

// Free the dynamically allocated memory

OPENSSL_free(number_str);

}

need to use a*b in our program. However, if they are big numbers, we cannot do that any more; instead,

we need to use an algorithm (i.e., a function) to compute their products.

There are several libraries that can perform arithmetic operations on integers of arbitrary size. In this

lab, we will use the Big Number library provided by openssl. To use this library, we will define each big

number as a BIGNUM type, and then use the APIs provided by the library for various operations, such as

addition, multiplication, exponentiation, modular operations, etc.

2.1 BIGNUM APIs

All the big number APIs can be found from https://linux.die.net/man/3/bn. In the following,

we describe some of the APIs that are needed for this lab.

• Some of the library functions requires temporary variables. Since dynamic memory allocation to cre ate BIGNUMs is quite expensive when used in conjunction with repeated subroutine calls, a BN CTX

structure is created to holds BIGNUM temporary variables used by library functions. We need to

create such a structure, and pass it to the functions that requires it.

BN_CTX *ctx = BN_CTX_new()

• Initialize a BIGNUM variable.

BIGNUM *a = BN_new()

• There are a number of ways to assign a value to a BIGNUM variable.

• Print out a big number.

EIE553 Lab 1 4

BN_sub(res, a, b);

BN_add(res, a, b);

/* bn_sample.c */

#include <stdio.h>

#include <openssl/bn.h>

#define NBITS 256

void printBN(char *msg, BIGNUM * a)

{

/* Use BN_bn2hex(a) for hex string

* Use BN_bn2dec(a) for decimal string */

char * number_str = BN_bn2hex(a);

printf("%s %s\n", msg, number_str);

OPENSSL_free(number_str);

}

int main ()

{

BN_CTX *ctx = BN_CTX_new();

BIGNUM *a = BN_new();

BIGNUM *b = BN_new();

BIGNUM *n = BN_new();

BIGNUM *res = BN_new();

• Compute res = a −b and res = a + b:

• Compute res = a ∗b. It should be noted that a BN CTX structure is need in this API.

BN_mul(res, a, b, ctx)

• Compute res = a ∗b mod n:

BN_mod_mul(res, a, b, n, ctx)

• Compute res = ac mod n:

BN_mod_exp(res, a, c, n, ctx)

• Compute modular inverse, i.e., given a, find b, such that a ∗ b mod n = 1. The value b is called

the inverse of a, with respect to modular n.

BN_mod_inverse(b, a, n, ctx);

2.2 A Complete Example

We show a complete example in the following. The program can be found from the Labsetup.zip file

that you can download from the lab’s webpage. In this example, we initialize three BIGNUM variables, a,

b, and n; we then compute a ∗b and (ab mod n).

EIE553 Lab 1 5

$ vim bn_sample.c

$ gcc bn_sample.c -lcrypto -o output

$ ./output

Compilation. We can use the following command to compile bn_sample.c (the character after - is the

letter £, not the number 1; it tells the compiler to use the crypto library).

Click “Open in Terminal”

Create bn_sample.c file

Paste your code into the file, Press Esc on your keyboard, input “: wq” to save file and quit.

Complie bn_sample.c

Run bn_sample.c

// Initialize a, b, n

BN_generate_prime_ex(a, NBITS, 1, NULL, NULL, NULL);

BN_dec2bn(&b, "273489463796838501848592769467194369268");

BN_rand(n, NBITS, 0, 0);

// res = a*b

BN_mul(res, a, b, ctx);

printBN("a * b = ", res);

// res = aˆb mod n

BN_mod_exp(res, a, b, n, ctx);

printBN("aˆc mod n = ", res);

return 0;

}

EIE553 Lab 1 6

p = F7E75FDC469067FFDC4E847C51F452DF

q = E85CED54AF57E53E092113E62F436F4F

e = 0D88C3

$ python3 -c ’print("A top secret!".encode("utf-8").hex())’

4120746f702073656372657421

3 Lab Tasks

NOTE: You must explicitly disclose the use of any GenAI tools (e.g., ChatGPT and DeepSeek) if utilized

in completing the tasks below.

3.1 Task 1: Deriving the Private Key (20 marks)

Let p, q, and e be three prime numbers. Let n = p*q. We will use (e, n) as the public key. Please

calculate the private key d. The hexadecimal values of p, q, and e are listed in the following. It should be

noted that although p and q used in this task are quite large numbers, they are not large enough to be secure.

We intentionally make them small for the sake of simplicity. In practice, these numbers should be at least

512 bits long (the one used here are only 128 bits).

Hint: The private key d (which is multiplicative inverse of e mod n) can be computed via the extended Euclidean

algorithm (introduced in Lecture 4). The pseudocode is

Input:

- Public key (N, e)

- Prime factors p and q of N (N = pq)

Output:

- Private key d

Steps:

1. Compute ϕ(N) = (p - 1) * (q - 1) // Euler's totient function

2. Use the Extended Euclidean Algorithm to find d such that:

(e * d) ≡ 1 mod ϕ(N)

Extended Euclidean Algorithm:

Function ExtendedEuclidean(a, b):

If b == 0:

Return (a, 1, 0) // gcd(a, b) = a, and coefficients x = 1, y = 0

Else:

(gcd, x1, y1) = ExtendedEuclidean(b, a mod b)

x = y1

y = x1 - (a // b) * y1

Return (gcd, x, y)

3. (gcd, d, _) = ExtendedEuclidean(e, ϕ(N))

4. If gcd != 1:

Return "No modular inverse exists (e and ϕ(N) are not coprime)"

Else:

Ensure d is positive by computing d = d mod ϕ(N)

Return d

3.2 Task 2: Encrypting a Message (20 marks)

Let (e, n) be the public key. Please encrypt the message "A top secret!" (the quotations are not

included). We need to convert this ASCII string to a hex string, and then convert the hex string to a BIGNUM

using the hex-to-bn API BN hex2bn(). The following python command can be used to convert a plain

ASCII string to a hex string.

SEED Labs 2.0 VM (Ubuntu 20.04.2 LTS):

SEED Labs 1.0 VM (Ubuntu 16.04 LTS):

EIE553 Lab 1 7

n = DCBFFE3E51F62E09CE7032E2677A78946A849DC4CDDE3A4D0CB81629242FB1A5

e = 010001 (this hex value equals to decimal 65537)

M = A top secret!

d = 74D806F9F3A62BAE331FFE3F0A68AFE35B3D2E4794148AACBC26AA381CD7D30D

C = 8C0F971DF2F3672B28811407E2DABBE1DA0FEBBBDFC7DCB67396567EA1E2493F

$ python3 -c

’print(bytes.fromhex("4120746f702073656372657421").decode("utf-8"))’

A top secret!

M = I owe you $2000.

M = Launch a missile.

S = 643D6F34902D9C7EC90CB0B2BCA36C47FA37165C0005CAB026C0542CBDB6802F

e = 010001 (this hex value equals to decimal 65537)

n = AE1CD4DC432798D933779FBD46C6E1247F0CF1233595113AA51B450F18116115

The public keys are listed in the followings (hexadecimal). We also provide the private key d to help

you verify your encryption result.

Requirement: In your lab report, you should change the message to "Your Name + Student ID" instead of using

"A top secret!" in the above demo.

3.3 Task 3: Decrypting a Message (20 marks)

The public/private keys used in this task are the same as the ones used in Task 2. Please decrypt the following

ciphertext C, and convert it back to a plain ASCII string.

You can use the following python command to convert a hex string back to to a plain ASCII string

(works in both VM versions).

Requirement: In your lab report, you should decrypt the ciphertext of "Your Name + Student ID" instead of

using "A top secret!" in the above demo.

3.4 Task 4: Signing a Message (20 marks)

The public/private keys used in this task are the same as the ones used in Task 2. Please generate a signature

for the following message (please directly sign this message, instead of signing its hash value):

Please make a slight change to the message M, such as changing $2000 to $3000, and sign the modified

message. Compare both signatures and describe what you observe.

Requirement: In your lab report, you should change the message to "Your PolyU email address" instead of

using "I owe you $2000" in the above demo.

3.5 Task 5: Verifying a Signature (20 marks)

Bob receives a message M = "Launch a missile." from Alice, with her signature S. We know that

Alice’s public key is (e, n). Please verify whether the signature is indeed Alice’s or not. The public key

and signature (hexadecimal) are listed in the following:

Suppose that the signature above is corrupted, such that the last byte of the signature changes from 2F

to 3F, i.e, there is only one bit of change. Please repeat this task, and describe what will happen to the

verification process.

$ python -c ’print("A top secret!".encode("hex"))’

4120746f702073656372657421

EIE553 Lab 1 8

$ openssl s_client -connect www.example.org:443 -showcerts

Certificate chain

0 s:/C=US/ST=California/L=Los Angeles/O=Internet Corporation for Assigned

Names and Numbers/OU=Technology/CN=www.example.org

i:/C=US/O=DigiCert Inc/OU=www.digicert.com/CN=DigiCert SHA2 High Assurance

Server CA

-----BEGIN CERTIFICATE-----

MIIF8jCCBNqgAwIBAgIQDmTF+8I2reFLFyrrQceMsDANBgkqhkiG9w0BAQsFADBw

MQswCQYDVQQGEwJVUzEVMBMGA1UEChMMRGlnaUNlcnQgSW5jMRkwFwYDVQQLExB3

......

wDSiIIWIWJiJGbEeIO0TIFwEVWTOnbNl/faPXpk5IRXicapqiII=

-----END CERTIFICATE-----

1 s:/C=US/O=DigiCert Inc/OU=www.digicert.com/CN=DigiCert SHA2 High

Assurance Server CA

i:/C=US/O=DigiCert Inc/OU=www.digicert.com/CN=DigiCert High Assurance

EV Root CA

-----BEGIN CERTIFICATE-----

MIIEsTCCA5mgAwIBAgIQBOHnpNxc8vNtwCtCuF0VnzANBgkqhkiG9w0BAQsFADBs

MQswCQYDVQQGEwJVUzEVMBMGA1UEChMMRGlnaUNlcnQgSW5jMRkwFwYDVQQLExB3

......

cPUeybQ=

-----END CERTIFICATE-----

3.6 (Optional) Task 6: Manually Verifying an X.509 Certificate (20 marks) (Optional)

In this task, we will manually verify an X.509 certificate using our program. An X.509 contains data about

a public key and an issuer’s signature on the data. We will download a real X.509 certificate from a web

server, get its issuer’s public key, and then use this public key to verify the signature on the certificate.

Step 1: Download a certificate from a real web server. We use the www.example.org server in

this document. Students should choose a different web server that has a different certificate than the

one used in this document (it should be noted that www.example.com share the same certificate with

www.example.org). We can download certificates using browsers or use the following command:

The result of the command contains two certificates. The subject field (the entry starting with s:) of

the certificate is www.example.org, i.e., this is www.example.org’s certificate. The issuer field (the

entry starting with i:) provides the issuer’s information. The subject field of the second certificate is the

same as the issuer field of the first certificate. Basically, the second certificate belongs to an intermediate

CA. In this task, we will use CA’s certificate to verify a server certificate.

If you only get one certificate back using the above command, that means the certificate you get is signed

by a root CA. Root CAs’ certificates can be obtained from the Firefox browser installed in our pre-built VM.

Go to the Edit ➔ Preferences ➔ Privacy and then Security ➔ View Certificates. Search

for the name of the issuer and download its certificate.

Copy and paste each of the certificate (the text between the line containing "Begin CERTIFICATE"

and the line containing "END CERTIFICATE", including these two lines) to a file. Let us call the first one

c0.pem and the second one c1.pem.

Step 2: Extract the public key (e, n) from the issuer’s certificate. Openssl provides commands to

extract certain attributes from the x509 certificates. We can extract the value of n using -modulus. There

is no specific command to extract e, but we can print out all the fields and can easily find the value of e.

EIE553 Lab 1 9

$ openssl x509 -in c0.pem -text -noout

...

Signature Algorithm: sha256WithRSAEncryption

84:a8:9a:11:a7:d8:bd:0b:26:7e:52:24:7b:b2:55:9d:ea:30:

89:51:08:87:6f:a9:ed:10:ea:5b:3e:0b:c7:2d:47:04:4e:dd:

......

5c:04:55:64:ce:9d:b3:65:fd:f6:8f:5e:99:39:21:15:e2:71:

aa:6a:88:82

$ cat signature | tr -d ’[:space:]:’

84a89a11a7d8bd0b267e52247bb2559dea30895108876fa9ed10ea5b3e0bc7

......

5c045564ce9db365fdf68f5e99392115e271aa6a8882

Step 3: Extract the signature from the server’s certificate. There is no specific opensslcommand to

extract the signature field. However, we can print out all the fields and then copy and paste the signature

block into a file (note: if the signature algorithm used in the certificate is not based on RSA, you can find

another certificate).

We need to remove the spaces and colons from the data, so we can get a hex-string that we can feed into

our program. The following command commands can achieve this goal. The tr command is a Linux utility

tool for string operations. In this case, the -d option is used to delete ":" and "space" from the data.

Step 4: Extract the body of the server’s certificate. A Certificate Authority (CA) generatesthe signature

for a server certificate by first computing the hash of the certificate, and then sign the hash. To verify the

signature, we also need to generate the hash from a certificate. Since the hash is generated before the

signature is computed, we need to exclude the signature block of a certificate when computing the hash.

Finding out what part of the certificate is used to generate the hash is quite challenging without a good

understanding of the format of the certificate.

X.509 certificates are encoded using the ASN.1 (Abstract Syntax Notation.One) standard, so if we can

parse the ASN.1 structure, we can easily extract any field from a certificate. Openssl has a command called

asn1parse used to extract data from ASN.1 formatted data, and is able to parse our X.509 certificate.

8:d=2 hl=2 l= 3 cons: cont [ 0 ]

10:d=3 hl=2 l= 1 prim: INTEGER :02

13:d=2 hl=2 l= 16 prim: INTEGER

:0E64C5FBC236ADE14B172AEB41C78CB0

... ...

1236:d=4 hl=2 l= 12 cons: SEQUENCE

1238:d=5 hl=2 l= 3 prim: OBJECT :X509v3 Basic Constraints

1243:d=5 hl=2 l= 1 prim: BOOLEAN :255

For modulus (n):

$ openssl x509 -in c1.pem -noout -modulus

Print out all the fields, find the exponent (e):

$ openssl x509 -in c1.pem -text -noout

EIE553 Lab 1 10

$ openssl asn1parse -i -in c0.pem -strparse 4 -out c0_body.bin -noout

$ sha256sum c0_body.bin

The field starting from 。 is the body of the certificate that is used to generate the hash; the field starting

from ＠ is the signature block. Their offsets are the numbers at the beginning of the lines. In our case, the

certificate body is from offset 4 to 1249, while the signature block is from 1250 to the end of the file. For

X.509 certificates, the starting offset is always the same (i.e., 4), but the end depends on the content length

of a certificate. We can use the -strparse option to get the field from the offset 4, which will give us the

body of the certificate, excluding the signature block.

Once we get the body of the certificate, we can calculate its hash using the following command:

Step 5: Verify the signature. Now we have all the information, including the CA’s public key, the CA’s

signature, and the body of the server’s certificate. We can run our own program to verify whether the

signature is valid or not. Openssl does provide a command to verify the certificate for us, but students are

required to use their own programs to do so, otherwise, they get zero credit for this task.

4 Submission

You need to submit a detailed lab report, with screenshots, to describe what you have done

and what you have observed. You also need to provide explanation to the observations that

are interesting or surprising. Please also list the important code snippets followed by

explanation. Simply attaching code without any explanation will not receive credits.

OCTET STRING

OBJECT

NULL