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sasc:laboratoare:03 [2017/03/06 12:46] dan.dragan |
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| - | ==== Exercise 1 (2p) ==== | + | ==== Exercise 1 (4p) ==== |
| In this exercise we'll try to break a Linear Congruential Generator, that may be used to generate "poor" random numbers. | In this exercise we'll try to break a Linear Congruential Generator, that may be used to generate "poor" random numbers. | ||
| Line 103: | Line 103: | ||
| </code> | </code> | ||
| + | ==== Exercise 2 (3p) ==== | ||
| - | ==== Exercise 2 - LFSR (2p) ==== | + | Let's use the experiment defined earlier as a pseudorandom generator ($\mathsf{PRG}$) as follows: |
| + | - Set a desired output length $n$ | ||
| + | - Obtain a random sequence $R$ of bits of length $n$ (e.g. using the Linear-congruential generator from Exercise 1) | ||
| + | - For each bit $r$ in the random sequence $R$ generated in the previous step, output a bit $b$ as follows: | ||
| + | * if the bit $r$ is $0$, then output a random bit $b \in \{0, 1\}$ | ||
| + | * if the bit $r$ is $1$, then output $1$ | ||
| + | |||
| + | a. Implement the frequency (monobit) test from [[http://csrc.nist.gov/publications/nistpubs/800-22-rev1a/SP800-22rev1a.pdf | NIST (see section 2.1)]] and check if a sequence generated by the above $\mathsf{PRG}$ (say $n=100$) seems random or not. | ||
| + | |||
| + | b. Run the test on a random bitstring (e.g. a string such as R used by the above $\mathsf{PRG}$), and compare the result of the test. | ||
| + | |||
| + | If the two results are different across many iterations, this test already gives you an attacker that breaks the $\mathsf{PRG}$. | ||
| + | |||
| + | <note tip>You may use a function like this to generate a random bitstring</note> | ||
| + | <code python> | ||
| + | import random | ||
| + | |||
| + | def get_random_string(n): #generate random bit string | ||
| + | bstr = bin(random.getrandbits(n)).lstrip('0b').zfill(n) | ||
| + | return bstr | ||
| + | </code> | ||
| + | |||
| + | <note tip>Also, in Python you may find the functions sqrt, fabs and erfc from the module math useful</note> | ||
| + | |||
| + | ==== Exercise 3 - LFSR (3p) ==== | ||
| In this exercise we'll build a simple Linear Feedback Shift Register (LFSR). LFSRs produce random bit strings with good statistical properties, but are very easy to predict. | In this exercise we'll build a simple Linear Feedback Shift Register (LFSR). LFSRs produce random bit strings with good statistical properties, but are very easy to predict. | ||
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| Using the above starting state and polynomial, generate $100$ random bits and run the monobit statistical test from the previous exercise to see if their frequency seems random. | Using the above starting state and polynomial, generate $100$ random bits and run the monobit statistical test from the previous exercise to see if their frequency seems random. | ||
| - | |||
| - | |||
| - | ==== Exercise 3 ==== | ||
| - | |||
| - | Let's analyse some substitution-permutation networks (SPN). | ||
| - | |||
| - | === SPN 1 (3p) === | ||
| - | |||
| - | We have the SPN from this figure: | ||
| - | {{:sasc:laboratoare:spn_1r_reduced_2s.png|}} | ||
| - | |||
| - | where S denotes the AES S-box (we'll discuss this in some detail during the next lecture), and 'Permutation' is a simple permutation block that simply shifts the input 4 bits to the right as in a queue. Both this S-box and the permutation are invertible and known by the attacker (you). Each input (x1, x2) is 8-bit (1 byte), as well as the keys k1, k2, and the outputs y1, y2. | ||
| - | |||
| - | - How can you find the key ? | ||
| - | - Given the message/ciphertext pair ('Hi' - as characters, 0xba52 - as hex number), find the key bytes k1 and k2. Print them in ascii. | ||
| - | |||
| - | <note tip> | ||
| - | For these exercises you can use the following helper/starter code: | ||
| - | </note> | ||
| - | |||
| - | <code> | ||
| - | import sys | ||
| - | import random | ||
| - | import string | ||
| - | import operator | ||
| - | |||
| - | # Rijndael S-box | ||
| - | sbox = [0x63, 0x7c, 0x77, 0x7b, 0xf2, 0x6b, 0x6f, 0xc5, 0x30, 0x01, 0x67, | ||
| - | 0x2b, 0xfe, 0xd7, 0xab, 0x76, 0xca, 0x82, 0xc9, 0x7d, 0xfa, 0x59, | ||
| - | 0x47, 0xf0, 0xad, 0xd4, 0xa2, 0xaf, 0x9c, 0xa4, 0x72, 0xc0, 0xb7, | ||
| - | 0xfd, 0x93, 0x26, 0x36, 0x3f, 0xf7, 0xcc, 0x34, 0xa5, 0xe5, 0xf1, | ||
| - | 0x71, 0xd8, 0x31, 0x15, 0x04, 0xc7, 0x23, 0xc3, 0x18, 0x96, 0x05, | ||
| - | 0x9a, 0x07, 0x12, 0x80, 0xe2, 0xeb, 0x27, 0xb2, 0x75, 0x09, 0x83, | ||
| - | 0x2c, 0x1a, 0x1b, 0x6e, 0x5a, 0xa0, 0x52, 0x3b, 0xd6, 0xb3, 0x29, | ||
| - | 0xe3, 0x2f, 0x84, 0x53, 0xd1, 0x00, 0xed, 0x20, 0xfc, 0xb1, 0x5b, | ||
| - | 0x6a, 0xcb, 0xbe, 0x39, 0x4a, 0x4c, 0x58, 0xcf, 0xd0, 0xef, 0xaa, | ||
| - | 0xfb, 0x43, 0x4d, 0x33, 0x85, 0x45, 0xf9, 0x02, 0x7f, 0x50, 0x3c, | ||
| - | 0x9f, 0xa8, 0x51, 0xa3, 0x40, 0x8f, 0x92, 0x9d, 0x38, 0xf5, 0xbc, | ||
| - | 0xb6, 0xda, 0x21, 0x10, 0xff, 0xf3, 0xd2, 0xcd, 0x0c, 0x13, 0xec, | ||
| - | 0x5f, 0x97, 0x44, 0x17, 0xc4, 0xa7, 0x7e, 0x3d, 0x64, 0x5d, 0x19, | ||
| - | 0x73, 0x60, 0x81, 0x4f, 0xdc, 0x22, 0x2a, 0x90, 0x88, 0x46, 0xee, | ||
| - | 0xb8, 0x14, 0xde, 0x5e, 0x0b, 0xdb, 0xe0, 0x32, 0x3a, 0x0a, 0x49, | ||
| - | 0x06, 0x24, 0x5c, 0xc2, 0xd3, 0xac, 0x62, 0x91, 0x95, 0xe4, 0x79, | ||
| - | 0xe7, 0xc8, 0x37, 0x6d, 0x8d, 0xd5, 0x4e, 0xa9, 0x6c, 0x56, 0xf4, | ||
| - | 0xea, 0x65, 0x7a, 0xae, 0x08, 0xba, 0x78, 0x25, 0x2e, 0x1c, 0xa6, | ||
| - | 0xb4, 0xc6, 0xe8, 0xdd, 0x74, 0x1f, 0x4b, 0xbd, 0x8b, 0x8a, 0x70, | ||
| - | 0x3e, 0xb5, 0x66, 0x48, 0x03, 0xf6, 0x0e, 0x61, 0x35, 0x57, 0xb9, | ||
| - | 0x86, 0xc1, 0x1d, 0x9e, 0xe1, 0xf8, 0x98, 0x11, 0x69, 0xd9, 0x8e, | ||
| - | 0x94, 0x9b, 0x1e, 0x87, 0xe9, 0xce, 0x55, 0x28, 0xdf, 0x8c, 0xa1, | ||
| - | 0x89, 0x0d, 0xbf, 0xe6, 0x42, 0x68, 0x41, 0x99, 0x2d, 0x0f, 0xb0, | ||
| - | 0x54, 0xbb, 0x16] | ||
| - | |||
| - | |||
| - | # Rijndael Inverted S-box | ||
| - | rsbox = [0x52, 0x09, 0x6a, 0xd5, 0x30, 0x36, 0xa5, 0x38, 0xbf, 0x40, 0xa3, | ||
| - | 0x9e, 0x81, 0xf3, 0xd7, 0xfb , 0x7c, 0xe3, 0x39, 0x82, 0x9b, 0x2f, | ||
| - | 0xff, 0x87, 0x34, 0x8e, 0x43, 0x44, 0xc4, 0xde, 0xe9, 0xcb , 0x54, | ||
| - | 0x7b, 0x94, 0x32, 0xa6, 0xc2, 0x23, 0x3d, 0xee, 0x4c, 0x95, 0x0b, | ||
| - | 0x42, 0xfa, 0xc3, 0x4e , 0x08, 0x2e, 0xa1, 0x66, 0x28, 0xd9, 0x24, | ||
| - | 0xb2, 0x76, 0x5b, 0xa2, 0x49, 0x6d, 0x8b, 0xd1, 0x25 , 0x72, 0xf8, | ||
| - | 0xf6, 0x64, 0x86, 0x68, 0x98, 0x16, 0xd4, 0xa4, 0x5c, 0xcc, 0x5d, | ||
| - | 0x65, 0xb6, 0x92 , 0x6c, 0x70, 0x48, 0x50, 0xfd, 0xed, 0xb9, 0xda, | ||
| - | 0x5e, 0x15, 0x46, 0x57, 0xa7, 0x8d, 0x9d, 0x84 , 0x90, 0xd8, 0xab, | ||
| - | 0x00, 0x8c, 0xbc, 0xd3, 0x0a, 0xf7, 0xe4, 0x58, 0x05, 0xb8, 0xb3, | ||
| - | 0x45, 0x06 , 0xd0, 0x2c, 0x1e, 0x8f, 0xca, 0x3f, 0x0f, 0x02, 0xc1, | ||
| - | 0xaf, 0xbd, 0x03, 0x01, 0x13, 0x8a, 0x6b , 0x3a, 0x91, 0x11, 0x41, | ||
| - | 0x4f, 0x67, 0xdc, 0xea, 0x97, 0xf2, 0xcf, 0xce, 0xf0, 0xb4, 0xe6, | ||
| - | 0x73 , 0x96, 0xac, 0x74, 0x22, 0xe7, 0xad, 0x35, 0x85, 0xe2, 0xf9, | ||
| - | 0x37, 0xe8, 0x1c, 0x75, 0xdf, 0x6e , 0x47, 0xf1, 0x1a, 0x71, 0x1d, | ||
| - | 0x29, 0xc5, 0x89, 0x6f, 0xb7, 0x62, 0x0e, 0xaa, 0x18, 0xbe, 0x1b , | ||
| - | 0xfc, 0x56, 0x3e, 0x4b, 0xc6, 0xd2, 0x79, 0x20, 0x9a, 0xdb, 0xc0, | ||
| - | 0xfe, 0x78, 0xcd, 0x5a, 0xf4 , 0x1f, 0xdd, 0xa8, 0x33, 0x88, 0x07, | ||
| - | 0xc7, 0x31, 0xb1, 0x12, 0x10, 0x59, 0x27, 0x80, 0xec, 0x5f , 0x60, | ||
| - | 0x51, 0x7f, 0xa9, 0x19, 0xb5, 0x4a, 0x0d, 0x2d, 0xe5, 0x7a, 0x9f, | ||
| - | 0x93, 0xc9, 0x9c, 0xef , 0xa0, 0xe0, 0x3b, 0x4d, 0xae, 0x2a, 0xf5, | ||
| - | 0xb0, 0xc8, 0xeb, 0xbb, 0x3c, 0x83, 0x53, 0x99, 0x61 , 0x17, 0x2b, | ||
| - | 0x04, 0x7e, 0xba, 0x77, 0xd6, 0x26, 0xe1, 0x69, 0x14, 0x63, 0x55, | ||
| - | 0x21, 0x0c, 0x7d] | ||
| - | |||
| - | def strxor(a, b): # xor two strings (trims the longer input) | ||
| - | return "".join([chr(ord(x) ^ ord(y)) for (x, y) in zip(a, b)]) | ||
| - | |||
| - | def hexxor(a, b): # xor two hex strings (trims the longer input) | ||
| - | ha = a.decode('hex') | ||
| - | hb = b.decode('hex') | ||
| - | return "".join([chr(ord(x) ^ ord(y)).encode('hex') for (x, y) in zip(ha, hb)]) | ||
| - | |||
| - | def bitxor(a, b): # xor two bit strings (trims the longer input) | ||
| - | return "".join([str(int(x)^int(y)) for (x, y) in zip(a, b)]) | ||
| - | | ||
| - | def str2bin(ss): | ||
| - | """ | ||
| - | Transform a string (e.g. 'Hello') into a string of bits | ||
| - | """ | ||
| - | bs = '' | ||
| - | for c in ss: | ||
| - | bs = bs + bin(ord(c))[2:].zfill(8) | ||
| - | return bs | ||
| - | |||
| - | def hex2bin(hs): | ||
| - | """ | ||
| - | Transform a hex string (e.g. 'a2') into a string of bits (e.g.10100010) | ||
| - | """ | ||
| - | bs = '' | ||
| - | for c in hs: | ||
| - | bs = bs + bin(int(c,16))[2:].zfill(4) | ||
| - | return bs | ||
| - | |||
| - | def bin2hex(bs): | ||
| - | """ | ||
| - | Transform a bit string into a hex string | ||
| - | """ | ||
| - | return hex(int(bs,2))[2:] | ||
| - | |||
| - | def byte2bin(bval): | ||
| - | """ | ||
| - | Transform a byte (8-bit) value into a bitstring | ||
| - | """ | ||
| - | return bin(bval)[2:].zfill(8) | ||
| - | |||
| - | |||
| - | def permute4(s): | ||
| - | """ | ||
| - | Perform a permutatation by shifting all bits 4 positions right. | ||
| - | The input is assumed to be a 16-bit bitstring | ||
| - | """ | ||
| - | ps = '' | ||
| - | ps = ps + s[12:16] | ||
| - | ps = ps + s[0:12] | ||
| - | return ps | ||
| - | |||
| - | def permute_inv4(s): | ||
| - | """ | ||
| - | Perform the inverse of permute4 | ||
| - | The input is assumed to be a 16-bit bitstring | ||
| - | """ | ||
| - | ps = '' | ||
| - | ps = ps + s[4:16] | ||
| - | ps = ps + s[0:4] | ||
| - | return ps | ||
| - | |||
| - | def spn_1r_reduced_2s(k, x): | ||
| - | """ | ||
| - | Performs an encryption with a substitution-permutation network. | ||
| - | Key k = {k1, k2}, total of 16 bits (2 x 8 bits) | ||
| - | Input x = {x1, x2}, total of 16 bits (2 x 8 bits) | ||
| - | Both k and x are assumed to be bitstrings. | ||
| - | |||
| - | Return: | ||
| - | a 16-bit bitstring containing the encryption y = {y1, y2} | ||
| - | """ | ||
| - | |||
| - | # Split input and key | ||
| - | x1 = x[0:8] | ||
| - | x2 = x[8:16] | ||
| - | k1 = k[0:8] | ||
| - | k2 = k[8:16] | ||
| - | |||
| - | #Apply S-box | ||
| - | u1 = bitxor(x1, k1) | ||
| - | v1 = sbox[int(u1,2)] | ||
| - | v1 = byte2bin(v1) | ||
| - | |||
| - | u2 = bitxor(x2, k2) | ||
| - | v2 = sbox[int(u2,2)] | ||
| - | v2 = byte2bin(v2) | ||
| - | |||
| - | #Apply permutation | ||
| - | pin = v1 + v2 | ||
| - | pout = permute4(pin) | ||
| - | |||
| - | return pout | ||
| - | | ||
| - | def spn_1r_full_2s(k, x): | ||
| - | """ | ||
| - | Performs an encryption with a substitution-permutation network. | ||
| - | Key k = {k1, k2, k3, k4}, total of 32 bits (4 x 8 bits) | ||
| - | Input x = {x1, x2}, total of 16 bits (2 x 8 bits) | ||
| - | Both k and x are assumed to be bitstrings. | ||
| - | |||
| - | Return: | ||
| - | a 16-bit bitstring containing the encryption y = {y1, y2} | ||
| - | """ | ||
| - | |||
| - | # Split input and key | ||
| - | x1 = x[0:8] | ||
| - | x2 = x[8:16] | ||
| - | k1 = k[0:8] | ||
| - | k2 = k[8:16] | ||
| - | k3 = k[16:24] | ||
| - | k4 = k[24:32] | ||
| - | |||
| - | #Apply S-box | ||
| - | u1 = bitxor(x1, k1) | ||
| - | v1 = sbox[int(u1,2)] | ||
| - | v1 = byte2bin(v1) | ||
| - | |||
| - | u2 = bitxor(x2, k2) | ||
| - | v2 = sbox[int(u2,2)] | ||
| - | v2 = byte2bin(v2) | ||
| - | |||
| - | #Apply permutation | ||
| - | pin = v1 + v2 | ||
| - | pout = permute4(pin) | ||
| - | |||
| - | #Apply final XOR | ||
| - | po1 = pout[0:8] | ||
| - | po2 = pout[8:16] | ||
| - | y1 = bitxor(po1, k3) | ||
| - | y2 = bitxor(po2, k4) | ||
| - | |||
| - | return y1+y2 | ||
| - | | ||
| - | def main(): | ||
| - | |||
| - | #Run reduced 2-byte SPN | ||
| - | msg = 'Hi' | ||
| - | key = '??' # Find this | ||
| - | xs = str2bin(msg) | ||
| - | ks = str2bin(key) | ||
| - | ys = spn_1r_reduced_2s(ks, xs) | ||
| - | print 'Two y halves of reduced SPN: ' + ys[0:8] + ' (hex: ' + bin2hex(ys[0:8]) + '), ' + ys[8:16] + ' (hex: ' + bin2hex(ys[8:16]) + ')' | ||
| - | |||
| - | #Run full 2-byte SPN | ||
| - | msg = 'Om' | ||
| - | key = '????' # Find this | ||
| - | xs = str2bin(msg) | ||
| - | ks = str2bin(key) | ||
| - | ys = spn_1r_full_2s(ks, xs) | ||
| - | print 'Two y halves of full SPN (2 bytes): ' + ys[0:8] + ' (hex: ' + bin2hex(ys[0:8]) + '), ' + ys[8:16] + ' (hex: ' + bin2hex(ys[8:16]) + ')' | ||
| - | |||
| - | |||
| - | |||
| - | if __name__ == "__main__": | ||
| - | main() | ||
| - | </code> | ||
| - | |||
| - | ==== SPN 2 (3p) ==== | ||
| - | |||
| - | Now we have a better SPN, where the output of the permutation is XOR-ed with another 2 key bytes, as in the following figure: | ||
| - | {{:sasc:laboratoare:spn_1r_full_2s.png|}} | ||
| - | |||
| - | - Try to find the key in this case, when given the following message/ciphertext pairs: ('Om', 0x0073), ('El', 0xd00e), ('an', 0x855b). Print the key in ascii. | ||
| - | |||
| - | <note tip>You may try some kind of brute-force search</note> | ||
