symmetric key cryptography tkkwon@snu.ac.kr

Symmetric Encryption Classic ciphers also known as (AKA) single key Secret key sender and recipient share a common key was only type prior to invention of public-key cryptography until second half of 1970’s

Symmetric Cipher Model there must be a secure mechanism for the distribution of this key a priori source: William Stallings

Requirements two requirements for secure use of symmetric encryption: a strong encryption algorithm a secret key known only to sender / receiver Y = EK(X) X = DK(Y) assume encryption algorithm is known Kerckhoffs’s Principle: A cryptosystem should be secure even if everything about the system, except the key, is public knowledge imply a secure channel to distribute the key

cryptographic attacks The language of the plaintext and the nature of the cipher are assumed to be known to the adversary. Ciphertext-only attack: The adversary knows only a number of ciphertexts. Known-plaintext attack: The adversary knows some pairs of ciphertext and corresponding plaintext. Chosen-plaintext attack: The adversary can choose a number of messages and obtain the ciphertexts Chosen-ciphertext attack: The adversary can choose a number of ciphertexts and obtain the plaintexts. * These attacks are also applicable to PKC

requirements for a secure cipher In cryptography, confusion and diffusion are two properties of the operation of a secure cipher [Claude Shannon] Diffusion means that if we change a character of the plaintext, then several characters of the ciphertext should change, and similarly, if we change a character of the ciphertext, then several characters of the plaintext should change Confusion means that the key does not relate in a simple way to the ciphertext. In particular, each character of the ciphertext should depend on several parts of the key

popular primitives Substitution Permutation/transposition exclusive-OR: 

first substitution (shift) cipher Caesar Cipher: E(x) = x+3 (mod 26) monoalphabetic cipher then, how to decrypt? D(y)? is it secure?

affine cipher another monoalphabetic substitution cipher for English alphabet, m is 26 below E(x) = (ax+b) mod m D(y) = a-1(y-b) mod m how many keys? 26 letters 12 numbers for a, 26 numbers for b is it secure?

Monoalphabetic Substitution Cipher The key space: all permutations of  = {A, B, C, …, Z} Encryption given a key : each letter X in the plaintext P is replaced with (X) Decryption given a key : each letter Y in the ciphertext C is replaced with -1(Y) Example: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z = B A D C Z H W Y G O Q X S V T R N M L K J I P F E U BECAUSE  AZDBJSZ Cipher alphabet is it secure?

frequency analysis Frequency analysis is based on the fact that, in any given stretch of written language, certain letters and combinations of letters occur with varying frequencies.

another hints to guess Most common English bigrams (frequency in 1000 words)

Vigenère Cipher polyalphabetic (polygraphic) substitution [A=0, B=1, …, Z=25], Zn= {0, 1, …, n-1}, P = C = (Z26)* Definition: Given m (key length), and K = (k1, k2, … , km) a key, Encryption: ek(p1, p2… pm) = (p1+k1, p2+k2…pm+km) (mod 26) Decryption: dk(c1, c2… cm) = (c1-k1, c2-k2 … cm- km) (mod 26) Example: Plaintext: C R Y P T O G R A P H Y Key: L U C K L U C K L U C K Ciphertext: N L A Z E I I B L J J I

Vigenère Cipher An attacker can figure out key length m? If the attacker finds the key length, then what happens?

Hill Cipher polygraphic substitution cipher based on linear algebra C1 = 9*p1 + 18*p2 + 10*p3 (mod 26) C2 = 16*p1 + 21*p2 + 1*p3 (mod 26) C3 = 5*p1 + 12*p2 + 23*p3 (mod 26) C1 9 18 10 p1 C2 = 16 21 1 p2 (mod 26) C3 5 12 23 p3 how to decrypt?

Hill cipher: A key is a matrix k11 k12 k13 k21 k22 k23 k31 k32 k33 Generalize to any size, larger blocks matrix multiplication can provide diffusion Matrix must be invertible is it secure? is vulnerable to a known-plaintext attack

transposition cipher aka permutation cipher columnar cipher

X-or() in cryptography Sender wants to send M to receiver M (Original plaintext): 1010 K (Key): 0011 M  K = 1001 (Encrypted ciphertext) 1001 transmitted Receiver already knows K (M  K)  K= 1001  0011 = 1010 = M -> original message is restored!

one time pad (OTP) The one-time pad, which is a provably secure cryptosystem, Gilbert Vernam in 1918. aka Vernam cipher The message is represented as a binary string (a sequence of 0’s and 1’s using a coding mechanism such as ASCII coding. The key is a truly random sequence of 0’s and 1’s of the same length as the message. The encryption is done by adding the key to the message modulo 2, bit by bit. This process is often called exclusive or, XOR ()

OTP: Example message =‘IF’ then its ASCII code =(1001001 1000110) key = (1010110 0110001) Encryption: 1001001 1000110 plaintext 1010110 0110001 key 0011111 1110110 ciphertext Decryption: 0011111 1110110 ciphertext 1001001 1000110 plaintext

OTP problems Key should be as long as plaintext key should not be reused Key distribution & Management difficult

Two types of symmetric ciphers Stream cipher Encrypts one bit at a time mimicking OTP e.g. RC4 Block cipher Encrypts a block of bits at a time e.g. DES, AES

stream cipher Generalization of one-time pad Stream cipher is initialized with key Key is “stretched” into long keystream Keystream  plaintext = ciphertext Stream cipher is a keystream generator Keystream Generator Keystream Generator =  Key Ciphertext Stream Plaintext Keystream Ki Keystream Ki Pi Å Å Pi Ci Plaintext Plaintext Ciphertext

stream cipher: LFSR linear feedback shift register (LFSR) S0 S1 S2 st+3 = st+1+ st S0 S1 S2 0 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 ----------------- 0 0 1 Initial fill determines the sequence of states Generates a periodic sequence …0010111... Maximal period 23-1=7

revisit confusion and diffusion confusion refers to making the relationship between the ciphertext and the symmetric key as complex and involved as possible; diffusion refers to dissipating the statistical structure of plaintext over bulk of ciphertext. This complexity is generally implemented through a series of substitutions and permutations The simplest way to achieve both diffusion and confusion is to use a substitution-permutation network

advanced encryption standard (AES) based on Rijndael cipher data block viewed as 4-by-4 table of bytes Such a table is called the current state 3 key lengths (128/192/256), 128 bit key is assumed here key is expanded to array of words each word is 128 bit long has 10 rounds in which state goes through the following transformations (called `layers’): SubBytes: byte substitution (1 S-box used on every byte) ShiftRows: shift rows (permute bytes between groups/columns) MixColumns: mix columns (uses matrix multiplication in GF(256)) AddRoundKey: add round key (XOR state with round key) First and last round are a little bit different

AES: 50,000 feet view

AES: SubBytes (S-box) a simple substitution of each byte a byte = two nibbles S-box has 16x16 entries: all possible 8-bit values each byte of state is replaced by byte indexed by row (left 4-bits) & column (right 4-bits)

AES: ShiftRows a circular byte shift in each row (permutation) 1st row is unchanged 2nd row does 1 byte circular shift to left 3rd row does 2 byte circular shift to left 4th row does 3 byte circular shift to left

AES: Mix Columns each column is processed separately each byte is replaced by a value dependent on all 4 bytes in the column effectively a matrix multiplication (Hill Cipher)

AES: Add Round Key XOR state with 128-bits of the round key

Modes of Operations message is typically longer than the block size DES and AES have multiple modes of operations depending on how consecutive blocks are processed five modes ECB CBC CFB OFB CTR

Electronic CodeBook (ECB) message is broken into independent blocks which are encrypted each block is a value which is substituted, like a codebook, hence name each block is encoded independently of the other blocks Ci = EK1 (Pi) uses: secure transmission of single values

Electronic Codebook Mode (ECB) 64 bit (8 byte) blocks in DES Ci-1 Ci Ci+1 Pi-1 Pi Pi+1 Ek Ek Ek Dk Dk Dk Ci-1 Ci Ci+1 Pi-1 Pi Pi+1 Ciphertext = (C0 C1 ... Cn)

Advantages and Limitations of ECB repetitions in message may show in ciphertext if aligned with message block particularly with data such graphics or with messages that change very little, which become a code-book analysis problem weakness due to encrypted message blocks being independent ECB mode is susceptible to substitution attacks main use is sending a few blocks of data ECB is not appropriate for any quantity of data, since repetitions can be seen, esp. with graphics, and because the blocks can be shuffled/inserted without affecting the en/decryption of each block.

Cipher Block Chaining (CBC) Two identical plain messages produce two different cipher messages. This prevents Chosen plaintext attack.

Cipher Block Chaining (CBC) message is broken into blocks but these are linked together in the encryption operation each previous cipher blocks is chained with current plaintext block, hence name use Initialization Vector (IV) to start process Ci = EK1(Pi XOR Ci-1) C-1 = IV uses: bulk data encryption, authentication To overcome the problems of repetitions and order independence in ECB, want some way of making the ciphertext dependent on all blocks before it. This is what CBC gives us, by combining the previous ciphertext block with the current message block before encrypting. To start the process, use an Initial Value (IV), which is usually well known (often all 0's), or otherwise is sent, ECB encrypted, just before starting CBC use. CBC mode is applicable whenever large amounts of data need to be sent securely, provided that its available in advance (eg email, FTP, web etc)

Cipher Block Chaining Mode (CBC) Encryption Decryption C0 = Ek(P0 Å IV) P0 = IV Å Dk(C0) Ci = Ek(Pi Å Ci-1) Pi = Ci-1 Å Dk(Ci) P0 Pi-1 Pi Pi+1 Ci-1 Ci Ci+1 IV Å Ci-2 Å Å Å Ek Ek Ek Ek Dk Dk Dk Ci-2 Å Å Å C0 Ci-1 Ci Ci+1 Pi-1 Pi Pi+1 Ciphertext = (IV C0 C1 ... Cn)

CBC – Error Propagation 1 bit error in the plaintext feed Will affect that block and all others Decryption will correct all but the flipped bit 1 bit error in the ciphertext feed Will affect two blocks CBC mode is self recovering

CBC – Initialization Vector (IV) If IV is same, every encryption of the same plaintext is the same. With the IV being a random number, the same plaintext will be all different IV can be in public, but should satisfy two req’s otherwise, some attacks possible two requirements for IV usage no IV is reused under the same key IV change should be unpredictable

Advantages and Limitations of CBC each ciphertext block depends on all previous blocks encrypting a block requires the finish of encryption of all the previous blocks no parallelism in encryption parallelism in decryption CBC is the generally used block mode. The chaining provides an avalanche effect, which means the encrypted message cannot be changed or rearranged without totally destroying the subsequent data. One issue is how to handle the last block, which may well not be complete. In general have to pad this block (typically with 0's), and then must recognise padding at other end - may be obvious (eg in text the 0 value should usually not occur), or otherwise must explicitly have the last byte as a count of how much padding was used (including the count). Note that if this is done, if the last block IS an even multiple of 8 bytes, will have to add an extra block, all padding so as to have a count in the last byte.

Cipher FeedBack (CFB) message is treated as a stream of bits can be a stream cipher added to the output of the block cipher result is feed back for next stage (hence name) standard allows any number of bits (1,8 or 64 or whatever) to be feedback denoted CFB-1, CFB-8, CFB-64 etc is most efficient to use all 64 bits (CFB-64) Ci = Pi XOR EK1(Ci-1) C-1 = IV If the data is only available a bit/byte at a time (eg. terminal session, sensor value etc), then must use some other approach to encrypting it, so as not to delay the info. Idea here is to use the block cipher essentially as a pseudo-random number generator (see stream cipher lecture later) and to combine these "random" bits with the message. Note as mentioned before, XOR is an easily inverted operator (just XOR with same thing again to undo). Again start with an IV to get things going, then use the ciphertext as the next input. As originally defined, idea was to "consume" as much of the "random" output as needed for each message unit (bit/byte) before "bumping" bits out of the buffer and re-encrypting. This is wasteful though, and slows the encryption down as more encryptions are needed. An alternate way to think of it is to generate a block of "random" bits, consume them as message bits/bytes arrive, and when they're used up, only then feed a full block of ciphertext back. This is CFB-64 mode, the most efficient. This is the usual choice for quantities of stream oriented data, and for authentication use.

Cipher Feedback Mode (CFB) Encryption Decryption C0 = P0 Å Ek(IV) P0 = Ek(IV) Å C0 Ci = Pi Å Ek(Ci-1) Pi = Ci Å Ek(Ci-1) Pi+1 Pi-1 Ci-1 P0 Pi Ci Ci+1 Ek IV Ci-2 Å Ek Å Ek Å Ek Å Å Ek Å C0 Ci-1 Ci Ci+1 Pi Pi+1 Ciphertext = (IV C0 C1 ... Cn)

CFB – Initialization Vector CFB must use an IV same requirements as CBC

CFB – Error Propagation CFB mode is self recovering one bit error in ciphertext corrupts some # of blocks CFB-8 -> 8 bytes are garbled Bit errors in the incoming cipher block (i.e. a byte) will cause bit error at the same bit positions in the first plaintext block. This cipher block will then be fed to the shift register and cause bit errors in the plaintext for as long as the erroneous bits stay in the shift register. Hence, for 8-bit CFB, the following 8 bytes will be garbled. After that, the system recovers, and all following bytes is decrypted correctly.

Advantages and Limitations of CFB appropriate when data arrives in bits/bytes: stream cipher most common stream mode encryption is not parallelizable keystream cannot be generated in advance decryption is parallelizable note that the block cipher is used in encryption mode at both ends CFB is the usual stream mode. As long as can keep up with the input, doing encryptions every 8 bytes. A possible problem is that if its used over a "noisy" link, then any corrupted bit will destroy values in the current and next blocks (since the current block feeds as input to create the random bits for the next). So either must use over a reliable network transport layer (pretty usual) or use OFB.

output feedback mode (OFB) can be a stream cipher IV is used as a seed to generate OTP actual encryption/decryption is only X-or fast no dependency parallel encryption/decryption

Output Feedback Mode (OFB) Encryption Decryption Note: The plaintext is never encrypted. IV and successive encryptions act as an one time pad generator. C0 = P0 Å Ek(IV) P0 = Ek(IV) Å C0 Ci = Pi Å Ek(Si-1) Pi = Ci Å Ek(Si-1)

OFB: advantages and limitations no error propagation 1 bit error in ciphertext affects only one bit in plaintext a (key)stream can be generated in advance fast due to parallelism if attacker knows plaintext and ciphertext, he can modify the plaintext

counter mode (CTR) can be a stream cipher IV is called a counter highly parallelizable no linkage between blocks very fast counter should not be repeated for the same key can decrypt from any arbitrary position unlike OFB attacker can get  of two plaintext blocks by taking the  of two ciphertext blocks like OFB

counter (CTR) mode

other usage of modes of operations: message authentication code (MAC) CBC can be used to check message integrity