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Universal Hash FamiliesPowerPoint Presentation

Universal Hash Families

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Universal hash families

- Family of hash functions
- Finite multiset H of string-valued functions, each h ∈ H having the same nonempty domain A ⊆{0,1}* and range B ⊆{0,1}*, for some constant b

- [Definition] -almost universal2 (-AU2) & -almost XOR universal2 (-AXU2)
- A family of hash functions H ={h: A{0,1}b} is -almost universal2 , written -AU2, if, for all distinct x, x’∈A, Prh∈ H [h(x)=h(x’)] .
- The family of hash functions H is -almost XOR universal2 , written -AXU2, if, for all distinct x, x’∈A, and for all c∈{0,1}b, Prh∈ H [h(x)h(x’)=c] .
- =maxxx’{Prh [h(x)=h(x’)]} : collision probability

- Principle measure of an AU2 : How small its collision probability is and how fast one can compute its functions

Composition of universal hash families

- Make the domain of a hash family bigger
- H ={h: {0,1}a{0,1}b}
- Hm ={h: {0,1}am{0,1}bm}
- its elements are the same as in H where h(x1 x2… xm), for | xi |=a, is defined by h(x1)|| h(x2)||…|| h(xm)

- [Proposition] If H is -AU2, then Hm is -AU2

- Make the collision probability smaller
- H1 ={h: A{0,1}b1}, H2 ={h: A{0,1}b2}
- H1 & H2 ={h: A{0,1}b1+ b2}
- its elements are pairs of functions (h1, h2)∈H1H2 and where (h1, h2)(x) is defined as h1(x)||h2(x)

- [Proposition] If H1 is 1-AU2 and H2 is 2-AU2, then H1 & H2 is 12-AU2

Composition of universal hash families(cont.)

- Make the image of a hash function shorter
- H1 ={h: {0,1}a{0,1}b}, H2 ={h: {0,1}b{0,1}c}
- H2○H1 ={h: {0,1}a{0,1}c}
- its elements are pairs of functions (h1, h2)∈H1H2 and where (h1, h2)(x) is defined as h2(h1(x))

- [Proposition] If H1 is 1-AU2 and H2 is 2-AU2, then H2○H1 is (1+2)-AU2

- Turn an AU2 family H1 and AXU2 family H2 into an AXU2 family H2○H1
- [Proposition] Suppose H1 ={h: AB} is 1-AU2 and H2 ={h: BC} is 2-AXU2. Then H2○H1 ={h: AC} is (1+2)-AXU2

Related researches

- Carter-Wegman(1979, 1981)
- Efficient authentication code under strongly universal hash functions
- Key observations
- Long messages can be authenticated efficiently using short keys if the number of bits in the authentication tag is increased slightly compared to ‘perfect’ schemes
- If a message is hashed to a short authentication tag, weaker properties are sufficient for the first stage of the compression
- Under certain conditions, the hash function can remain the same for many plaintexts, provided that hash results is encrypted using a one-time pad

- Stinson(1994)
- Improves the works by Wegman-Carter and establishes an explicit link between authentication codes and strongly universal hash functions

- Johansson-Kabatianskii-Smeets(1996)
- Establish a relation between authentication codes and codes correcting independent errors

Related researches(cont.)

- Krawczyk(1994, 1995)
- Propose universal hash functions that are linear with respect to bitwise XOR
- Makes it easier to reuse the authentication code
- Encrypt the m-bit hash result for each new message using a one-time pad

- Simple and efficient constructions based on polynomials and LFSR

- Shoup(1996)
- Propose and analyze the constructions based on polynomials over finite fields

- Rogaway : Bucket hashing (1995)
- Halevi-Krawczyk : MMH (1997)
- Make optimal used of the multiply and accumulate instruction of the Pentium MMX processor

- Black-Halevi-Krawczyk-Krovertz-Rogaway : UMAC (1999)
- Further improved the performance on high end processors

Constructions

- Bucket hashing
- is an -AU introduced by Rogaway
- Defining the Bucket Hash Family B
- word size w(1), parameters n(1), N(3)
- domain D={0,1}wn, range R={0,1}wN
- Let h Band let X=X1Xnbe the string we want to hash, where each |Xi|=w. Then h(X) is defined by the following algorithm. First, for each j{1,,N}, initialize Yjto 0w. Then, for each i{1,, n} and k hi, replace Ykby Yk Xi. When done, set h(X) = Y1||Y2||||YN.
- Pseudocode
for j 1 to N do Yj 0w

for i 1 to n do

Yhi1 Yhi1 Xi

Yhi2 Yhi2 Xi

Yhi3 Yhi3 Xi

return Y1||Y2||||YN

Constructions(cont.)

- Bucket Hashing with Small Key Size
- N=2s/L
- Each hash function hB’[w,M,N] is specified by a list of length M
- each entry contains L integers in the interval [0, N-1]

- L arrays are introduced, each containing N buckets
- Next, each array is compressed to s/L words, using a fixed primitive element GF(2s/L)
- The hash result is equal to the concatenation of the L compressed arrays, each containing s/L words

Constructions(cont.)

- Hash Family Based on Fast Polynomial Evaluation
- is based on polynomial evaluation over a finite field
- q = 2r, Q = 2m = 2r+s, n = 1+2s, : a linear mapping from GF(Q) onto GF(q)
- Q = q0m, q = q0r , q0 : a prime power

- fa(x) =a0 + a1x + + an-1xn-1
- x, y, a0, a1, , an-1 GF(Q), z GF(q)

- H = {hx,y,z : hx,y,z(a) = hx,y,z(a0, a1, , an-1) = (y fa(x)) + z}

Constructions(cont.)

- Hash Family Using Toeplitz Matrices
- Toeplitz matrices are matrices with constant values on the left-to-right diagonals
- A Toeplitz matrix of dimension n m can be used to hash messages of length m to hash results of length n by vector-matrix multiplication

- The Toeplitz construction uses matrices generated by sequences of length n + m - 1 drawn from -biased distributions
- -biased distributions are a tool for replacing truly random sequences by more compact and easier to generate sequences
- The lower , the more random the sequence is

- Krawczyk proves that the family of hash functions associated with a family of Toeplitz-matrices corresponding to sequences selected from a -biased distribution is -AXU with = 2-n +

- Toeplitz matrices are matrices with constant values on the left-to-right diagonals

Constructions(cont.)

- Evaluation Hash Function
- is one of the variants analyzed by Shoup
- The input (of length tn) : viewed as a polynomial M(x) of degree < t over GF(2n)
- The key : a random element GF(2n)
- the hash result : equal to M() GF(2n)
- This family of hash functions is -AXU with = t/2n

Constructions(cont.)

- Division Hash Function
- represents the input as a polynomial M(x) of degree less than tn over GF(2)
- The hash key : a random irreducible polynomial p(x) of degree n over GF(2)
- The hash result : m(x) xn mod p(x)
- This family of hash functions is -AXU with = tn/2n
- The total number of irreducible polynomials of degree n is roughly equal to 2n/n

Constructions(cont.)

- MMH(Multilinear Modular Hashing) hashing
- consists of a (modified) inner product between message and key modulo a prime p (close to 2w, with w the word length; below w = 32)
- is an -AXU2, but with xor replaced by subtraction modulo p
- The core hash function maps 32 32-bit message words and 32 32-bit key words to a 32-bit result
- The key size is 1024 bits and = 1.5/ 230
- For larger messages, a tree construction can be used
- the value of and the key length have to be multiplied by the height of the tree

- This algorithm is very fast on the Pentium Pro, which has a multiply and accumulate instruction
- On a 32-bit machine, MMH requires only 2 instructions per byte for a 32-bit result

Comparing the Hash Functions(cont.)

- Scheme A
- the input : divided into 32 blocks of 8 Kbyte
- each block is hashed using the same bucket hash function with N = 160
- results in an intermediate string of 20480 bytes

- the input : divided into 32 blocks of 8 Kbyte
- Scheme B
- the input : divided into 64 blocks of 4 Kbyte
- each block is hashed using the same bucket hash function with short key(s=42, L=6, N=128)
- results in an intermediate string of 10752 bytes

- the input : divided into 64 blocks of 4 Kbyte
- Scheme C
- the input is divided into 64 blocks of 4 Kbyte
- each block is hashed using a 331024 Toeplitz matrix, based on a -biased sequence of length 1056 generated using an 88-bit LFSR
- The length of the intermediate string is 8448 bytes

- the input is divided into 64 blocks of 4 Kbyte

Comparing the Hash Functions(cont.)

- Scheme D
- the input : hashed twice using the polynomial evaluation hash function with = 2-15 resulting in a combined value of 2-30
- W = 5
- The performance is slightly key dependent. Therefore an average over a number of keys has been computed.

- the input : hashed twice using the polynomial evaluation hash function with = 2-15 resulting in a combined value of 2-30
- Scheme E
- this is simply the evaluation hash function with t = 32768
- the resulting value of is too small
- However, choosing a smaller value of n that is not a multiple of 32 induces a performance penalty

- this is simply the evaluation hash function with t = 32768
- Scheme F
- the input : divided into 2048 blocks of 128 bytes
- each block is hashed twice using MMH
- the length of the intermediate string is 16384 bytes
- It is not possible to obtain a value of closer to 2-32 in an efficient way

- the input : divided into 2048 blocks of 128 bytes

Message authentication based on Universal hashing

- Message authentication based on Universal hashing
- Wegman-Carter approach
- The parties share a secret key k=(h,P)
- P : infinite random string
- h : function drawn randomly from a strongly universal2 family of hash functions H
- H is strongly universal2 if, for all xx’, the random variable h1(x)||h2(x), for h ∈ H, is uniformly distributed

- To authenticate a message x, the sender transmits h(x) xored with the next piece of the pad P

- The parties share a secret key k=(h,P)
- Standard cryptographic technique
- use of a pseudorandom function family, F
- [Theorem] Assume H is -AXU2, and that F is replaced by the truly random function family R of functions. In this case, if an adversary makes q1 queries to the authentication algorithm S and q2 queries to the verification algorithm V, the probability of forging a MAC is at most q2

- Wegman-Carter approach

Universal hashing MAC

- Why Universal hashing MAC?
- The speed of a universal hashing MAC depends on the speed of the hashing step and encrypting step
- The encryption does not take long
- hash function compresses messages => the encrypting message is short

- The combinatorial properties of the universal hash function family is mathematically proven
- needs no “over-design” or “safe margin” the way a cryptographic primitive would

- Universal hashing MAC makes for desirable security properties
- can select a cryptographically conservative design for the encrypting step
- can pay with only a minor impact on speed
- the cryptographic primitive is applied only to the much shorter hashed image of the message
- security and efficiency are not conflicting requirements

UMAC

- The UMAC algorithm
- species how the message, key, and nonce determine an authentication tag
- The sender
- will need to provide the receiver with the message, nonce, and tag

- The receiver
- can then compute what “should be” the tag for this particular message and nonce, and see if it matches the received tag

- employs a subkey generation process in which the shared (convenient-length) key is mapped into UMAC's internal keys
- subkey generation is done just once, at the beginning of a communication session during which the key does not change, and so subkey-generation is usually not performance-critical
- UMAC depends on a few different parameters

UMAC(cont.)

- An illustrative special case of UMAC
- Subkey generation:
- Using a PRG, map Key to K = K1K2 K1024 and to A
- each Ki : a 32-bit word, |A| = 512

- Using a PRG, map Key to K = K1K2 K1024 and to A
- Hashing the message Msg to HM = NHXKey(Msg):
- Let Len be |Msg| mod 4096, encoded as a 2-byte string
- Append to Msg the minimum number of 0 bits to make |Msg| divisible by 8
- Let Msg = Msg1 || Msg2 || || Msgt where each Msgi is 1024 words except for Msgt, which has between 2 and 1024 words
- Let HM = NHK(Msg1) || NHK(Msg2) || || NHK(Msgt) || Len

- Computing the authentication tag:
- The tag is Tag = HMAC-SHA1A(HM || Nonce)

- Subkey generation:

UMAC(cont.)

- Definition of NH
- blocksize n 2, wordsize w 1
- domain : A = {0, 1}2w {0, 1}2w {0, 1}nw
- range : B = {0, 1}2w
- a random function in NH[n,w] is given by a random nw-bit string K
- Uw : {0, , 2w-1}, U2w : {0, , 22w -1}
- for integers x, ylet (x +wy) denote (x + y) mod 2w
- M A and M = M1 Ml, |M1| = = |Ml|= w
- K = {0, 1}nwandK = K1 Kn, |K1| = = |Kn|= w

UMAC(cont.)

- where miUw is the number that Mi represents (as an unsigned integer)
- where kiUw is the number that Ki represents (as an unsigned integer)
- the right-hand side of the above equation is understood to name the (unique) 2w-bit string which represents (as an unsigned integer) the U2w-valued integer result

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