Patent Application: US-1605308-A

Abstract:
an alternative scheme to the classical boneh - franklin scheme simplifies the generation and the use of the asymmetric keys . the alternative scheme takes advantage of the discovery that simpler calculations resulting in exponents of reduced size can be used as part of boneh - franklin type scheme . the alternative scheme thus provides a traceable encryption scheme which allows for fast , secure cryptographic calculations to be made while providing the necessary level of security required for reliable tracing capabilities to be achieved .

Description:
we now present a traceable private key public component generation process which allows deriving public components which offer a significantly improved decryption speed . previously , we noted that the components γ ( i ) can be computed using the recursive formula eq . ( 3 ); this operation is typically feasible in the broadcasting center , but not in a receiver . we can furthermore note that , working in a usual security configuration of 2 80 operations , the elements of a public component γ ( i ) have all a length of 160 bits . this new method works as follows : in the key generation process described previously , the step 1 is replaced by 1 ′ we compute the i - th fast boneh - franklin traceable private key public component as being the following 2k - valued vector over z / qz : γ ( i ) =( 1 , i mod q , i 2 mod q , . . . , i 2k - 1 mod q ). ( 12 ) the method presented below results in rather small exponent sizes which can drastically speed up the ciphertext decryption in the receiver : re - writing ( 11 ) as m = s ( ∏ p j i j - 1 ) θ i ⁢ = s ( ( ( p 2 ⁢ k i ⁢ p 2 ⁢ k - 1 ) i ⁢ p 2 ) i ⁢ p 1 ) θ i ( 13 ) we can transform , for instance for l = 2 20 , 2k + 1 modular exponentiations with 160 - bit exponents by 2k modular exponentiations with 20 - bit exponents and one 160 - bit exponentiation . this is more than a 7 - times speedup . according a particular embodiment of the invention , q is higher than 2 127 in order to avoid generic attacks against the discrete logarithm problem . a further advantage of this method is that a receiver can compute the public component of the decryption key without the need to evaluate the recursive formula of eq . ( 3 ). in practical scenarios , there might be a situation where an attacker might have 2k secret components θ i at his disposal . this part of the invention describes specifically how to the system can be protected in such a case . we start by describing an attack that might occur in practice and allow the attacker to derive every private key in the system . let us suppose than an adversary has managed to get 2k private elements θ i , for 1 ≦ s ≦ 2k . the vectors in i − ={ γ ( 1 ) , γ ( 2 ) , . . . , γ ( l ) } are assumed to be public . then , we can rewrite eq . ( 9 ) over z / qz as θ ( i a ) - 1 = ∑ j = 1 2 ⁢ k ⁢ r j ⁢ γ j ( i a ) ∑ j = 1 2 ⁢ k ⁢ r j ⁢ α j = ∑ j = 1 2 ⁢ k ⁢ ω j ⁢ γ j ( i a ) ( 22 ) with ω j = r j / σr j α j ; note that the ω j are unknown coefficients to an adversary . however , with 2k private elements , we have a system of 2k linear equations with 2k variables with a single solution revealing the values of ω j to the adversary using a simple gaussian reduction . from those coefficients , the adversary can compute any other private key θ i v in the system not only the adversary will be able to create many untraceable combinations of keys , but he will be also able to distribute newly derived keys so that innocent users ( whose keys were a priori never compromised ) will be accused of treachery . we now present a traceable key generation process which allows deriving traceable keys resistant to pirates able to gather 2k keys or more . this new method works as follows : 1 ″ we compute the i - th fast boneh - franklin public component of the traceable private key as being the following 2k - valued vector over z / qz : γ ( i ) =( 1 , ζ mod q , ζ 2 mod q , . . . , ζ 2k - 1 mod q ). ( 14 ) where ζε r z / qz is drawn independently and uniformly at random for each γ ( i ) . d ( i ) =( θ i γ 1 ( i ) , . . . , θ 2k γ 2k ( i ) ) ( 15 ) in tamper - proof memory and hence the abovementioned public component becomes secret . a possible variant would consist in deriving ζ from i by processing i and / or additional information with a cryptographically secure pseudo - random function ( or permutation ) parametered by a secret key . to encrypt a message mεg q , the standard boneh - franklin encryption procedure requires to generate a random value a ε r z / qz and the ciphertext is defined as being the ( 2k + 1 )- valued vector in most practical situations , the message m consists in a symmetric session key k , which is then used to encrypt some content , since m is of limited length ( no more than 20 bytes , usually ). furthermore , one possibly needs a hash function mapping a group element to a symmetric key . we propose to bypass these intermediate steps and to use one of the two following possible variants to encrypt any type of message faster than the standard boneh - franklin scheme , but keeping the same tracing and security properties . 1 . to encrypt a message mε { 0 , 1 }* ( i . e ., a bitstring of arbitrary length ), we first generate a random value a ε r z / qz and the ciphertext is defined as being the ( 2k + 1 )- valued vector ( m ⊕ prf ( n , y a ), h 1 a , . . . , h 2k a ) ( 17 ) where prf (., .) denotes a cryptographically secure pseudo - random function . for instance , it can be hmac - sha1 , hmac - sha256 or a block cipher evaluated on a counter and where y a is considered as being the symmetric key and n is a nonce value ( e . g ., a counter incremented sufficiently many times to generate enough key stream ). here , the xor operation ⊕ could be replaced by any group law . 2 . to encrypt a message mε { 0 , 1 }*, we first generate a random value a ε r z / qz and the ciphertext is defined as being the ( 2k + 1 )- valued vector ( e ( m , y a ), h 1 a , . . . , h 2k a ) ( 18 ) where e (., .) is a block cipher or any symmetric encryption scheme based on a block cipher , and where y a is considered as being the key . a possible variant would consist in mapping the y a value to a key using a hash function . another possible variant is an encryption scheme e (., .) requiring additional information , like an initial vector . in pay - tv systems , the use of traceable asymmetric keys is an advantage in terms of fighting against piracy . the pay - tv receiver ( or the security module thereof ) is loaded with a private key i . e ., the public component γ ( i ) and the secret component θ i . each pay - tv receiver , such as a set top - box , multimedia device or wireless portable device ( dvb - h ), comprises at least one private key . the secret component is preferably stored in a secure container such as a sim card , smartcard of any type of tamper - proof memory . in a practical example , a video / audio data packet pspacket will be encrypted in the following way , assuming we are working with a multiplicative group and hmac - sha256 as the function prf ( see formula ( 17 )): generate uniformly distributed random value a compute h 1 a , h 2 a , . . . h 2k a using 2k last elements of the public key ( see formula ( 8 )), compute y a using the first element of the public key , divide the pspacket into chunk packets of 256 bits possibly remaining a residual packet of less than 256 bits , initialize an index to an arbitrary constant ( usually 0 ), for each chunk , computing the hmac - sha256 of the index with y a as key , the index being updated for each chunk , and applying an xor function ( or any group operation ) with the respective chunk in case that a residual chunk exists , adjusting the hmac - sha256 value by extracting the number of bit corresponding to the number of bits of the residual chunk before applying the xor function . transmitting to the receiver , the result values after the xor function and the h 1 a , h 2 a , . . . , h 2k a in the receiver side , the received values h 1 a , h 2 a , . . . , h 2k a are considered as 2k values i . e . p 1 , p 2 , . . . p 2k . in order to extract the audio / video data pspacket , the following steps will be executed : y a = ( ∏ j = 1 2 ⁢ k ⁢ p j γ j ( i ) ) θ i using the γ ( i ) public component of the private key , and θ i is the secret component of the private key , executing the same hmac - sha256 operation as made on the sender side , by defining an index in the same way as defined during the encryption operation . in this way , the broadcasting center can send a global , encrypted version of audio / video packet to all receivers ; those receivers decrypt the packets using their own private key . a pirate willing to implement an unofficial ( unlawful ) receiver will necessarily have to embed a unique private key ( or a mix of several private keys ) in order to decrypt the packets . having such a rogue receiver in hands , the pay - tv operator can then recover the pirate private key ( s ) and possibly revoke it ( them ) using another mechanism and / or possibly take legal or any other action against the person having purchased the original ( broken ) receiver ( s ), provided such a link exists . instead of mixing the packets with hmac result , the packets are encrypted with a standard symmetric encryption scheme using a key k , this key being used at the mixing step with the hmac result . according to another embodiment , the encrypted packet is obtained by encrypting the said packet with a symmetric encryption scheme using the y a value as a key ( e . g . tdes in cbc mode ). according to an alternative embodiment , a hashing function is first applied to the y a value before being used as a key . this is preferably the case when the size of the y a value is different than the size of the symmetric encryption scheme key . another possible field of application concerns the protection of software against piracy . we may assume that a software is sold together with a hardware dongle containing a different private key for every package . this dongle is able to decrypt a global ciphertext contained in the software and getting a piece of information which is necessary to the use of the software . if a pirate is willing to clone dongles and sell them , he must embed at least a private key . getting such a pirate dongle in hands , the software seller can then recover the involved private key ( s ) and take legal or any other action against the person having purchased the original ( broken ) dongle ( s ), provided such a link exists . a . fiat and m . naor , “ broadcast encryption ”, crypto &# 39 ; 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