Patent Publication Number: US-9906368-B2

Title: General encoding functions for modular exponentiation encryption schemes

Description:
TECHNICAL FIELD 
     Various embodiments disclosed herein relate generally to data encoding and digital signature and, more particularly but not exclusively, to encoding for RSA-based digital signature schemes. 
     BACKGROUND 
     Digital signature schemes commonly apply encoding processes to data prior to signing a message to ensure that the data conforms to a pre-chosen alphabet to which the scheme applies. For example, a standard RSA algorithm using a key 3-tuple of &lt;N, e, d&gt; first converts each symbol to a corresponding value between 0 and N. Other encoding schemes may be used for different sets of input values to enable signature schemes that exploit various mathematical properties of the values in the input value set. 
     After encoding, the encoded values are used to creature a signature for transmission. Some encryption algorithms, such as RSA, utilize a modular exponentiation function to create such a signature. As an example RSA processes an encoded value a using the private key pair &lt;N, d&gt; by computing a d  mod N. RSA would then verify the signature value, b, using the public key pair &lt;N, e&gt; by first computing be mod N. A decoding function is then applied to the resulting value to retrieve the original value. 
     SUMMARY 
     A brief summary of various embodiments is presented below. Some simplifications and omissions may be made in the following summary, which is intended to highlight and introduce some aspects of the various embodiments, but not to limit the scope of the invention. Detailed descriptions of a preferred embodiment adequate to allow those of ordinary skill in the art to make and use the inventive concepts will follow in later sections. 
     Various embodiments described herein relate to a method of encoding data and related device and non-transitory machine-readable storage medium, the method including: determining a set of digits, X, representative of a value to be encoded; determining a set of factor values, S, to be used in generating an encoded value, wherein the set of factor values, S, is a set of input value factors for a modular exponentiated process; for a given digit, x, of the set of digits, X, determining at least one factor value, s, of the set of factor values, S, corresponding to the given digit, x; and including the at least one factor value, s, in an encoded value. 
     Various embodiments described herein relate to a method of verifying a digital signature and related device and non-transitory machine-readable storage medium, the method including: receiving a message, m, and a digital signature from a sending party; verifying the digital signature using a public key associated with the sending party to produce a reference signature, a; creating a message digest, h, based on the received message; encoding the message digest, h, to produce an encoded digest, v, including: determining a set of digits, X, representative of the message digest, h, determining a set of factor values, S, to be used in generating an encoded value, for a given digit, x, of the set of digits, X, determining at least one factor value, s, of the set of factor values, S, corresponding to the given digit, x, and including the at least one factor value, s, in the encoded digest, v; comparing the reference signature, a, to the encoded digest, v, to determine whether the sending party is authentic. 
     Various embodiments described herein relate to a method of signing data and related device and non-transitory machine-readable storage medium, the method including: determining a set of digits, X, representative of a value to be signed; determining a set of factor values, S, to be used in generating a signature; for a given digit, x, of the set of digits, X, determining at least one factor value, s, of the set of factor values, S, corresponding to the given digit, x; generating at least one signature factor value, l, equivalent to the at least one factor value, s, raised to the power of a private key, d; and including the at least one encrypted factor value, l, in an encoded value. 
     Various embodiments are described wherein: the encoded value is an encoded and signed value; and including the at least one factor value, s, in the encoded value includes: retrieving at least one signature factor value, l, from a lookup table based on the at least one factor value, s, and including the at least one signature factor value, l, in the encoded and signed value. 
     Various embodiments are described wherein the encoded and signed value is a product of signature factor values and including the at least one signature factor value, l, in the encoded and signed value includes multiplying a working value for the encoded and signed value by the at least one signature factor value, l. 
     Various embodiments are described wherein determining a set of digits, X, representative of a value to be encoded includes: for a given radix, r, determining a radix-r representation of the value to be encoded, wherein the set of digits, X, include the digits of the radix-r representation and wherein each digit in the set of digits X is less than the radix, r. 
     Various embodiments are described wherein: the set of factor values, S, is an ordered set, and determining at least one factor value, s, of the set of factor values, S, corresponding to the given digit, x, includes: determining the factor value, s i , located at a position within the set of factor values, S, that corresponds to the position of the given digit, x, within the set of digits, X. 
     Various embodiments are described wherein determining at least one factor value, s, of the set of factor values, S, corresponding to the given digit, x, further includes: raising the determined factor value, s i , to the power of the given digit, x, to calculate the at least one factor value, s. 
     Various embodiments are described wherein the encoded value is a product of factor values and including the at least one factor value, s, in an encoded value includes multiplying a working value of the encoded value by the at least one factor value, s. 
     Various embodiments are described wherein the steps of determining at least one factor value, s, of the set of factor values, S, corresponding to the given digit, x i , and including the at least one factor value, s, in an encoded value are performed for each digit in the set of digits, X. 
     Various embodiments are described wherein: determining a set of digits, X, representative of a value to be encoded includes, for a value xε /2 n    and radix, r, determining a set of digits,
 
 x   i   εX  
 
as
 
             x   =       ∑     i   =   0       R   -   1       ⁢           ⁢       x   i     ⁢     r   i               where 
             R   =     ⌈     n       log   2     ⁢   r       ⌉           
and x i  is less than r and greater than or equal to zero; and determining at least one factor value, s, of the set of factor values, S, corresponding to the given digit, x, and including the at least one factor value, s, in an encoded value together include calculating an encoded value as
 
               ∏     i   =   0       R   -   1       ⁢           ⁢     s   i     x   i             
where s i  is the factor value at position i within the set of factor values S.
 
     Various embodiments are described wherein: the encoded value is a signed and encoded value; determining a set of digits, X, representative of a value to be encoded includes, for a value xε /2 n    and radix, r, determining a set of digits,
 
 x   i   εX  
 
as
 
             x   =       ∑     i   =   0       R   -   1       ⁢           ⁢       x   i     ⁢     r   i               where 
             R   =     ⌈     n       log   2     ⁢   r       ⌉           
and x i  is less than r and greater than or equal to zero; and determining at least one factor value, s, of the set of factor values, S, corresponding to the given digit, x, and including the at least one factor value, s, in an encoded value together include calculating an encoded value as
 
               ∏     i   =   0       R   -   1       ⁢         L   ⁡     [     i   +   1     ]         x   i       ⁢   mod   ⁢           ⁢   N           
where L[i+1] is an element within a lookup table L located at position i+1 and corresponding to a signature value of at least one factor value, s, in the set of factor values, S, the signature value corresponds to the at least one factor value, s, raised to the power of an exponent portion of a private key, d, and N is a modulus portion of the private key.
 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In order to better understand various embodiments, reference is made to the accompanying drawings, wherein: 
         FIG. 1  illustrates an example of a hardware system for implementing the encoding and signature schemes described herein; 
         FIG. 2  illustrates an example of a method for signing data; 
         FIG. 3  illustrates an example of a method for encoding data; 
         FIG. 4  illustrates an example of a alternative method for encoding and signing data; 
         FIG. 5  illustrates an example of a method for creating and transmitting a digital signature; 
         FIG. 6  illustrates an example of a method for verifying a received digital signature; and 
         FIG. 7  illustrates an example of a system for providing a user device secure content and a software application that processes the secure content. 
     
    
    
     To facilitate understanding, identical reference numerals have been used to designate elements having substantially the same or similar structure or substantially the same or similar function. 
     DETAILED DESCRIPTION 
     The description and drawings presented herein illustrate various principles. It will be appreciated that those skilled in the art will be able to devise various arrangements that, although not explicitly described or shown herein, embody these principles and are included within the scope of this disclosure. As used herein, the term, “or” refers to a non-exclusive or (i.e., and/or), unless otherwise indicated (e.g., “or else” or “or in the alternative”). Additionally, the various embodiments described herein are not necessarily mutually exclusive and may be combined to produce additional embodiments that incorporate the principles described herein. 
     In view of the growing contexts and applications for encryption, such as applications on untrusted platforms, recent efforts have been devoted to the concept of “white box cryptography,” wherein cryptographic schemes are developed to be secure even when the cryptographic implementation is laid open to an attacker. White-box cryptography is concerned with the design and analysis of software implementations of cryptographic algorithms engineered to execute on untrusted platforms. Particularly, this is the scenario where the user of a particular device can decrypt messages (with a secret key) which are encrypted with his public key but is unable to extract or derive sufficient information to recover this secret key. Furthermore, it is assumed in such implementations that the user can be the attacker: e.g. the attacker has full access to the software implementation, can pause, alter and resume the execution of the software implementation at any time 
     For example, in digital rights management systems, it is desirable to provide a content-consumer with the ability to easily authenticate themselves as a party that is entitled to access the content. It is also desirable, however, to prevent that content-consumer from sharing credentials with other parties for the purpose of provided those other parties with access to the same content that is only licensed to that original content-consumer. 
     One white-box approach to this scenario is to provide the content-consumer with the ability to digitally sign messages using a private key, d, assigned to the content-consumer without actually giving the private key, d, to the content-consumer. To that end, the content-consumer may be provided, instead, with a lookup table of pre-computed exponentiated values based on the private key, d. In various systems, for example, the look-up table may be provided to the content-consumer by, for example, a central digital rights management server for use in authenticating the content-consumer to one or more media servers serving the protected content. The content-consumer may then use this lookup table to compute digital signatures in spite of not knowing the value of their private key, d. It would be desirable to build upon these efforts to simplify the scheme and reduce the resources devoted to execution such as, for example, reducing the size of the look-up table. 
     It will be appreciated that, while various examples described herein are explained in the context of digital signature schemes, various aspects described herein may be adapted to data encryption schemes wherein data is encrypted with a public key and retrieved using a private key. 
       FIG. 1  illustrates an example of a hardware system  100  for implementing the encoding and signature schemes described herein. The hardware system  100  may correspond to virtually any device that may participate in a digital signature scheme such as, for example, a personal computer, laptop, tablet, mobile communications device, server, blade, smart card, near field communication (NFC) device, or other device. For example, the hardware system may correspond to a set-top box for receiving and rendering digital content or a server for providing digital content. Various applications of the method described herein will be apparent such as, for example, digital rights management (DRM), banking applications, and generally protecting cryptographic keys in devices such as mobile phones and television set-top boxes. 
     As shown, the device  100  includes a processor  120 , memory  130 , user interface  140 , network interface  150 , and storage  160  interconnected via one or more system buses  110 . It will be understood that  FIG. 1  constitutes, in some respects, an abstraction and that the actual organization of the components of the device  100  may be more complex than illustrated. 
     The processor  120  may be any hardware device capable of executing instructions stored in the memory  130  or the storage  150 . As such, the processor may include a microprocessor, field programmable gate array (FPGA), application-specific integrated circuit (ASIC), or other similar devices. 
     The memory  130  may include various memories such as, for example L1, L2, or L3 cache or system memory. As such, the memory  130  may include static random access memory (SRAM), dynamic RAM (DRAM), flash memory, read only memory (ROM), or other similar memory devices. 
     The memory  130  may include various memories such as, for example L1, L2, or L3 cache or system memory. As such, the memory  130  may include static random access memory (SRAM), dynamic RAM (DRAM), flash memory, read only memory (ROM), or other similar memory devices. 
     The user interface  140  may include one or more devices for enabling communication with a user such as an administrator. For example, the user interface  140  may include a display, a mouse, and a keyboard for receiving user commands. In some embodiments, the user interface  140  may include a command line interface or graphical user interface that may be presented to a remote terminal via the network interface  150 . 
     The network interface  150  may include one or more devices for enabling communication with other hardware devices. For example, the network interface  150  may include a network interface card (NIC) configured to communicate according to the Ethernet protocol. Additionally, the network interface  150  may implement a TCP/IP stack for communication according to the TCP/IP protocols. Various alternative or additional hardware or configurations for the network interface  150  will be apparent. 
     The storage  160  may include one or more machine-readable storage media such as read-only memory (ROM), random-access memory (RAM), magnetic disk storage media, optical storage media, flash-memory devices, or similar storage media. In various embodiments, the storage  160  may store instructions for execution by the processor  120  or data upon with the processor  120  may operate. 
     For example, as shown, the storage  160  includes a modular exponentiation algorithm  162  for use in signing or encrypting data. The modular exponentiation algorithm  162  additionally utilizes an encoding algorithm  164  to encode data prior to encryption or signing and in other cases, as will be explained below. In some embodiments, the modular exponentiation algorithm  162  and encoding algorithm  164  may be combined into a single algorithm, an example of which will be described below with respect to  FIG. 4 . In various embodiments, the modular exponentiation algorithm  162  may utilize a lookup table  166  of encrypted factor value such as, for example, where the modular exponentiation algorithm  162  is a white-box implementation. Various applications of the modular exponentiation and encoding algorithms  162 ,  164  will be apparent; an example of a digital signature algorithm  168  will be described in greater detail below with respect to  FIGS. 5-6 . 
     It will be apparent that various information described as stored in the storage  160  may be additionally or alternatively stored in the memory  130 . In this respect, the memory  130  may also be considered to constitute a “storage device” and the storage  160  may be considered a “memory.” Various other arrangements will be apparent. Further, the memory  130  and storage  160  may both be considered to be “non-transitory machine-readable media.” As used herein, the term “non-transitory” will be understood to exclude transitory signals but to include all forms of storage, including both volatile and non-volatile memories. 
     While the hardware device  100  is shown as including one of each described component, the various components may be duplicated in various embodiments. For example, the processor  120  may include multiple microprocessors that are configured to independently execute the methods described herein or are configured to perform steps or subroutines of the methods described herein such that the multiple processors cooperate to achieve the functionality described herein. In other embodiments, such as those embodiments wherein the device  100  is implemented in a cloud computing environment, the various components may be physically located in diverse machines. For example, the processor  120  may include a first microprocessor in a first data center server and a second microprocessor in a second data center server. Various additional arrangements will be apparent. 
     Various modular exponentiation methods described herein may be premised on the prior selection of a set of factor values, S, from which encoded values to be processed are constructed. The set of factor values, S, may be virtually any ordered and increasing set of integers greater than zero. In mathematical terms, let S be a predefined set of m non-zero positive integer values, S={s 1 , . . . , s m }, that is sorted such that s i &lt;s j  whenever 0&lt;i&lt;j≦m. The set of values, V, that may be processed according to the modular exponentiation method for a given set S is then the set of values that may be constructed from those values in S such as by multiplying the chosen factors in S together: 
     
       
         
           
             V 
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                         v 
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                     α 
                     i 
                   
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                     ℤ 
                     
                       &gt; 
                       0 
                     
                   
                 
               
               } 
             
           
         
       
     
     Exponentiated cryptographic schemes sign (and verify) data (at least partially) by raising the value to be signed/verified to a predetermined value, such as a predetermined public key exponent, e, or private key exponent, d. As will be understood, the calculation of an exponentiated value may be broken down into smaller operations by calculating the exponentiated factors of the value, and then finding the product of these exponentiated values. In other words, the commutative property is used to break the original calculation down into several smaller sub-calculations. According to the schemes detailed herein, because the value to be encrypted or signed is a member of the set V (i.e., is a value that includes as factors values in the set S), the resultant value may be computed by computing the encrypted or signature values of the constituent factors (i.e., those factors in S). In mathematical notation, 
               b   d     =         (       ∏   i     ⁢     b   i       )     d     =       ∏   i     ⁢     b   i   d               
where b i εS are divisors of b and, as such, bεV.
 
     In white-box implementations, a lookup table may be provided with precomputed values for each factor in S raised to the power of the private key exponent, d, modulo N, the key modulus:
 
 L [ i ] =s   i   d  mod  N  for  iε{ 1, . . . , m}.  
 
Given the look-up table L, the base to be processed b, the fixed modulus N, and the definition of the subset V, the user is able to compute the signature value b d  mod N without knowing the private key exponent, d, as follows. First, if b=1, then b d  mod N=1. Otherwise, the user retrieves the l&gt;0 divisors b i εS of b (e.g., using trial division). The signed value may then be computed as
 
               ∏     i   =   0       ℓ   -   1       ⁢       L   ⁡     [     ι   ⁡     (     b   i     )       ]       ⁢   mod   ⁢           ⁢   N           
where t: S→{1, . . . , m} is a function that returns the index of an element in S. In other words, l(x)=i if and only if x=s i .
 
     It will be apparent that the selection of factors to include in S for a given implementation of the signature and encoding methods plays a role in the observed properties of the digital signature system. For example, if the factors are chosen as prime numbers or to be otherwise coprime to all other factors in the set S, then no information will be lost in the signature (i.e., the data can be verified to a single value). However, such a selection would lead to a lookup table that includes an entry for each element in the set S because no combination of elements in the set may be used to reconstruct another element through multiplication. Alternatively, when the elements in the set S are not entirely co-prime, the size of the lookup table may be reduced. For example, if Sε{2, 4}, the lookup table may omit a dedicated entry for the value 4, because the signed value 4 d  may be calculated using the entry for the value 2: 4 d =2 d *2 d . Such a non-coprime selection for S, however, may result in some degree collision between signed values, such that the information may not be easily retrieved without inferring the correct value from a set of possible verified values. Such selections, however, may nonetheless be useful. For example, some uses of the signature and encoding methods described herein (e.g., the example digital signature scheme described with respect to  FIGS. 5-6 ) may operate without decoding any data. 
       FIG. 2  illustrates an example of a method  200  for signing data in accordance with the above-described example. In various embodiments, the method  200  may correspond to the modular exponentiation algorithm  162  and may be performed by a processor such as the processor  120 . The method  200  begins in step  205  and proceeds to step  210  where the processor encodes the data to be signed according to an encoding function such as one of the encoding functions from the classes described herein. For example, the processor may encode the data according to the example method described with respect to  FIG. 3 . Next, in step  215 , the processor initializes a working signature value, E, to a value of 1 and, in step  220 , the processor determines a factor s i  of the value v to be signed. For example, the processor may utilize trial division by attempting to divide the current working value by each potential factor in the set of factors S. Once a factor s i  is identified, the working value of the value v to be signed may be set equal to the previous working value divided by the identified factor (to account for the fact that the identified factor has been accounted for in the signed value). 
     In step  225 , the processor looks up the signature factor l i  corresponding to the identified factor s i  in the lookup table. It will be appreciated that, in various embodiments wherein a lookup table is not used, the processor may instead calculate the signature factor in step  225  as l i =s i   d . Then, in step  230 , the processor includes the signature factor l i  in the working signature value E. In step  235 , the processor determines whether the most recently identified factor s i  is the final factor of the value to be signed v. For example, the processor may determine whether the working value of v is equal to 1. If not, additional factors remain to be processed and the method returns to step  2220 . Otherwise, the processor sets the final signature value equal to the signature value, E, modulo N in step  240 . The method then ends in step  245 . 
     It will be appreciated that the example method  200  may be modified in various manners to achieve a similar result. For example, step  220  may be modified to calculate all factors of the value to be signed in a single pass. In such embodiments, step  235  may loop back to step  225  instead of step  220  when additional factors remain to be processed. As another alternative, the modulus operation of step  240  may performed between steps  230  and  235  to reduce the size of the working signature value. E. Various additional modifications will be apparent. 
     According to various embodiments described herein, the encoding functions used (e.g. in step  210  of method  200 ) may belong to a class of encoding functions that digests values  /2 n    to V. For example, one could write any value xε /2 n    in a radix-r representation as x=Σ i=0   R-1 x i r i  where 
             R   =     ⌈     N       log   2     ⁢   r       ⌉           
and 0≦x i &lt;r for an integer radix r&gt;1. Next, the encoding may be defined as
 
               ∑     i   =   0       R   -   1       ⁢       x   i     ⁢       r   i     ⟶       ∏     i   =   0       R   -   1       ⁢     s   i     x   i                   
where s i εS and therefore R=|S| (i.e., the number of elements, or cardinality, of S is equal to R).
 
       FIG. 3  illustrates an example of a method  300  for encoding data in accordance with the above-described example. In various embodiments, the method  300  may correspond to the encoding algorithm  164  and may be performed by a processor such as the processor  120 . The method  300  begins in step  305  and proceeds to step  310  where the processor determines a set of digits X in a value x for a given radix r. In various embodiments, the radix r may be fixed in the encoding implementation. Various algorithms for determining the set of digits X in view of the foregoing disclosure will be apparent. 
     In steps  315  and  320 , the processor initializes the encoded value, v, and working index, i, to values of 1 and 0, respectively. Next, in step  325 , the processor calculates a factor f i  corresponding to digit i from the values in S. For example, according to the foregoing example, the processor may determine the value s i  εS located at position i, and raises the factor to the power of the i th  digit in X. 
     After determining the factor f i  corresponding to digit i, the processor includes the factor in the working encoded value in step  330 . For example, the processor may multiply the current working encoded value by the factor f i . The processor then increments the working index i in step  335 . Then, in step  340  the processor determines whether additional digits remain to be processed by determining whether i is less than the number of digits, R. If so, the method  300  loops back to step  325 . Otherwise, the method  300  proceeds to end in step  345 . 
     In various alternative embodiments, the encoding and signature methods may be combined into a single method. In other words, the factors may be signed as they are determined as part of the encoding algorithm. In mathematical notation, 
                 ENCODE   ⁡     (   x   )       d     =         ∏     i   =   0       R   -   1       ⁢       (     s   i     x   i       )     d       =         ∏     i   =   0       R   -   1       ⁢       (     s   i   d     )       x   i         =       ∏     i   =   0       R   -   1       ⁢       L   ⁡     [     i   +   1     ]         x   i                   
where L[i+1] is the value located at position i+1, within the lookup table, which is indexed beginning at an index of “1.” Alternatively, when the private key exponent d is known or when the public key exponent e is to be used for signature, the second product above may be used to calculate the signature value by raising each factor s i , to the power of (d*x i ).
 
       FIG. 4  illustrates an example of a alternative method  400  for encoding and signing data according to the above-described example. In various embodiments, the method  400  may correspond to both the modular exponentiation algorithm  162  and encoding algorithm  164  and may be performed by a processor such as the processor  120 . The method  400  begins in step  405  and proceeds to step  410  where the processor determines a set of digits X in a value x for a given radix r. In various embodiments, the radix r may be fixed in the encoding implementation. Various algorithms for determining the set of digits X in view of the foregoing disclosure will be apparent. 
     In steps  415  and  420 , the processor initializes the signature value, E, and working index, i, to values of 1 and 0, respectively. Next, in step  425 , the processor calculates a base signature factore value l i  for the i th  in S from the lookup table (or, alternatively, computes the base signature factor value l i  using a known private or public key exponent). Next, in step  430 , the processor calculates the signature factor value l to be included in the signature value E. For example, according to the foregoing example, the processor may raise the factor l to the power of the i th  digit in X. 
     After determining the signature factor l corresponding to digit i, the processor includes the factor in the working encoded value in step  435 . For example, the processor may multiply the current working encoded value by the factor l. The processor then increments the working index i in step  440 . Then, in step  445  the processor determines whether additional digits remain to be processor by determining whether i is less than the number of digits, R. If so, the method  400  loops back to step  425 . Otherwise, the method  400  proceeds to end in step  450 . 
     As noted above, the encoding and signature methods described herein may be used in various diverse applications. For example, methods described herein may be used as part of a digital signature scheme.  FIG. 5  illustrates an example of a method  500  for creating and transmitting a digital signature. The method  500  may correspond to at least a transmission aspect of the digital signature algorithm  168  and may be performed by a processor such as the processor  120 . 
     The method  500  begins in step  505  and proceeds to step  510  where the processor computes a message digest from a message to be signed and transmitted. For example, the processor may use a predetermined hash algorithm to produce the message digest. Next, in step  515 , the processor encodes the message digest to produced an encoded message digest. The processor may encode the digest using any of the encoding algorithms described herein such as, for example, the encoding algorithm described above with respect to  FIG. 3 . Then, in step  520 , the processor computes a signature from the encoded message digest by signing the encoded message digest using a private key exponent. For example, the processor may use a modular exponentiation algorithm such as that described above with respect to  FIG. 2 . In various embodiments, steps  515  and  520  may be combined by, for example, using the example algorithm described above with respect to  FIG. 4 . After generating the signature, the processor may send the message and signature to a recipient in step  525  and the method  500  may proceed to end in step  530 . 
       FIG. 6  illustrates an example of a method  600  for verifying a received digital signature. The method  600  may correspond to at least a reception aspect of the digital signature algorithm  168  and may be performed by a processor such as the processor  120 . 
     The method begins in step  605  and proceeds to step  610  where the processor receives a message and signature to be verified. For example, the processor may receive a message and signature transmitted by a sending device executing step  525  of method  500 . Next, in step  615 , the processor verifies the signature using the sender&#39;s public key to produce a reference (encoded) value. Then, in step  620 , the processor computes a message digest using, for example, a hash algorithm. The processor proceeds to encode the message digest in step  625  according to any one of the encoding functions described herein such as, for example, the encoding function described with respect to  FIG. 3 . In various embodiments, step  620 ,  625  may perform the same operations as performed in steps  510 ,  515  of method  500 . 
     Next, in step  630 , the processor compares the reference value to the encoded message digest and to determine if the values match. If so, the processor determines in step  635  that the message and signature is verified. Otherwise, the processor determines in step  640  that the signature is not verified. The method  600  then proceeds to step  645 . 
     It will be noted that, according to example method  600 , the signature may be verified without decoding the reference value. Instead, the reference value is compared to an encoded message digest to determine equivalency. As such, various functional embodiments may omit implementing any decoding function and, instead, operate with only an encoding function. Further, because no decoding function is used, sets of S that yield colliding encoded results across the possible input values (i.e., information loss on encode) may nonetheless be used to provide some degree of certainty that the reference message digest was created using the private key corresponding to the known public key (and is therefore verified). Specifically, if the encoding algorithm produces the same output value on two subsequent executions, it is likely (if not certain) that the same input value was used both times. 
     As a simple example of an encoding function in the above-described family, the radix r may be set equal to N, thereby yielding a set of S that includes only a single element τε( /N )\{±1}. The set of input values, V, would then become
 
 V={τ   α   :τεS,αε     &gt;0 }
 
The index function i becomes the trivial function i(x)=1. The encode function then becomes
 
ENCODE( x )=τ x  mod  N.  
 
and the lookup table may be created with a single element L[1]=τ d . When used according to a signature scheme, such as the signature schemes described above, the generation step may be simplified to
 
 t=L [1 =(τ d   =( ) d (mod  N ).
 
For a message m and has function  . Signature verification simplifies to verifying that
 
 t   e = 
 
holds.
 
       FIG. 7  illustrates an example of a system for providing a user device secure content and a software application that processes the secure content. The system includes a content server  700 , application server  720 , user devices  750 ,  752 , and a data network  740 . The user devices  750 ,  752  may request access to secure content provided by the content server  700  via data network  740 . The data network can be any data network providing connectivity between the user devices  750 ,  752  and the content server  700  and application server  720 . The user devices  750 ,  752  may be one of a plurality of devices, for example, set top boxes, media streamers, digital video recorders, tablets, mobile phones, laptop computers, portable media devices, smart watches, desktop computers, media servers, etc. 
     The user request for access may first require the downloading of a software application that may be used to process the secure content provided by the content server  700 . The software application may be downloaded from the application server  720 . The software application may be obscured using the techniques described above as well as operate as described above. Once the user devices  750 ,  752  install the software application, the user device may then download secure content from the content server  700  and access the secure content using the downloaded software application. For example, the downloaded software application may perform decryption of encrypted content received from the content server. In other embodiments, the software application may perform other secure operations, such as for example, encryption, digital signature generation and verification, etc. 
     The content server  700  may control the access to the secure content provided to the user devices  750 ,  752 . As a result when the content server  700  receives a request for secure content, the content server  700  may transmit the secure content to the requesting user device. Likewise, the application server  720  may control access to the software application provided to the user devices  750 ,  752 . As a result when the content server  720  receives a request for the software application, the application server  720  may transmit the software application to the requesting user device. A user device requesting the software application or secure content may also be authenticated by the respective servers, before providing the software application or secure content to the user device. 
     The content server  700  may include a processor  702 , memory  704 , user interface  706 , network interface  710 , and content storage  712  interconnected via one or more system buses  708 . It will be understood that  FIG. 7  constitutes, in some respects, an abstraction and that the actual organization of the components of the device  700  may be more complex than illustrated. 
     The processor  702  may be any hardware device capable of executing instructions stored in memory  704  or storage  712 . As such, the processor may include a microprocessor, field programmable gate array (FPGA), application-specific integrated circuit (ASIC), or other similar devices. 
     The memory  704  may include various memories such as, for example L1, L2, or L3 cache or system memory. As such, the memory  704  may include static random access memory (SRAM), dynamic RAM (DRAM), flash memory, read only memory (ROM), or other similar memory devices. 
     The user interface  706  may include one or more devices for enabling communication with a user such as an administrator. For example, the user interface  706  may include a display, a mouse, and a keyboard for receiving user commands. 
     The network interface  710  may include one or more devices for enabling communication with other hardware devices. For example, the network interface  710  may include a network interface card (NIC) configured to communicate according to the Ethernet protocol. Additionally, the network interface  710  may implement a TCP/IP stack for communication according to the TCP/IP protocols. Various alternative or additional hardware or configurations for the network interface  710  will be apparent. 
     The content storage  712  may include one or more machine-readable content storage media such as read-only memory (ROM), random-access memory (RAM), magnetic disk storage media, optical storage media, flash-memory devices, or similar storage media. In various embodiments, the content storage  712  may store content to be provided to users. 
     The application server  720  includes elements like those in the content server  700  and the description of the like elements in the content server  700  apply to the application server  720 . Also, the content storage  712  is replaced by application storage  732 . Further, it is noted that the content server and applications server may be implemented on a single server. Also, such servers may be implemented on distributed computer systems as well as on cloud computer systems. 
     As will be understood, the modular exponentiation, encoding, or digital signature methods described herein may be deployed and utilized within the system of  FIG. 7  or similar systems in various manners. For example, the user devices  750 ,  752  may be provided by a manufacturer or other seller preconfigured to transmit signed messages to the content server  700  to request the provision of content. Alternatively, the user devices  750 ,  752  may not be fully preconfigured for such operation; instead, the application server  720  may communicate with the user devices  750 ,  752  to effect such configuration. For example, the application server may transmit code instructions for implementing the methods described herein or data defining one or more lookup tables. 
     According to the foregoing, various embodiments enable the removal of decoding functions and, instead, define a generalized family of encoding functions. Such encoding functions may be used, for example, in a digital signature scheme that compares encoded hashed values instead of the hashed values themselves. The resulting scheme is much simpler and allows for using much smaller look-up table than in other efforts. Various additional benefits will be apparent in view of the foregoing. 
     It should be apparent from the foregoing description that various embodiments of the invention may be implemented in hardware. Furthermore, various embodiments may be implemented as instructions stored on a non-transitory machine-readable storage medium, such as a volatile or non-volatile memory, which may be read and executed by at least one processor to perform the operations described in detail herein. A machine-readable storage medium may include any mechanism for storing information in a form readable by a machine, such as a personal or laptop computer, a server, or other computing device. Thus, a non-transitory machine-readable storage medium excludes transitory signals but may include both volatile and non-volatile memories, including but not limited to read-only memory (ROM), random-access memory (RAM), magnetic disk storage media, optical storage media, flash-memory devices, and similar storage media. 
     It should be appreciated by those skilled in the art that any block diagrams herein represent conceptual views of illustrative circuitry embodying the principles of the invention. Similarly, it will be appreciated that any flow charts, flow diagrams, state transition diagrams, pseudo code, and the like represent various processes which may be substantially represented in machine readable media and so executed by a computer or processor, whether or not such computer or processor is explicitly shown. 
     Although the various embodiments have been described in detail with particular reference to certain aspects thereof, it should be understood that the invention is capable of other embodiments and its details are capable of modifications in various obvious respects. As is readily apparent to those skilled in the art, variations and modifications can be effected while remaining within the spirit and scope of the invention. Accordingly, the foregoing disclosure, description, and figures are for illustrative purposes only and do not in any way limit the invention, which is defined only by the claims.